Belief-Free Review-Strategy Equilibrium Without Conditional Independence Takuo Sugaya y Princeton University May 20, 2012
Abstract We study repeated games with imperfect private monitoring. We obtain a characterization, which depends only on the parameters of the stage game, of the set of belief-free review-strategy equilibrium payo¤s in the limit as the discount factor converges to unity. Our characterization is valid for a generic monitoring technology if the number of private signals is su¢ ciently large and the number of players is no less than four. In addition, we show a su¢ cient condition for the characterized set to contain a Pareto e¢ cient payo¤ pro…le.
[email protected] The author is thankful to Stephen Morris for his encouragement and guidance. The author is also grateful to Dilip Abreu, Marco Battaglini, Je¤rey Ely, Kyna Fong, Johannes Hörner, Andriy Norets, Ichiro Obara, Yuliy Sannikov, Satoru Takahashi, Yuichi Yamamoto and three anonymous referees for their valuable comments. The usual disclaimer of course applies. y
1
1
Introduction
The main …nding in the theory of repeated games is that long term relationships enhance cooperation. In fact, the central result is probably the folk theorem: any feasible and individually rational payo¤ can be sustained in equilibrium when players are su¢ ciently patient. Fudenberg and Maskin (1986) establish the folk theorem under perfect monitoring, that is, when players can observe the opponents’actions directly. Fudenberg et al. (1994) extend the folk theorem to imperfect public monitoring, where players cannot observe the opponents’actions directly, but observe public noisy signals about the opponents’actions.1 Recently, it has been shown that these results are robust to the introduction of private monitoring.2 Hörner and Olszewski (2006) show the robustness of the folk theorem to almost perfect monitoring and Hörner and Olszewski (2009) establish the robustness of the folk theorem to almost public monitoring.3 On the other hand, when monitoring is neither almost perfect nor almost public, almost all the results are attained only with conditionally independent monitoring: players can obtain no information on what their opponents have observed by observing their own private signals conditional on an action pro…le. Combining the idea of review strategies in Radner (1985) and Abreu et al. (1991) and that of belief free equilibria in Piccione (2002), Ely and Välimäki (2002) and Ely et al. (2005),4 Matsushima (2004) shows the folk theorem in the two-player prisoners’ dilemma with conditionally independent monitoring, which is extended by Yamamoto (2007), who shows that an e¢ cient payo¤ is attainable in the N player prisoners’dilemma. Finally, incorporating the idea of the block equilibria in Hörner and Olszewski (2006) and Yamamoto (2009), Yamamoto (2012) obtains the characterization of the equilibrium payo¤ set implementable by so-called belief-free review-strategy equilibria 1
See also Fudenberg and Levine (1994). See Kandori (2002) and Mailath and Samuelson (2006) for the survey of private monitoring. 3 See also Mailath and Morris (2002) and Mailath and Morris (2006). 4 Kandori and Obara (2006) use a similar concept to analyze a private strategy in public monitoring. Kandori (2011) considers “weakly belief-free equilibria,” which is a generalization of belief-free equilibria. Apart from a typical repeated-game setting, Takahashi (2010) and Deb (2011) consider the community enforcement and Miyagawa et al. (2008) consider the situation where a player can improve the precision of monitoring by paying cost. 2
2
(henceforth BFRSE) for a general N -player game with conditionally independent monitoring. The main idea of BFRSE is as follows. The in…nite periods are regarded as a sequence of T -period review phases. In each review phase, suppose it is optimal to take a constant action. Then, however noisy the monitoring is, taking T su¢ ciently large, players can statistically infer the opponents’ actions with arbitrarily high power by aggregating information for T periods. The remaining task is to verify the optimality of a constant action. It has been done by Yamamoto (2012) with conditionally independent monitoring. This paper constructs an equilibrium where the important properties that Yamamoto (2012) uses to show the optimality of a constant action are satis…ed without assuming conditionally independent monitoring a priori. The importance of conditional independence in Matsushima (2004), Yamamoto (2007) and Yamamoto (2012) is explained as follows.5 To statistically infer the opponents’actions by pooling the information, it is important to create an equilibrium where each player takes a constant action in each review phase. Consider a prisoners’-dilemma example, in which each player is likely to be punished in the following review phase if a lot of signals indicating defection are pooled in the current review phase by the opponents. Suppose a player takes cooperation initially in the current review phase. If signals are conditionally independent, after any realization of the signals, it is optimal to stick to the cooperation since the observed signals have no information about whether she will be punished or not. With conditionally dependent signals, however, in the periods near to the end of the current review phase, under some realization of the signals, she considers that, judging from her own signal observations and correlation, the opponents have already observed many signals indicating defection and they will punish her regardless of their signal observations in the remaining periods of the current review phase, which destroys the incentive to continue cooperation. Let us call this problem the “statistical inference problem.” As long as each player (say, player i) has one monitor (say, player j) who monitors player 5
Ely et al. (2005) also characterize the set of BFRSE payo¤s in the two-by-two game.
3
i only by player j’s own signals, this statistical inference problem inevitably occurs with correlated signals. Suppose player j creates a statistic to review player i from player j’s own signal and vice versa. Then, the player whose support of signals is no smaller than that of the other player can statistically infer the other player’s review from her histories (see in Section 5). The idea of this paper is that, without assuming conditionally independent monitoring directly, if player j can also use the signals of the other players
(i; j), then player j has
su¢ cient information to create statistics to monitor player i in such a way that player i cannot update the distribution of those statistics from her signal observations (conditional independence property). Intuitively, after main rounds corresponding to the review phases in Matsushima (2004), players communicate to inform player j of signal observations by players (i; j). Then, the freedom that player j can use to construct the statistic is the combination of players
i’s signals, which can be very large compared to the support of player i’s signal
distribution. To formalize this idea, we proceed in three steps. First, we assume that players could communicate with cheap talk. In this step, there are two di¢ culties. (i) Player j needs to construct statistics to review player i using the messages from players
(i; j) so that the
conditional independence property is satis…ed. This is possible if the signals of players
i
have enough information compared to player i’s actions and signals. (ii) We have to give an incentive for players to tell the truth. For example, it could be of player n 2
(i; j)’s
interest to tell a lie so that player j believes that player i has deviated if the action pro…le to punish player i gives player n a higher payo¤. As usual in mechanism design, if we have no less than three players to review player i, it is possible to give incentives to tell the truth. That is why we assume the number of players is no less than four. Second, we dispense with cheap talk communication devices but assume that public randomization devices would be available. Cheap talk communication devices are special since they are (i) instantaneous, (ii) payo¤ irrelevant and (iii) perfect. When players communicate by taking actions, communication becomes (i) taking time, (ii) payo¤ relevant and (iii)
4
imperfect and private. Because of (i) and (ii), if players
(i; j) sent the messages about the
signal observations in all the periods, then it would a¤ect the equilibrium payo¤. Hence, after a review phase, player j randomly picks a small proportion of the phase by public randomization and players
(i; j) send the messages about the signal observations only in the
picked periods. Then, (i) the periods used by player j to review player i are short enough to keep the communication short but are long enough to aggregate information about player i’s action. (ii) Since the communication is short, the di¤erence in instantaneous utilities can be canceled out by the movement of the continuation payo¤. Further, (iii) to increase the precision of the message, each player repeats the message many times (the total communication should be still short enough). Third, we dispense with public randomization. In this case, player j picks periods to monitor player i randomly by herself and then sends the message about the picked periods by taking actions. Again, we want to make sure that (i) it does not take too long to send this message, that (ii) we can cancel out the di¤erence in the instantaneous utilities by the movement of the continuation payo¤, and that (iii) player j repeats the message to increase the precision of the message. Finally, therefore, we have characterized the set of BFRSE payo¤s without conditional independence, communication or public randomization devices. Let us …nish the introduction by reviewing other papers to attain the e¢ ciency in repeated games with private monitoring.6 Fong et al. (2010) derive conditions for e¢ ciency to be attainable in the two-player prisoners’ dilemma. They show that there is an open set of parameters that satisfy their conditions. Neither their result nor ours contains the other as a special case. Their conditions are not generic while ours are. However, our conditions require no less than four players while Fong et al. (2010) consider a two-player game. Moreover, the main ideas are di¤erent. Fong et al. (2010) construct an equilibrium where the statistical inference problem occurs only with a small probability on the equilibrium path. On the other hand, this paper constructs an equilibrium that is completely free from the statistical inference problem. 6
Our review is close to that in Fong et al. (2010).
5
Several papers such as Sekiguchi (1997) and Bhaskar and Obara (2002) focus on beliefbased techniques. In such equilibria, players’ strategies involve statistical inference about the past history of the play. Results in those papers are limited to prisoners’dilemma with almost perfect monitoring. Another approach is to introduce explicit communication. Versions of the folk theorem have been proven by Compte (1998), Kandori and Matsushima (1998), Aoyagi (2002), Fudenberg and Levine (2007) and Obara (2009). Since some practical economic settings make communication impossible or costly, their result might not be applicable. For example, in Stigler (1964)’s oligopoly example, anti-trust laws make communication illegal. Hörner and Olszewski (2006) also argue that “communication reintroduces an element of public information that is somewhat at odds with the motivation of private monitoring as a robustness test”to the lack of public signals. Note that the result of this paper does not rely on explicit communication. Similar results are obtained without discounting in Lehrer (1990). Although the average payo¤ in BFRSE under discounting with T -period review phases converges to T -period timeaverage payo¤ when the discount factor converging to unity, our equilibrium construction is signi…cantly di¤erent from papers without discounting since we need to take care of the incentives for gains and losses in …nite periods. The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 de…nes the belief-free review-strategy equilibrium. Section 4 states the assumptions and the main result that characterizes the set of the belief-free review-strategy equilibrium payo¤s. After intuitively explaining the equilibrium construction to support the characterized set in Section 5, we construct an equilibrium in Section 6. Section 7 presents su¢ cient conditions for the characterized set to support e¢ ciency payo¤s. Technical proofs are relegated to the Appendix.
6
2
Model
The stage game is given by I; (Ai ;
i ; gi )i2I
is the …nite set of player i’s pure actions,
i
; q . I = f1; 2; : : : ; N g is the set of players, Ai
is the …nite set of player i’s private signals, and
gi : Ai 4 and jAi j 2. Let i ! R is player i’s ex post utility function. We assume N Q Q A Ai and i be the set of action pro…les and signal pro…les, respectively. For i2I
I
i2I
I, A
I,
I,
a
I
2A
I
and !
I
2
I
are de…ned as usual.
In every stage game, players choose an action pro…le a a signal pro…le !
(! 1 ; : : : ; ! N ) 2
probability function q ( j a). For I
(a1 ; : : : ; aN ) 2 A and then
is distributed according to the conditional joint I, q
I
( j a) is the marginal conditional distribution
of ! I . Given an action ai 2 Ai and a private signal ! i 2
i,
player i receives an ex post
payo¤ gi (ai ; ! i ). Thus, her expected payo¤ conditional on an action pro…le a 2 A is denoted P by i (a) = !2 q (! j a) gi (ai ; ! i ). For each a 2 A, let (a) represent the payo¤ vector (
i
(a))i2I .
For a whole paper, we assume that the signal distribution q satis…es full support: Assumption 1 The signal distribution q satis…es full support: for all a 2 A and ! 2 , we have q(! j a) > 0. Consider the in…nitely repeated game in which the discount factor is
2 (0; 1). Let
ai; and ! i; denote the performed action and the observed private signal respectively in period by hti h0i
by player i. Player i’s private history up to and including period t
1 is given
(ai; ; ! i; )t =1 . Hence, player i at the beginning of period t has observed hti 1 . Let
0, let Hit be the set of all hti . Then, a strategy for player i is 1 S de…ned to be a mapping si : Hit ! 4(Ai ). Let Si be the set of all strategies for player t=0 Q i and let S Si . Also, for any strategy si 2 Si , history hti 2 Hit and action ai 2 Ai , ; and for each t
i2I
let si j hti be player i’s continuation strategy after hti and si (hti ) [ai ] be the probability for
si to take ai after hti . In addition, let si j (hti ; ai ) represent player i’s strategy s~i 2 Si such that s~i (h0i ) = ai (in the …rst period, player i takes ai ) and for any hi 2 Hi with s~i j hi = si j ht+ where hit+ = (hti ; hi ) (in period i 7
+1
1,
2, player i takes the strategy as
if player i with si observed hti ; hi ).7 In words, si j (hti ; ai ) denotes the continuation strategy after history hti but the play in the …rst period is replaced with the pure action ai . Finally, let wi (s) represent player i’s expected average payo¤ by a strategy pro…le s 2 S, that is, P1 t 1 wi (s) = (1 )E i (at ) j s . t=1
3
Belief-Free Review-Strategy Equilibrium
Our objective is to characterize the belief-free review-strategy equilibrium (BFRSE) payo¤ set as discount factor
goes to unity. In this section, we de…ne BFRSE, which is the same
as Yamamoto (2012). First, we de…ne a review strategy pro…le: De…nition 1 Let (tl )1 l=0 be a sequence of integers with t0 = 0, tl > tl
1
for all l
1. A
strategy pro…le s 2 S is a review strategy pro…le with (tl )1 l=0 if si hti for all l 2 N, t 2 ftl
1
1
(1)
[ai;t 1 ] = 1
+ 2; : : : ; tl g, i 2 I and hti = (ai; ; ! i; )t =1 2 Hit .
In this de…nition, an in…nitely repeated game is divided into in…nitely repeated review phases. The lth review phase is from tl
1
+ 1 to tl . (1) implies that, except for the …rst
period of the review phase, player i takes the same action as in the previous period, that is, once player i decides an action ai;tl +1 in the initial period of phase l, then player i sticks to that action within the phase. Second, we de…ne a belief-free review-strategy equilibrium (BFRSE): De…nition 2 Let s 2 S be a review strategy pro…le with a sequence (tl )1 l=0 . s is belief-free if t
si j hil
1
2 BR(s
t
i
j h l i 1 ; a i ) for all a
7
t
i
with s i (h l i 1 ) [a i ] > 0
(2)
Since we assume full support of private signals in Assumption 1, Nash equilibria and sequential equilibria are realization equivalent (See Sekiguchi (1997)). Hence, neglecting player i’s action s~i h0i = ai when we construct hti ; hi does not a¤ect the set of equilibrium payo¤s.
8
for all i 2 I, l 2 N and htl
1
2 H tl 1 .
Imagine that player i is at the beginning of the lth review phase (period tl t
1 + 1)
t
observed hil 1 . (2) requires that, given other players’history h l i 1 and action a t
be taken in the lth review phase, player i’s continuation strategy si j hil
1
i
and has
that will
be optimal. Note
that a belief-free review-strategy pro…le is a Nash equilibrium by de…nition. Since Sekiguchi (1997) shows that with Assumption 1, the set of Nash equilibrium payo¤s is equivalent to that of sequential equilibrium payo¤s, the set of BFRSE payo¤s is included in that of sequential equilibrium payo¤s. The concept of BFRSE, used by Matsushima (2004) and Yamamoto (2012), is the combination of review strategies in Radner (1985) and Abreu et al. (1991) and belief free equilibria in Piccione (2002), Ely and Välimäki (2002) and Ely et al. (2005). In Section 4, we characterized the set of BFRSE payo¤s, which is equal to the characterized set in Yamamoto (2012) and a generalization of Ely et al. (2005). In Section 6, we construct an BFRSE that supports the characterized set, generalizing the equilibrium in Yamamoto (2012). The constructions in both this paper and Yamamoto (2012) incorporate the idea of the block equilibria in Hörner and Olszewski (2006).
4
Assumptions and Result
Our objective is to characterize the set of BFRSE payo¤s as discount factor
goes to unity.
First, we derive a non-trivial upper bound for the set of BFRSE payo¤s.
4.1
Upper Bound of the BFRSE Payo¤ Set
If we did not restrict our attention to BFRSE, then the upper bound applicable for all the monitoring structures would be the set of feasible and individually rational payo¤s.8 For BFRSE, we have the following counterparts for the feasible and individually rational payo¤s. 8
Except for the discussion of correlated minimax (See, for example, Gossner and Hörner (2010)), we know that the folk theorem holds for almost perfect monitoring by Hörner and Olszewski (2006).
9
First, we de…ne the counterpart of the feasible payo¤ set. To de…ne the set, we introduce several notations. A non-empty set A A is a regime generated from A if A has a product Q structure, that is, A = Ai with Ai Ai for all i 2 I. As we will see, A is the set of i2I
actions taken on the equilibrium path.
Let J be the set of all regimes generated from A. Suppose players can access to a public randomization device, which selects a “recommended action set”A from J with probability p(A).9 For any probability distribution p 2 4J , we can consider the “feasible payo¤ set” under the constraint that players obey the recommendation to take an action pro…le in A (that is, an action pro…le that will be taken after A is recommended, which is denoted by a (A), satis…es a (A) 2 A): V (p)
co
(
X
A2J
p (A) (a (A)) j a (A) 2 A, 8A 2 J
)
,
where coB denotes the convex hull of a set B. Second, we de…ne the counterpart of the individually rational payo¤ set. For each i and A 2 J , consider the situation that players
i punish player i. Suppose players
i are
required to take actions included in A but player i is free to deviate to any action. Then, the lowest payo¤ that players
i can guarantee regardless of player i’s action ai 2 Ai is v i (A)
a
min max i 2A i
ai 2Ai
i
This is the “minimax payo¤”when actions by players
(3)
(a) : i are restricted to a
On the other hand, consider the situation that players
i
2 A i.
i reward player i. Suppose every
player is required to take actions included in A. Then, the highest payo¤ that players
i
can guarantee regardless of player i’s action ai 2 Ai is vi (A) 9
a
max min i 2A i
ai 2Ai
i
(a) :
Our result does not reply on the existence of public randomization devices.
10
(4)
This is an object unique to BFRSE. See Ai as the set of player i’s actions taken with a positive probability. By (2), regardless of players
i’s history, player i needs to be indi¤er-
ent with all the actions in Ai . Imagine that players
i have the best history for player i,
that is, player i’s continuation payo¤ is highest. This means that players
i cannot reward
player i by increasing the continuation payo¤ (since player i is currently in the best situation). Hence, player i’s payo¤ of taking ai 2 Ai should be equal to minai 2Ai is, players
(a) (that
i ‡attens player i’s value for all ai 2 Ai by decreasing the continuation payo¤
if player i takes ai 2 = arg minai 2Ai arg maxa
i
i 2A i
minai 2Ai
i
i
(a)) given that players
i take the best action pro…le
(a).
For notational convenience, for each i 2 I and A 2 J , let ai (A) 2 A and ai (A) 2 A be such that ai i (A) 2 A
i
and ai i (A) 2 A
v i (A) = max
ai 2Ai
i
i
solve the above problem, that is,
ai ; ai i (A) ; vi (A) = min
ai 2Ai
i
ai ; ai i (A) :
(5)
Note that aii (A) 2 Ai and aii (A) 2 Ai are arbitrary. In total, [v i (A) ; vi (A)] restricts player i’s payo¤ given A. If A is picked by p, in expecP P tation, [pv i ; pvi ] with pv i A2J p (A) v i (A) and pvi A2J p (A) vi (A) restricts player i’s value given p.
Together with the “feasible payo¤ set”V (p), V (p)\
Q
[pv i ; pvi ] is the upper bound given
i2I
p. Considering all the possible p’s, let F BFRSE
[
p24J
V (p) \
Q
(6)
[pv i ; pvi ] :
i2I
Three remarks are in order: …rst, if there may not exist such A with v i (A)
vi (A) for
all i 2 I. In such a case, F BFRSE is de…ned to be empty. Second, F BFRSE is convex by Q de…nition: if v and v 0 are in F BFRSE , then there exist p and p0 with v 2 V (p) \ [pv i ; pvi ] Q Q i2I and v 0 2 V (p0 ) \ [p0 v i ; p0 vi ]. For all t 2 (0; 1), tv + (1 t)v 0 2 V (~ p) \ [~ pv i ; p~vi ] with i2I
p~ = tp + (1
i2I
t)p0 . Third, by using di¤erent regimes, we can attain asymmetric equilibrium
payo¤s in the repeated game. For example, in the prisoners’ dilemma, if one player has 11
Ai = Ai and the others have Aj being equal to {cooperation}, then player i has high v i (A) and vi (A) while the other players have low v j (A) and vj (A). See Ely et al. (2005) for the detailed discussion of the regimes. The following proposition shows that F BFRSE is the upper bound. Proposition 1 (Proposition 1 in Yamamoto (2012)) If Assumption 1 is satis…ed, then F BFRSE is an upper bound for BFRSE payo¤ set as the discount factor converges to unity: F BFRSE :
lim E ( ) !1
Proposition 1 in Yamamoto (2012) proves lim
!1
E( )
F BFRSE for any monitoring
structure. Then, Yamamoto (2012) proceeds to show that the converse is also true with conditionally independent monitoring, that is, lim lim
!1
F BFRSE , thereby showing
E( )
E ( ) = F BFRSE . The object of this paper is to give generic su¢ cient conditions for
the result lim
4.2
!1
!1
F BFRSE , without assuming conditionally independent monitoring.
E( )
Assumptions E( )
F BFRSE . First, we
assume that each player j taking aj can statistically identify players
j’s actions, that is,
In this subsection, we give su¢ cient conditions to show lim
!1
for each aj , the distribution of player j’s signals given (aj ; a j ) is linearly independent with respect to a j : Assumption 2 For all j 2 I and aj 2 Aj , the collection (qj ( j aj ; a j ))a
j 2A j
of j
j j-
dimensional vectors is linearly independent. Second, suppose again that player j taking aj tries to infer players
j’s actions a j . Fix
two players i 6= j and l 6= i; j arbitrarily. Player j uses the information except for players i and l, !
(i;l) .
Then, the conditional distribution of !
i’s perspective and q
(i;l) (! (i;l)
(i;l)
is q
(i;l) (! (i;l)
j aj ; a j ; ! i ) from player
j aj ; a j ; ! l ) from player l’s perspective. We assume that 12
the collection (q of
(i;l)
(i;l) (
j aj ; a j ; ! i ))a
j 2A j ;! i 2
i
and (q
(i;l) (! (i;l)
j aj ; a j ; ! l ))a
j 2A j ;! l 2 l
-dimensional vectors is linearly independent:
Assumption 3 For all j; i; l 2 I with j 6= i 6= l 6= j and aj 2 Aj , if the collection q aj ; a j ; ! i ))a
j 2A j ;! i 2
i
and (q
(i;l) (! (i;l)
j aj ; a j ; ! l ))a
j 2A j ;! l 2 l
of
(i;l)
(i;l) (
j
-dimensional
vectors is linearly independent. See Section 5 for how we use this assumption. We will also compare this assumption with conditionally independent monitoring. Third, we assume that F BFRSE has full dimensionality: Assumption 4 A stage game payo¤ structure satis…es the full dimensionality condition: F BFRSE = N . Intuitively speaking, with Assumption 4, the set of BFRSE payo¤s has an enough freedom to incentivize players to obey the equilibrium strategy.10
4.3
Characterization of the BFRSE Payo¤ Set
We will show that Assumptions 1, 2, 3 and 4 are enough to show lim
!1
E( )
F BFRSE .
That is, together with Proposition 1, we state the main result as follows: Theorem 1 If N
4 and Assumptions 1, 2, 3 and 4 are satis…ed, then F BFRSE is the limit
set of BFRSE payo¤s as the discount factor converges to unity: F BFRSE = lim E ( ) : !1
(7)
Since Assumptions 1, 2 and 3 are generic if the number of players is no less than four and jA j j
j
jj
and jA j j
(j i j + j l j)
(i;l)
for all j; i; l 2 I, we can derive the following
corollary from Theorem 1: 10
See Yamamoto (2012) for the cases where the full dimensionality condition is violated.
13
Corollary 1 If N
4, jA j j
j
jj
for all j and jA j j
(j i j + j l j)
j; i; l 2 I, then generically, Assumption 4 implies F BFRSE = lim
!1
(j i j + j l j)
(i;l)
The rest of the paper is devoted to the proof of lim (2012) shows lim ori, we show lim
!1 !1
!1
for all
E ( ).
Proof. Assumption 1 is generic. Assumption 2 is generic if jA j j Assumption 3 is generic if jA j j
(i;l)
j
jj
for all j and
for all j; i; l 2 I. E( )
F BFRSE . While Yamamoto
E( )
F BFRSE assuming conditionally independent monitoring a pri-
E( )
F BFRSE with Assumptions 1, 2, 3 and 4, without conditionally
independent monitoring.
5
The Role of Assumption 3: Recovery of the Conditional Independence Property
As mentioned, Matsushima (2004), Yamamoto (2007) and Yamamoto (2012) assume that monitoring is conditionally independent: q(!
i
j a; ! i ) = q(!
i
j a) for all i, a and ! i .
In this section, we …rst explain how Matsushima (2004), Yamamoto (2007) and Yamamoto (2012) use conditional independence to construct BFRSE that supports F BFRSE . Second, we explain how we “endogenously”recover the conditional independence property used in their construction from Assumption 3. The idea of BFRSE is as follows. Even if the monitoring is far from perfect, if we aggregate information for a T -period review phase with long T and make players take a constant action within a phase, player can infer the opponents’actions accurately. Speci…cally, imagine that player i is supposed to take a “cooperative action” ai 2 Ai . Player j monitors player i by creating a “random event”
for all a ~
i
2A
i
j (fai g)
2 f0; 1g such that
8 < q if a ~ i = ai ; 3 Pr(f j (fai g) = 1gj~ a) = : q if a ~i 6= ai 2
with q3 > q2 .
14
(8)
Throughout the paper, in general, we identify a random event j
: Aj
j
calculates
j
2 f0; 1g with a function
! [0; 1] as Yamamoto (2012). After taking aj and observing ! j , player j
j (aj ; ! j )
2 [0; 1]. After that, player j draws a random variable according to the
uniform distribution on [0; 1]. If the realization of this uniform random variable is less than j (aj ; ! j ),
then
random event
j j
= 1; otherwise,
j
= 0. Then,
j (aj ; ! j )
denotes the probability that the
is counted conditional on (aj ; ! j ). Hence, Pr(f
j
= 1gj~ a) = E[
The creation of the random event satisfying (8) (or, function satisfying E[
a]. j (aj ; ! j )j~
a] j (fai g)(aj ; ! j )j~
q3 if a ~i = ai and q2 otherwise) is possible since Assumption 2 implies that player j can identify the opponents’ actions. Player j constructs the “score” by taking the summation of j (fai g)
for the review phase. If the score is high, then player j thinks that player i obeys
the equilibrium strategy while if it is low, then player j thinks that player i does not obey the equilibrium strategy. If the former is the case, then player i “passes the review”while if the latter is the case, then player i “fails the review”and players
i will punish player i by
ai i (A) (given A). By the law of large numbers, the score that player j can expect under player i’s cooperation is close to q3 T . That is, if players below q3 T , then players Hence, players
i do not punish player i even if the score is far
i punish player i too less often to support a low equilibrium payo¤.
i should not let player i pass the review if the score is slightly below q3 T .
If the monitoring is conditionally independent, since player i does not have any additional information about the score during the review phase, by the law of large numbers, player i believes that the realization of the score is always around q3 T after any history where player i has followed the equilibrium. Hence, it is optimal for player i to constantly take ai to escape from the punishment, as desired. However, suppose player i’s signals and player j’s signals are correlated. Imagine, in the middle of the review phase, player i observes a lot of player i’s signals indicating player j’s realized score has been very low, judging from the correlation between ! i and ! j . Then, player i believes that the score will be far below q3 T regardless of player i’s continuation play and then may start to deviate. This is the statistical inference problem mentioned in the Introduction.
15
=
One may think that if player j carefully constructs a function
j,
then we can escape
from the statistical inference problem, which turns out to be false as long as player j only uses her own signal ! j . To prevent the statistical inference problem, we need to construct a function 1. E
j
: Aj
j
a j (fai g)j~
! R such that, conditional on a ~ i ,11
gives di¤erent values between a ~i = ai and a ~i 6= ai (that is,
j (fai g)
identi…es player i’s deviation); 2. E
a; ! i j (fai g)j~
with a ~i = ai gives the same value for all ! i (that is,
j (fai g)
satis…es
the conditional independence property on the equilibrium path). Since Condition 1 imposes jAi j 1 constraints and Condition 2 imposes j i j 1 constraints, there are jAi j + j i j
2 constraints. On the other hand, given a ~
the signal distribution of ! j should be on the simplex over
j,
i
(in particular a ~j ), since
j
has j
jj
1 degrees of
freedom. Hence, to satisfy Conditions 1 and 2 for a generic monitoring, we need to have j
jj
1
jAi j + j i j
2 for all i and j. Since jAi j
2, this implies j
jj
> j i j, which cannot
be satis…ed for all i and j. Therefore, as long as player j only uses her own signal ! j , the statistical inference problem is inevitable. Given this impossibility, it is natural to consider the situation where player j can use the information owned by the other players. As we will see, at the end of the review phase, players
j send the messages about their signal observations to player j. Assumption 3
implies that, for each (j; i; l), if player j can use the information of players j can construct a new signal yj; identify players
(i;l)
j’s actions and yj;
2 Yj; (i;l)
(i;l)
from !
(i;l)
such that yj;
(i; l), then player
(i;l)
can statistically
satis…es the conditional independence property for
players i and l: Lemma 1 If Assumption 3 is satis…ed, then for all j; i; l 2 I with j 6= i 6= l 6= j and aj 2 Aj , player j taking aj can construct a new signal yj; 11
(i;l)
2 Yj;
(i;l)
from !
(i;l)
such that
Since De…nition 2 requires that the strategy is optimal conditional on the opponents’ action in the current phase, we condition that player i take a ~ i.
16
1. yj;
(i;l)
can statistically identify players
aj ; a j ))a a
j
j 2A j
2 A j . Here, pj;
j constructs yj; 2. yj;
of Yj;
(i;l)
(i;l)
(i;l)
(i;l)
j’s actions: the collection (pj;
(i;l) (yj; (i;l)
j
-dimensional vector is linearly independent with respect to
is the distribution of yj;
from !
(i;l) ,
determined by q and how player
(i;l) ;
satis…es the conditional independence property for players i and l: pj;
for all a, yj;
(i;l) (yj; (i;l)
(i;l) ,
j a; ! i ) = pj;
(i;l) (yj; (i;l)
j a; ! l ) = pj;
j a)
! i and ! l .
That is, player j can endogenously create a statistic (signal) yj; identify player
(i;l) (yj; (i;l)
(i;l)
from !
(i;l)
that can
j’s actions with the conditional independence property. Then, player j can
review player i without causing the statistical inference problem. Let us explain why player j excludes player l’s information. Suppose players will play a 2 A in the next review phase if player i passes the review while they will play ai (A) to punish player i if player i fails. Suppose further that player l’s payo¤ is higher in the latter case. If player j always used player l’s information, then player l would tell a lie so as not for player i to pass the review. To prevent such an incentive to tell a lie, player j reviews player i by both yj; player l is excluded) and yj;
(i;l0 )
(i;l)
(where
(where player l0 is excluded) with l 6= l0 . Since there are
at least four players, there are at least two players l and l0 in addition to players i and j. By cross-checking information between players l and l0 , player j can review player i, keeping player l’s incentive to tell the truth. For player l’s incentive, it is important for player l not to be able to infer from ! l how player j monitors player i according to yj;
(i;l)
(Condition 2
of Lemma 1). One may realize that conditionally independent monitoring does not satisfy Assumption 3. However, yj;
(i;l)
= ! j (player j uses her own signal directly) satis…es Conditions 1 and 2
of Lemma 1 and so the equilibrium construction below is valid for conditionally independent monitoring. 17
Let us …nish this section by proving Lemma 1: Proof. Fix j; i; l 2 I with j 6= i 6= l 6= j and aj 2 Aj arbitrarily. Assumption 3 guarantees that there exists jA j j-dimensional vector (a) the collection (E
j; (i;l) (aj ; ! (i;l) )
2 RjA
j; (i;l) (aj ; ! (i;l) )
j aj ; a
)a
j
j 2A j
jj
such that
of jA j j-dimensional vectors is
linearly independent; (b) the conditional independence property holds: for all a 2 A, ! l 2 j a; ! l = E
j; (i;l) (aj ; ! (i;l) )
E
j; (i;l) (aj ; ! (i;l) )
To see why, consider a (jA j j j i j+jA j j j l j) aj ; a j ; ! i ) (1 (i;l)
vector) for all a
(i;l)
vector) for all a
v(a j ) for each a
j
2A
j j
2 A j ; !l 2
j l.
j a; ! i = E matrix Q
(i;l)
2 A j ; !i 2
i
l
and ! i 2
j; (i;l) (aj ; ! (i;l) )
(i;l) (aj )
and q
(i;l) (
stacking q
Assumption 3 guarantees that there exists a
(i;l)
j 2A j
(i;l) (aj )Q (i;l) (aj )
is linearly independent.
jA j j matrix Q
j; (i;l) (aj ; ! (i;l) )
(i;l) (aj )
such that
jA j j matrix
being equal to the row corresponding to !
(i;l)
(i;l) (aj ).
By applying the a¢ ne transformation, we can make sure that the elements are non-negative) and of
j
is equal to v(a j ). That is, Conditions (a) and (b) are satis…ed if we take
jA j j-dimensional vector of Q
(i;l) (
jA j j vector
the row corresponding to (a j ; ! i ) or (a j ; ! l ) of a (jA j j j i j + jA j j j l j) Q
ja :
j aj ; a j ; ! l ) (1
In addition, arbitrarily …x a 1
such that the collection of (v(a j ))a
i,
j; (i;l) (aj ; ! (i;l) )
Let Yj;
(i;l)
j; (i;l) (! (i;l) )
1
j; (i;l) (aj ; ! (i;l) )
1 (the summation of jA j j elements
is no more than 1).
f1; 2; :::; jA j j + 1g. Player j after taking aj and having !
draws a variable yj;
(i;l)
2 Yj;
(i;l)
being equal to the yth element of ing probability 1
0 (all
j; (i;l) (! (i;l) )
such that yj;
(i;l)
j; (i;l) (aj ; ! (i;l) )
(i;l)
= y 2 f1; 2; :::; jA j jg with probability and yj;
(i;l)
= jA j j + 1 with the remain-
1. Then, from Conditions (a) and (b), yj;
Conditions 1 and 2.
18
randomly
(i;l)
satis…es
6
Equilibrium Construction
Take any v 2 int(F BFRSE ). We construct an equilibrium to support v for su¢ ciently large with Assumptions 1, 2, 3 and 4. In the next subsection, we give an overview of the equilibrium construction. After that, we proceed in steps. First, we follow Yamamoto (2012), assuming conditionally independent monitoring. We identify what properties driven from conditionally independent monitoring are used in his construction. Second, we consider a general monitoring satisfying Assumptions 1, 2 and 3. In this step, we assume players would communicate with cheap talk to inform player j of ! 1 guarantees that, using !
(i;l) ,
(i;l) .
Lemma
player j reviews player i, satisfying the properties identi…ed
in the …rst step. Third, we dispense with cheap talk and players communicate by actions. In this step, we assume that public randomization devices were available. As we have mentioned, while the cheap talk is instantaneous, payo¤ irrelevant and precise, the communication by actions takes time, is payo¤ relevant and is imprecise. Players randomly pick a small fraction of periods in a phase to be used for the review by public randomization and communicate signal observations in the picked periods. To increase the precision of the communication, players repeat the messages. Ex ante, since no players know which period will be used for the review, we can maintain players’incentive to follow the equilibrium strategy. Ex post, since players communicate signals in only a small fraction of a phase, the length of communication is su¢ ciently short. Finally, we dispense with public randomization devices. When player j reviews player i by !
(i;l) ,
player j decides what periods to be used for the review and sends the message
about the picked periods to players
(j; i; l) by taking actions. Then, players
(j; i; l) inform
player j of their signal observations in the periods that they believe player j selected. Since player j sends the message by taking actions, players
(j; i; l) may mis-interpret player j’s
message. Since such a mistake happens only with an ex ante small probability, this does not a¤ect players’incentives. 19
For the rest of the paper, let “player i”refer to player i (mod N ) whenever i 2 = f1; : : : ; N g.
6.1
Overview
In the equilibrium construction below, we see the in…nite repeated game as a sequence of Tb -period block games. Tb will be speci…ed in Section 8.4.1. Throughout the paper, we use period t for the tth period of the current block whenever it is apparent what block we are considering.12 In each block game, each player i is either in the “good state” G or in the “bad state” B. When player i is in state G, she plays a Tb -period repeated game strategy sG i during the current block game. On the other hand, when player i is in state B, she plays a Tb -period B G repeated game strategy sB i . These “block-game strategies” fsi gi2I and fsi gi2I are chosen
in such a way that player i earns a high block-game payo¤ wi if player i and earns a low block-game payo¤ wi if player i of players
(i
1). That is, player i
1 is in state G,
1 is in state B, regardless of the state
1’s state controls player i’s payo¤. Therefore, the
B strategy sG i is a “good”strategy and si is a “bad”strategy for player i + 1.
At the end of each block game, a player transits over two states G and B so that the following conditions are satis…ed: (i) given players
i’s states, player i is indi¤erent between
being in good state and being in bad state and is not willing to deviate to a block-game B strategy siTb 6= sG i ; si ; (ii) for each j 6= i
1, player j’s state does not a¤ect player i’s
continuation payo¤; (iii) player i’s continuation payo¤ from a block game is high (wi ) if player i
1’s current state is good and it is low (wi ) if player i
1’s current state is bad.
From (i), a player is indi¤erent between being in state G and in state B independently of the opponents’states. This assures that a continuation play from the beginning of each block game is an equilibrium given any state pro…le. Moreover, (ii) and (iii) imply that player i
1 can solely control player i’s continuation payo¤ through a choice of states and
hence player i
1 does not need to know the state of the other players in order to punish
12 On the contrary, in De…nitions 1 and 2, we use t for the tth period from the beginning of the repeated game, rather than the current block.
20
or reward player i. In particular player i
1 chooses state G if she wants to reward player
i, while she chooses state B if she wants to punish. We use xi 2 fG; Bg to denote player i’s state and x 2 fG; BgN to denote the state pro…le. Let Xi = fG; Bg and X = fG; BgN be the set of possible states of player i and state pro…les, respectively. 6.1.1
Actions, Regimes and Payo¤s
We specify the targeted equilibrium payo¤s wi and wi . Take any v = (v1 ; : : : ; vN ) 2 int(F BFRSE ). By de…nition of F BFRSE in (6), there exists p 2 4J with v included in the Q interior of V (p) \ i2I [pv i ; pvi ]. Let (wi )i2I and (wi )i2I be vectors of real numbers such Q that wi < vi < wi and such that the hyper-rectangle i2I [wi ; wi ] is included in the interior Q Q of V (p) \ i2I [pv i ; pvi ]. It su¢ ces to show that i2I [wi ; wi ] is sustained in BFRSE for su¢ ciently large .
~ a sequence (A1 ; : : : ; AK~ ) As Yamamoto (2012) shows, there exist a natural number K, ~
of regimes, and 2N sequences (ax;1 ; :::; ax;K )x2X of action pro…les such that ~ ~ K K 1 X 1 X k v i A < wi < vi < wi < vi Ak for all i 2 I; ~ ~ K k=1 K k=1 ~ ax;k 2 Ak for all x 2 X and k 2 f1; : : : ; Kg; 8 ~ K < < w if x = B 1 X i 1 i x;k x for all x 2 X and i 2 I: a w i i ~ : > w if x = G K k=1
i
(9) (10) (11)
i 1
(9) implies the following: by (3), v i Ak is player i’s minimax payo¤ when players
i
are restricted to pure actions in Ak . The …rst inequality of (9) implies that, if players
i
~ periods, restricting themselves to Ak in each period k, then player i’s minimax play i for K time-average payo¤ is less than the lower bound of the targeted payo¤ wi . On the other hand, by (4) and the last inequality of (9), if players
~ periods, restricting i reward play i for K
themselves to Ak in each period k, then player i’s time-average payo¤ is more than the upper bound of the targeted payo¤ wi , as long as player i picks ai from Aki (can be less if player i 21
takes an action not included in Aki but it will not be incentive compatible to take such an action). (10) implies that, for each x, taking ax;k in each period k is consistent with the regime i ~
sequence (A1 ; : : : ; AK ): player i in period k takes the action from the regime in period k, ~
Ak . (11) implies that player i’s time-average payo¤ of such a sequence (ax;1 ; :::; ax;K ) is high if player i
1’s state is G and low if player i
1’s state is B.
Intuitively, in the block game, given that the state pro…le of players is x, players take the ~
cycle of (ax;1 ; :::; ax;K ) until some player i’s deviation is con…rmed. After the con…rmation of player i’s deviation, players
~
i take the cycle of (ai i (A1 ); :::; ai i (AK )) if xi
~
of (ai i (A1 ); :::; ai i (AK )) if xi
1
1
= B and that
= G.
Then, on the equilibrium path, player i’s payo¤ is no more than wi (no less than wi , respectively) if xi
1
= B (if xi
1
= G, respectively). In addition, if player i deviates (and
if we assume that the deviation is con…rmed quickly with a high probability by the review), then player i will be punished (rewarded, respectively) by ai i (Ak ) (ai i (Ak ), respectively) and player i’s payo¤ is no more than wi (no less than wi , respectively). In total, player i’s payo¤ in the block game is no more than wi if xi wi if xi
1
1
= B after any deviation and no less than
= G as long as player i takes ai 2 Aki .
Since player i
1 with xi
1
= B (with xi
1
= G, respectively) can increase (decrease,
respectively) player i’s continuation payo¤ by transiting to xi
1
= G (if xi
1
= B, respec-
tively) with a higher probability, this implies that player i’s payo¤ in the block game is equal to wi if xi
= B after any deviation and wi if xi Q Therefore, we can sustain i2I [wi ; wi ] in BFRSE. 6.1.2
1
1
= G for any strategy with ai 2 Aki .
Blocks, Rounds and Phases
The actual equilibrium construction is more complicated: since xi is player i’s private information, players communicate to coordinate on x. In addition, players need to review player i’s action in private monitoring and coordinate on punishment ai i (Ak ) or reward ai i (Ak ) properly.
22
For these purposes, a block game is divided into rounds and each round is further divided into phases.13 Speci…cally, with conditionally independent monitoring,14 each block consists of a signaling round, a con…rmation round, K pairs of a main round and a supplemental round, and a report round. Signaling, con…rmation, supplemental and report rounds are regarded as “communication stages,” where players disclose their private information by actions to coordinate the future play. Unlike cheap talk, actions take time and are payo¤ relevant. However, since the length of the communication stages is su¢ ciently shorter than that of main rounds, the di¤erence in instantaneous utilities can be canceled out by the movement of the continuation payo¤ without a¤ecting the equilibrium payo¤. Hence, in the intuitive explanation below, we neglect the instantaneous utility in the communication stages. Now, we brie‡y explain the structure of the block game given players’state pro…le x. Signaling and Con…rmation Rounds These rounds are used for communication. As suggested above, what actions players should take in the main rounds depends on the state pro…le x. Since xi is player i’s private information, players communicate to coordinate on x. Intuitively, player i reveals whether she is in state G or in state B by taking actions. Based on the history in these rounds, each player j “con…rms” the current state pro…le being x(j) = (xi (j))N i=1 2 X. Loosely speaking, by the law of large numbers, with a high probability, players can coordinate on the true x successfully: x(j) = x for all j 2 I. First Main and Supplemental Rounds If player i has con…rmed that the current state x(i);1
pro…le is x(i), then player i plays an action ai
in the …rst main round. At the same time,
each player reviews the opponents. In the …rst supplemental round, each player reveals whether she thinks the opponents deviated in the …rst main round. Based on the history of the …rst supplemental round, each player “con…rms” who deviated (or no player deviated) in the …rst review round. Intuitively, by the law of large numbers, if a player deviates, then 13 As we have mentioned in Section 3, the strategy pro…le should be belief free at the beginning of each phase and players should take a constant aciton within a phase. 14 With conditionally dependent monitoring, we need to insert additional rounds. See Section 6.5.1.
23
the deviation of the player is con…rmed by all the players with a high probability. kth Main and Supplemental Rounds Players’ play from the second main rounds is determined recursively as follows. If player i has not con…rmed any player’s deviation until x(i);k
the previous supplemental round, then player i plays an action ai
~ in (10)). If (mod K
player i has con…rmed some player j’s deviation, then player i punishes player j by taking aji (Ak ) if player i con…rmed that the state of player j
1 (controller of player j) is B
(xj 1 (i) = B) and rewards player j by taking aji (Ak ) if player i con…rmed that the state of player j
1 is G (xj 1 (i) = G). ~
Note that, if x(i) = x for all i, then players play the cycle of (ax;1 ; :::; ax;K ) until some player i’s deviation is con…rmed. After the con…rmation of player i’s deviation, players ~ K
take the cycle of (ai i (A1 ); :::; ai i (A )) if xi xi
1
1
= B and that of (ai i (A1 ); :::; ai i (A )) if
= G. Hence, if player i’s deviation is properly con…rmed, then player i’s payo¤ in
the block game is no more than wi if xi xi
1
i
~ K
1
= B after any deviation and no less than wi if
= G, as mentioned in Section 6.1.1.
Report Round By the law of large numbers, in the signaling, con…rmation and supplemental rounds, the messages transmit correctly with a high probability. However, there is a small probability that a message transmits wrongly. The report round is used for communication and each player reports what she has observed in the signaling, con…rmation and supplemental rounds to deal with this small probability of mis-transmission of the messages. 6.1.3
Message Exchange
Below, we will o¤er a more detailed explanation of each round. In general, when player i sends a message m to player j from a …nite message space M , we encode the messages in B M into sequences of two di¤erent actions faG i ; ai g and choose a one-to-one map from M B to the set of …nite sequences of faG i ; ai g. In general, the length of the sequence is of order
log2 jM j.15 Throughout the paper, we neglect the integer problem since it is handled by 15
See Section 8.2 for the explicit protocol.
24
replacing each variable s that should be an integer with minn2N n. To increase the precision n s
of the message, player i repeats each action (the component of the message) for su¢ ciently many times. Then, by the law of large numbers, player j can infer player i’s message correctly with an ex ante high probability. In addition, from Assumption 2, this is true regardless of players
(i; j)’s deviation. Hence, we can omit the veri…cation of players
(i; j)’s incentives
while player i sends a message to player j. Note that these two properties are true whether the monitoring is conditionally independent or dependent.
6.2
Conditionally Independent Monitoring
We assume that the monitoring is conditionally independent. Then, the equilibrium construction by Yamamoto (2012) works. We identify what properties driven from the conditionally independent monitoring are used in his construction, expecting that we endogenously re-create these properties from Assumption 3 without conditional independence. 6.2.1
Signaling and Con…rmation Rounds
As explained above, players coordinate on the state pro…le in these rounds: based on the history in these rounds, for each player i 2 I, player j 2 I creates the inference of player i’s state xi (j) 2 Xi . Let x(j) be player j’s inference of the state pro…le. Speci…cally, the signaling round takes N T periods, where each player i sends the message B xi by taking axi i 2 faG i ; ai g, and the ith T periods are used to infer player i’s state xi . (Player
i repeats xi for N T periods but the periods used for the inference of player i’s state and those for player j’s state do not overlap if i 6= j.) Each player j 2 fi
1; i; i + 1g creates player j’s inference of player i’s state m0i;j 2
fG; B; Eg as follows (the inferences by players each period t, player j 2 fi
(i
1; i + 1g can construct
1; i; i + 1) are redundant): as (8), in G j;t (fai g)
2 f0; 1g such that
8 < q if a ~ i = aG 3 i ; Pr(f j;t (faG g) = 1gj~ a ) = i : q if a ~i 6= aG 2 i 25
(12)
for all a ~
i
2A
i
with q3 > q2 . If the time average of
G j;t (fai g)
is very close to q3 , then player
j infers player i’s message is G: m0i;j = G. If the time average of
G j;t (fai g)
is very close to
q2 , then player j infers player i’s message is B: m0i;j = B. Otherwise, player j infers player i’s message is E: m0i;j = E. (Since the average is close to neither the one under a ~ i = aG i nor the one under a ~ i = aB i , player j infers an error.) On the other hand, player i herself infers her state straightforwardly: m0i;i = xi . For players j; j 0 2 fi
1; i; i + 1g, we say
players j and j 0 agree if m0i;j = m0i;j 0 2 fG; Bg, that is, if they infer the same state of player i, which is not E; players j and j 0 disagree if fG; Bg 3 m0i;j 6= m0i;j 0 2 fG; Bg, that is, if one infers xi = G and the other infers xi = B. Note that if one of them infers E, then they neither agree nor disagree. In the con…rmation round, for each i 2 I and j 2 fi
1; i; i + 1g, we assign 2T periods
for player j to send m0i;j 2 fG; B; Eg.16 If m0i;j = G, then player j takes aG j for 2T periods, G 0 if m0i;j = B, then player j takes aB j for 2T periods, and if mi;j = E, then player j takes aj
for the …rst T periods and aB j for the second T periods. Each player n 2
j constructs the inference m0i;j (n). Player n creates
2T periods to infer player j’s message. If the time average of
G n;t (faj g)
G n;t (faj g)
for the
during the …rst
T periods and that during the second T periods are both close to q3 , then player n infers m0i;j (n) = G (since it is likely that player j takes aG i for 2T periods). Similarly, if the time average of
G n;t (faj g)
during the …rst T periods and that during the second T periods are
both close to q2 , then player n infers m0i;j (n) = B. Otherwise, player n infers m0i;j (n) = E. Player j herself straightforwardly infers the message: m0i;j (j) = m0i;j . According to (m0i;j (n))j2fi
1;i;i+1g ,
player n infers xi (n) = G if either (i-a) m0i;i 1 (n) =
m0i;i+1 (n) = G, (i-b) m0i;i+1 (n) = m0i;i (n) = G, or (i-c) m0i;i 1 (n) = m0i;i (n) = G, or (ii) “m0i;i+1 (n) = G and m0i;i 1 (n) = E” or “m0i;i+1 (n) = E and m0i;i 1 (n) = G.” Otherwise, 16
See Section 6.2.3 for the details of the structure.
26
player n infers xi (n) = B. That is, player n infers xi = G if player n infers either that (i) there are at least two players agreeing on G or that (ii) one of players fi xi is G and the other says that there is an error (players i We say that player j 2 fi some player n 2
1; i + 1g says that
1 and i + 1 do not disagree).
1; i; i + 1g is “pivotal”in the coordination of xi if there exists
j whose inference of player j’s message in the con…rmation round m0i;j (n)
has an impact on that player n’s inference of player i’s state xi (n). In other words, if player j is not pivotal, then regardless of player j’s strategy in the con…rmation round, players
j
infer xi in the same way. We repeat this procedure for each player i. From player i’s perspective, players repeat the procedure for each j 2 I. In the report round, players (xj )j2 i , (m0j;j 1 ; m0j;j ; m0j;j+1 )j2I (if i 2 fj
i send their states and inferences
1; j; j + 1g, then exclude player i’s inference
m0j;i ) and ((m0j;j 1 (n); m0j;j (n); m0j;j+1 (n))j2I )n2
i
to player i
1 (a controller of player i) via
actions. Since we can take care of the probability of mis-transmission of the messages in the report round in Section 6.2.4, we proceed as if player i If one of the following events happens, then player i
1 knew these variables precisely. 1 will change the continuation play
from the next block so that player i is indi¤erent between any action pro…le sequence in the current block: (a) in the signaling round, there exists player j 2 who is not player i mis-infers xj : for some j 2
i such that one of the receivers of xj i and j 0 2
(i; j), m0j;j 0 6= xj ;
(b) in the con…rmation round, the communication is erroneous among players j 0; n 2
i: for some
i, m0j;j 0 (n) 6= m0j;j 0 for some j 2 I (including i);
(c) in the con…rmation round, player i is pivotal in the coordination of xj for some j 2 I (including i). We …rst verify that (a), (b) and (c) do not a¤ect player i’s incentive. Second, we list the possible outcomes of the signaling and con…rmation rounds. Third, we verify players’ incentive. Finally, we make sure that with an ex ante high probability, the coordination goes well. 27
(a), (b) and (c) are independent of player i’s strategy in the con…rmation round Since m0j;j 0 is determined in the con…rmation round and the distribution of m0j;j 0 (n) given m0j;j 0 is independent of player i’s strategy by (12), (a) and (b) are independent of player i’s strategy. Given m0j;j 0 and conditioning that (b) is not the case (otherwise, player i has been indi¤erent), (c) is independent of player i’s strategy. Note that Yamamoto (2012) does not require conditional independence in this step. (a), (b) and (c) are almost independent of player i’s strategy in the signaling round The distribution of m0j;j 0 given xj is independent of player i’s strategy by (12). In addition, regardless of the realization of m0j;j 0 , by the law of large numbers, we will have m0j;j 0 (n) = m0j;j 0 and (b) occurs only with a negligible probability. Hence, (a) and (b) are almost independent. Hence, we can condition that neither (a) nor (b) is the case. Note that, for j 2
i, player i is not pivotal in the coordination of xj : if neither (a) nor
(b) is the case, then each player n 2
i thinks that players j and j 0 2 fj
1; j + 1g \ i (a
receiver of xj who is not player i) agree on the true xj , which means player i is not pivotal. Player i is pivotal in the coordination of xi only if players i
1 and i + 1 disagree. We
show that this happens only with a negligible probability regardless of player i’s history in the signaling round. Suppose player i believes that player j 2 fi that is, the time average of
G j (fai g)
1; i + 1g has m0i;j = G,
is close to q3 . By conditional independence, player i
should take aG i for most of the time in the signaling round. Hence, player i believes that the G j 0 (fai g)
time average of
is close to q3 for j 0 = fi
or E with a high probability. That is, players fi
1; i + 1g \ j and that m0i;j 0 is either G 1; i + 1g do not disagree.
Yamamoto (2012) uses conditional independence only to show that player i cannot create a situation where she is pivotal in the coordination of xi . List of possible outcomes Given above, the following is the list of the possible outcomes of the signaling and con…rmation rounds: (i) players
i mis-infer the state of some player j 2
xj (n) 6= xj ; 28
i: there exist j; n 2
i with
(ii) players
i disagree on the inferences for player i’s state: there exist n; n0 2
i with
xi (n) 6= xi (n0 ); (iii) players
i coordinate on the same state pro…le and the inference of x
x(n) = x~ 2 X with x~
i
=x
i
for all n 2
i
is correct:
i.
As we will verify in Section 6.2.2, if (iii) is the case, regardless of players
i’s inference
xi and player i’s own inference of the state pro…le x(i), player i’s payo¤ is equal to wi (wi , respectively) if xi
1
= G (B, respectively). This implies that player i is indi¤erent between
any inference x(i) at the end of the con…rmation round. Player i’s incentives Since (a), (b) and (c) are independent of player i’s strategy in the con…rmation round, player i is indi¤erent between any strategy in the con…rmation round. In addition, (a), (b) and (c) are almost independent of player i’s strategy in the signaling round, player i from the beginning of the block game conditions that none of them is the case. This implies player i will condition that (iii) is the case. Then, player i’s payo¤ is fully determined by xi 1 , which means player i has an incentive to follow the equilibrium path.17 Regular outcomes Since player i has an incentive to take axi i in the signaling round and every player tells the truth in the con…rmation round, from the perspective of the beginning of the block, by the law of large numbers, with an ex ante high probability, each player j 2 fi
1; i; i + 1g has m0i;j = xi , each player n 2 I has m0i;j (n) = xi for all j, no player is
pivotal, and players coordinate on xi . Note that this observation is about ex ante probability and does not use conditional independence. In summary, the following three properties are important: 1. player i in the signaling round believes that she will be pivotal only with a negligible probability regardless of the history; 17
In the signaling round, the probability of having (b) or (c) is small yet di¤erent for di¤erent strategies of player i. Hence, we need to slightly adjust player i’s incentive in the signaling round. On the other hand, player i is completely indi¤erent between any strategy in the con…rmation round.
29
2. if players
i fail to coordinate on the same x~ with x~
i
= x i , then either (a), (b) or
(c) is satis…ed; 3. if players
i coordinate on the same x~ with x~
i’s payo¤ is determined solely by xi
1
i
= x i , then regardless of x(i), player
(this is yet to be shown in Section 6.2.2);
Only 1 requires conditional independence and for the rest of the properties, Assumption 2 is su¢ cient and conditional independence is not necessary. 6.2.2
kth Main and Supplemental Rounds
In the kth main and supplemental rounds, each player’s action is determined recursively as follows: at the beginning of the kth main round, each player i either has not con…rmed the deviation of any player or has con…rmed the deviation of some player j 2 I.
x(i);k
If the former is the case (for k = 1, this is always the case), then player i takes ai
~ in (10)) and reviews the other players in the kth main round. In the kth sup(mod K plemental round, players communicate their review in the kth main round. Based on this communication, player i decides whether or not to con…rm some player’s deviation. (At most one player’s deviation is con…rmed.) ~ in If the latter is the case, then player i punishes player j by taking aji (Ak ) (mod K ~ if xj 1 (i) = G in the (9)) if xj 1 (i) = B and rewards player j by taking aji (Ak ) (mod K) kth main round. (Remember that player j
1 is player j’s controller.) Regardless of the
communication in the kth supplemental round, player i keeps con…rming player j’s deviation. From the above discussion, player i can condition that players x~ = x(n) for all n 2
i with x~
i
i coordinate on the same
= x i . Regardless of x~i and player i’s constant action
included in Aki , if (i) player i’s deviation from axi~;k is properly con…rmed and (ii) players
i’s
deviations are not wrongly con…rmed, then player i’s payo¤ in the block game is equal to wi if xi
1
= B and wi if xi
1
= G, as mentioned in Section 6.1.2.
Conditional independence is important for both (i) and (ii). For (i), with conditional independence, player i cannot update the belief about players 30
i’s signals. Hence, regardless
of player i’s signal observations in the kth main round, player i believes that if she deviates for a non-negligible fraction of the main round, her deviation will be inferred with a high probability by the law of large numbers. For (ii), we want to make sure that, for each player j 2 the review of player j and mislead players
i, player i cannot manipulate
j to con…rm player j’s deviation. As we will
see below, to con…rm player j’s deviation, there need to be more than one player who think player j deviated. With conditional independence, player i believes that players have coordinated on x~, do not think there is a deviator among
i, who
i. Hence, there will be at
most one player (player i) who accuses player j, which means player j’s deviation will not be con…rmed. Speci…cally, Yamamoto (2012) creates the following review, modifying Hörner and Olszewski (2006): in the kth main round, each player j 2 I monitors two players j
1 and
j + 1. By Assumption 2, in each period t in the kth main round, player j can construct x(j);k x(j);k j;t (faj 1 ; aj+1 g)
2 f0; 1g to monitor players j
8 > > q if a ~j > < 1 x(j);k x(j);k a) = Pr(f j;t (faj 1 ; aj+1 g) = 1gj~ q3 if a ~j > > > : q
1 and j + 1: with q1 < q2 < q3 , we have x(j);k 1
1
= aj
1
6= aj
x(j);k x(j);k j;t (faj 1 ; aj+1 g)
viation and the high realization of
x(j);k
and a ~j+1 = aj+1 ;
(13)
otherwise.
2
That is, the low realization of
x(j);k 1
x(j);k
and a ~j+1 6= aj+1 ;
statistically indicates player j + 1’s de-
x(j);k x(j);k j;t (faj 1 ; aj+1 g)
statistically indicates player j
1’s
deviation.18 Based on
x(j);k x(j);k j;t (faj 1 ; aj+1 g),
deviation, mkj 2 fj
player j constructs the inference of players j 1 and j +1’s
1; j + 1; 0g. Speci…cally, if the time average of
x(j);k x(j);k j;t (faj 1 ; aj+1 g)
is
much lower than q2 , then player j infers player j + 1 has deviated: mkj = j + 1. If it is much higher than q2 , then player j infers player j j infers neither player j 18
Aj
1 has deviated: mkj = j
1. Otherwise, player
1 nor j + 1 has deviated: mkj = 0.
As in (8), we identify a random event j (faj j ! [0; 1], following Yamamoto (2012).
1 ; aj+1 g)
31
2 f0; 1g with a function
j (faj 1 ; aj+1 g)
:
In the kth supplemental round, each player j 2 I sends the message about mkj and each player n creates the inference mkj (n). The communication is done as in the con…rmation round, that is, given mkj , the distribution of mkj (n) is independent of player i’s strategy for all j; n 2
i and by the law of large numbers, from the perspective of the main round,
mkj (n) = mkj with a high probability. Based on (mkj (n))j2I , player n “newly con…rms”player i’s deviation if and only if player n has not yet con…rmed any player’s deviation in the previous rounds and infers that there is a unique player i whose two monitors i
1 and i + 1 agree that player i has deviated
(mki 1 (n) = mki+1 (n) = i). As in the con…rmation round, in the report round, players i send (mkj )j2 to player i
1. If one of the following events happens, then player i
i
and ((mkj (n))j2I )n2
1 will change the
continuation play from the next block so that player i will be indi¤erent between any action pro…le sequence from the next main round: (a) in the main round, there exists player j 2
i such that player j infers the deviation of
some player who is not player i: for some j; j 0 2
i, mkj = j 0 . Intuitively, if there exists
another player to be punished or rewarded, then player i is exempted from following axi~;k+1 ; (b) in the supplemental round, the communication is erroneous among players some j; n 2
i: for
i, mkj (n) 6= mkj .
As in Section 6.2.1, we verify that (a) and (b) do not a¤ect player i’s incentive, list the possible outcomes, verify players’ incentive, and make sure that with an ex ante high probability, the review goes well. (a) and (b) are independent of player i’s strategy in the supplemental round Since mkj is determined and the distribution of mkj (n) is independent of player i’s strategy, (a) and (b) are independent of player i’s strategy. Note that Yamamoto (2012) does not require conditional independence in this step. 32
i
(a) and (b) are almost independent of player i’s strategy in the main round Regardless of the realization of mkj , by the law of large numbers, we will have mkj (n) = mkj with a high probability. Hence, (b) is almost independent. Note that Yamamoto (2012) does not require conditional independence in this step. Let us consider (a): suppose players
i have not con…rmed any player’s deviation yet. (In
other words, focus on the main round where neither (a) nor (b) has happened before.) Player i conditions that players player j 2
i coordinate on the same x~. Therefore, players
i uses the same x~ to construct
x ~;k x ~;k j;t (faj 1 ; aj+1 g).
i take ax~;ki and
By conditional independence,
after any history in the kth main round, player i believes that mkj is either i or 0. Therefore, player i after any history in the kth main round believes that players
i’s deviation will not
be con…rmed. Here, Yamamoto (2012) uses conditional independence. List of possible outcomes Given above, the following is the list of the possible outcomes of the main and supplemental rounds: (i) some player n 2
i con…rm a deviation of some player j 2
i: there exist j; n 2
i
i disagree on the con…rmation of player i’s deviation: there exist n; n0 2
i
with mkj 1 (n) = mkj+1 (n) = j; (ii) players
with (mki 1 (n); mki+1 (n)) 6= (mki 1 (n); mki+1 (n)); (iii) players
i do not con…rm a deviation of players
viation: for all n; j 2
i and coordinate on player i’s de-
i, either mkj 1 (n) 6= j or mkj+1 (n) 6= j, and for all n; n0 2
i,
(mki 1 (n); mki+1 (n)) = (mki 1 (n0 ); mki+1 (n0 )); As we will verify later in this subsection, if (iii) is the case, regardless of players i’s inference of (mkj (i))j2I , player i’s payo¤ from the next main round will be constant. This implies that player i is indi¤erent between any inference (mkj (i))j2I . Player i’s incentives Since (a) and (b) are independent of player i’s strategy in the supplemental round, player i is indi¤erent between any strategy in the supplemental round. 33
In addition, since (a) and (b) are almost independent of player i’s strategy in the main round, player i from the beginning of the kth main round conditions that neither (a) nor (b) is the case. This implies that player i will condition that (iii) is the case. Therefore, in the kth main round, player i believes that the distribution of (mki 1 ; mki+1 ) is solely determined by player i’s action frequency by conditional independence, and that (mki 1 (n); mki+1 (n)) = (mki 1 ; mki+1 ) for all n 2
i with a high probability by the law of large numbers. That is,
player i in the kth main round believes that player i’s deviation will be con…rmed if and only if player i deviates for a non-negligible fraction of the kth main round.19 Let us verify that if (iii) is the case, regardless of players i’s inference of x(i) and (mkj (i))j2I , player i’s payo¤ from the next main round will be constant. Remember that player i in the k + 1th main round conditions that players x~
= x i , and that either all the players
i
kth supplemental round or all the players
i coordinate on the same x~ with
i con…rm player i’s deviation as a result of the i con…rm no deviation.
In the …rst case, player i will be punished (rewarded, respectively) if xi ~
1
= B (G,
~
respectively). As seen in (3) and (4), player i will yield v i (Ak ) (vi (Ak ), respectively) for the ~ review round with k~ kth
~
k + 1 as long as player i will take a constant action from Ak .
In the second case, if either (A) player i deviates in the k + 1th main round, (B) x(i) 6= x~ (player i’s inference of the state pro…le is di¤erent), or (C) player i mistakenly con…rms some player’s deviation in the kth supplemental round, then player i’s deviation will be con…rmed as a result of the k + 1th supplemental round with a high probability. By the discussion ~ ~ ~ main round of the …rst case, player i will yield v i (Ak ) (vi (Ak ), respectively) for the kth
with k~
k + 2 (note that player i’s deviation will be con…rmed in the k + 1th supplemental
round, not kth supplemental round). (D) Otherwise, players will take ax~;k+1 . Player i 1 with xi
1
= B (G, respectively) increases (decreases, respectively) the probability of xi
in the next block depending on player i
1
=G
1’s history in the k + 1th main round so player
i’s payo¤ should be the same among (A), (B), (C) and (D) as long as player i will take a 19
In the main round, the probability of having (b) is small yet di¤erent for di¤erent strategies of player i. Hence, we need to slightly adjust player i’s incentive in the main round. On the other hand, player i is completely indi¤erent between any strategy in the supplemental round.
34
constant action from Ak+1 in the k + 1th main round. By backward induction and (9), player i
1 can make sure that player i’s value at the
beginning of the …rst main round is equal to wi if xi
1
= B and wi if xi x(i);k
Regular outcomes Since player i has an incentive to take ai
1
= G.
(or aji (Ak ) and aji (Ak ) if
some player j’s deviation has been con…rmed), from the perspective of the beginning of the block, by the law of large numbers, with an ex ante high probability, x(i) = x(j) = x for all i; j, no player’s deviation is con…rmed, neither (a) nor (b) is the case, and players will take ax;k+1 in the next main round. In summary, the following three properties are important: 1. player i in the kth main round believes that mkj is either i or 0 regardless of the history; 2. player i in the kth main round believes that the distribution of (mki 1 ; mki+1 ) is solely determined by player i’s action frequency; Conditional independence is important for 1 and 2. 6.2.3
Structure of the Block
Before explaining the structure of the report round, let us summarize the structure of the block. Remember that, from De…nition 1, players need to take a constant action within a phase. Hence, some rounds need to be further divided into several phases to satisfy this requirement. The length of each phase is of order T , where T is a large number to apply the law of large numbers. Signaling Round This round is one phase which consists of N T periods. Each player i sends xi by taking axi i for N T periods. Players infer player i’s state sequentially: using the …rst T periods, players creates (m01;j )j2fN;1;2g (the inferences of player 1’s state), using the second T periods, players creates (m02;j )j2f1;2;3g (the inferences of player 2’s state), and so on until using the N th T periods, players creates (m0N;j )j2fN state). 35
1;N;1g
(the inferences of player N ’s
Con…rmation Round This round is divided into 6N phases, six for each i. Each phase consists of T periods. Players communicate about xi sequentially: in the …rst six phases of the con…rmation round, players communicate about x1 ; in the second six phases, players communicate about x2 , and so on, until in the N th six phases, players communicate about xN . Within the six phases where players communicate about xi , players i sequentially send (m0i;j )j2fi
1;i;i+1g :
player i
1, i and i + 1
1 spends two phases to send m0i;i 1 , then player
i spends two phases to send m0i;i , and …nally player i + 1 spends two phases to send m0i;i+1 . Speci…cally, when player j 2 fi
1; i; i + 1g sends m0i;j 2 fG; B; Eg, if m0i;j = G, then player
0 B j takes aG j for two phases (2T periods), if mi;j = B, then player j takes aj for two phases
(2T periods), and if m0i;j = E, then player j takes aG j for the …rst phase (T periods) and 20 While another player is sending a message, player i aB j for the second phase (T periods).
takes aG i . kth Main Round This round is one phase, which consists of KT periods. In the kth x(i);k
main round, each player i takes ai by the end of the k
~ if no player’s deviation has been con…rmed (mod K)
1th supplemental round. If player j’s deviation has been con…rmed,
~ if player i infers xj 1 (i) = B (the controller of player then player i takes aji (Ak ) (mod K) ~ if player i infers xj 1 (i) = G (the controller of j is in the bad state) and aji (Ak ) (mod K) player j is in the good state). kth Supplemental Round This round is divided into 2N phases, two for each i. Each phase consists of T periods. Players communicate about mki sequentially: in the …rst two phases of the supplemental round, player 1 spends two phases to send mk1 ; in the second two phases, player 2 spends two phases to send mk2 , and so on, until in the N th two phases, player N spends two phases to send mkN . Within the two phases where player i sends mki 2 fi
1; i + 1; 0g, if mki = i
1, then player i takes aG i for two phases (2T periods), if
k m0i = i + 1, then player i takes aB i for two phases (2T periods), and if mi = 0, then player B j takes aG j for the …rst phase (T periods) and aj for the second phase (T periods). While 20
If player j is equal to player i, then m0i;j = xi 2 fG; Bg and only …rst two cases are relevant.
36
another player is sending a message, player i takes aG i . 6.2.4
Report Round
As we have seen, so that player i
1 can control player i’s incentive, player i
know the information owned by players ((m0j;j 0 (n))j;j 0 2I )n2
(i 1;i) ,
(mkj )j2
(i 1;i)
1; i) such as (xj )j2
(i
and ((mkj (n))j2I )n2
(i 1;i) ,
(i 1;i) .
1 needs to
((m0j;j 0 )j2I )j 0 2
Each player j 2
can send the messages about these pieces of information to player i
(i 1;i) ,
(i 1; i)
1 by taking actions.
Speci…cally, the report round consists of 2N (4+3N +K+N K) phases, 2(4+3N +K+N K) for each player i. Players communicate sequentially: in the …rst 2(4 + 3N + K + N K) phases of the report round, player 1 sends the messages; in the second 2(4 + 3N + K + N K) phases, player 2 sends the messages, and so on, until in the N th 2(4 + 3N + K + N K) phases, player N sends the messages. Within the 2(4 + 3N + K + N K) phases where player i sends the messages, player i sends xi , (m0j;i )j2fi
(m0j;j 0 (i))j2I;j 0 2fj
1;i;i+1g ,
1;j;j+1g ,
mki and (mkj (i))j2I ,
sequentially, as in the signaling, con…rmation and supplemental rounds.21 It takes 1 |{z} xi
+
1 |{z}
one phase is su¢ cient to send xi 2fG;Bg
3N |{z}
(m0j;j 0 (i))j2I;j 0 2fj
+|{z} K mki
1;j;j+1g
2 |{z}
+
3 |{z}
(m0j;i )j2fi 1;i;i+1g
2 |{z}
2 |{z}
two phases are su¢ cient to send m0j;i 2fG;B;Eg
two phases are su¢ cient to send m0j;j 0 (i)2fG;B;Eg
+
two phases are su¢ cient to send mki 2fi 1;i+1;0g
= 2(4 + 3N + K + N K)
NK |{z}
(mkj (i))j2I
2 |{z}
two phases are su¢ cient to send mkj (i)2fG;B;Eg
phases. Since the cardinality of this information is independent of T and player i repeats each action (the component of the message) for T periods, each player j can make sure that player 1 infers player j’s messages correctly with probability no less than 1
i 21
exp( O(T )) by
Note that while players simultaneously send x in the signaling round, each player sequentially sends xi in the report round.
37
the law of large numbers. Since each player j (that is, player j players
1 uses players
(j
1; j)’s messages to control player j’s incentive
1’s state in the next block), in total, each player i’s messages only a¤ect
i. Hence, the truthtelling incentive is satis…ed. In addition, since player i
signal can identify player j 2
(i
1’s
1; i)’s action (the component of the message) regardless
of player i’s deviation by Assumption 2, player i cannot manipulate the communication between players i
1 and j.
The remaining problem is that, since player i
1’s signal is noisy, although players
1; i) repeat the messages to increase the precision of the communication, there exists
(i
a small but positive probability that player i
1 makes a mistake to infer the messages.
Consider the following adjustment of the continuation strategy from the next block to deal with the probability of mistakes: let Mi
1
be the set of information possibly sent to player
1 in the report round. Remember that jMi 1 j is …nite and independent of T . Let
i Pi
1
be the jMi 1 j
jMi 1 j matrix whose (k; k 0 ) element represents the probability that
1 receives the messages corresponding to the element k 0 of Mi
player i
1
given that players
1; i) send the messages corresponding to the element k of Mi 1 . Since jMi 1 j is …nite
(i
and independent of T and all the messages transmit correctly with probability no less than 1
exp( O(T )), lim Pi
T !1
1 1
= E (identity matrix).
Suppose that the messages in the report round would transmit without a mistake (that is, Pi
1
would be E). In such an ideal situation, player i
1 with current state xi
1
would
take to the generous strategy from the next block with probability Pr( x0i
1
= G j xi 1 ; hTi b 1 ; k)
(14)
after observing history hTi b 1 and receiving the messages corresponding to the kth element of Mi 22
1
in the report round.22 In addition, let Pr(xi 1 ; hTi b 1 ) be the vector stacking all
Since hTi b 1 includes the history in the report round, hTi b 1 has enough information to pin down k, but we
38
Pr( x0i
1
= G j xi 1 ; hTi b 1 ; k) with respect to k.
Now consider the situation where the probability of a mistake is denoted by Pi 1 . Instead of using (14), player i
1 uses kth element of Pi 11 Pr(xi 1 ; hTi b 1 )
when player i 1 receives kth element of Mi is the same as in the situation where Pi
1
1
in the report round. Then, player i’s incentive
would be E. That is, player i before the report
round can assume that the messages in the report round will transmit correctly, as desired. Note that this property is valid whether or not the monitoring is conditionally independent.
6.3
Conditionally Dependent Monitoring with Cheap Talk
In this subsection, we consider the case with conditionally dependent monitoring. For simplicity, we assume that players can communicate by cheap talk, which will be dispensed with in the sequel. Below, we explain how we show the properties derived in Section 6.2 without conditional independence by using the idea in Section 5. 6.3.1
Signaling and Con…rmation Rounds
As we have seen in Section 6.2, we use conditional independence only to show that, after any history in the signaling round, player i’s posterior on the event that player i will be pivotal is negligible. Speci…cally, player i believes that the time average of player i G i 1 (fai g)
and that of player i + 1’s score
G i+1 (fai g)
1’s score
are close to each other since, with
conditional independence, from player i’s perspective, their distribution is determined only by player i’s action frequency. However, with conditionally dependent monitoring, after some history, player i may receive player i’s signals indicating that the realizations of and
G i+1 (fai g)
G i 1 (fai g)
are very di¤erent and that player i may be pivotal if she deviates optimally
in the remaining periods. factor out k explicitly to focus on k.
39
To make sure that player i believes that she will not be pivotal after any history in the signaling round, we modify the inference in the signaling round as follows: as with conditional independence, player i takes axi i for N T periods so that players fi
1; i; i + 1g construct the
inference of xi from the ith T periods. Since conditional independence is important, player j 2 fi
1; i + 1g constructs the inference by collecting the information from players
(i; j)
so that the conditional independence property holds, as mentioned in Section 5. Speci…cally, after the signaling round, players exchange their signal observations by cheap talk, so that each player j 2 fi
1; i + 1g knows players
this information, player j constructs jIj j constructs yj;
(i;l)
2 inferences (m0i;j;l )l2
1; i + 1g constructs m0i;j;l , player l 2
When player j 2 fi
j’s signal observations. Based on
de…ned in Lemma 1. Since yj;
(i;l)
(i;j)
with m0i;j;l 2 fG; B; Eg.
(i; j) is excluded. That is, player
identi…es player i’s action, player j can
construct m0i;j;l 2 fG; B; Eg as with conditionally independent monitoring, replacing yj with yj;
(i;l) .
As with conditionally independent monitoring, player i has only one straightforward
inference of xi : mi;i = xi . We say players fi for some l 2
1; i + 1g “disagree”if there are player j 2 fi
(i; j) and player j 0 2 fi
1; i + 1g with m0i;j 0 ;l0 = B for some l0 2
(Note that j and j 0 can be the same. For example, if m0i;i players fi
1; i + 1g with m0i;j;l = G
1;i+1
= G and m0i;i
1;i+2
(i; j 0 ).
= B, then
1; i + 1g disagree.)
In the con…rmation round, each player j sends (m0i;j;l )l2
(i;j)
(if j = i, m0i;i ) to the other
players so that each player n 2 I constructs the inferences m0i;i (n) and ((m0i;j;l (n))l2
(i;j) )j2fi 1;i+1g .
The communication protocol is the same as in conditionally independent monitoring. Hence, each player cannot manipulate the communication among the other players and the messages transmit precisely with an ex ante high probability. Based on m0i;i (n) and ((m0i;j;l (n))l2 if player n infers players fi ((m0i;j;l (n))l2
(i;j) )j2fi 1;i+1g ,
player n construct xi (n) as follows:
1; i + 1g do not disagree, then player n only uses
(i;j) )j2fi 1;i+1g :
– if there exist j 2 fi
1; i + 1g and l 2 40
(i; j) with m0i;j;l (n) = G, then xi (n) = G;
– if there exist j 2 fi
1; i + 1g and l 2
– if m0i;j;l (n) = E for all j 2 fi if player n infers players fi
(i; j) with m0i;j;l (n) = B, then xi (n) = B;
1; i + 1g and l 2
(i; j), then xi (n) = B;23 and
1; i + 1g disagree, then player n only uses m0i;i (n): xi (n) =
m0i;i (n). As in Section 6.2.1, we say that player j 2 fi 1; i; i+1g is “pivotal”in the coordination of xi if, for some player n 2
j, player n’s inferences of player j’s messages in the con…rmation
round have an impact on player n’s inference of player i’s state xi (n). As with conditionally independent monitoring, we repeat this procedure for each player i. Player i
1, the controller of player i, will know players
i’s inferences in the sig-
naling and con…rmation rounds from the messages in the report round: (xj )j2 i , (m0j;j )j2 i , ((m0j;j
1;l )l2 (j;j 1) )j2I;j 16=i ,
((m0j;j
1;l )l2 (j;j 1) )j2I
player i
((m0j;j+1;l )l2
and ((m0j;j+1;l )l2
(j;j+1) )j2I;j+16=i , (j;j+1) )j2I
and player n’s inference of (m0j;j )j2I ,
with n 2
i. Based on this information,
1 makes player i indi¤erent between any action pro…le sequence if one of the
following conditions is satis…ed: (a) in the signaling round, there exists player j 2
i such that one of the receivers of
xj who is not player i mis-infers xj when player i is excluded: for some j 2 j 0 2 (j
i and
1; j + 1) \ i, player j 0 infers m0j;j 0 ;i 6= xj ;
(b) in the con…rmation round, the communication about m0j;j 0 ;i or m0j 0 ;j 0 is erroneous among players
i; for some j; n 2
or for some j 0 ; n 2
i and j 0 2 (j 1; j +1)\ i, player n has m0j;j 0 ;i (n) 6= m0j;j 0 ;i
i, m0j 0 ;j 0 (n) 6= m0j 0 ;j 0 ;
(c) in the con…rmation round, player i is pivotal in the coordination of xj for some j 2 I (including i). As before, we verify that (a), (b) and (c) do not a¤ect player i’s incentive. 23
If player n infers that players fi
1; i + 1g do not disagree, then these are only possible cases.
41
(a), (b) and (c) are independent of player i’s strategy while players communicate about signal observations in the signaling round or they are in the con…rmation round Since player i is excluded in the construction of m0j;j 0 ;i , player i after the signaling round takes m0j;j 0 ;i as given. Hence, (a) and (b) are independent of player i’s strategy. (a), (b) and (c) are almost independent of player i’s strategy in the signaling round Regardless of the realization of m0j;j 0 ;i , player n will infer player j 0 ’s message correctly with a high probability. Hence, (a) and (b) are almost independent. Hence, we can condition that neither (a) nor (b) is the case. Note that, for j 2
i, player i is not pivotal in the coordination of xj : if neither (a) nor
(b) is the case, then each player n 2
i infers that
m0j;j (n) = m0j;j since (b) is not the case = xj by de…nition and that m0j;j 0 ;i (n) = m0j;j 0 ;i since (b) is not the case = xj since (a) is not the case, which means player n infers xj (n) = xj regardless of player i’s message in the con…rmation round. Player i is pivotal in the coordination of xi only if players i 1 and i+1 disagree. Without assuming conditional independence a priori, Lemma 1 guarantees that this happens only with a negligible probability: since the conditional independence property holds in the signaling round for how player j 2 fi
1; i + 1g monitors player i’s action by yj;
(i;l)
for all l 2
(i; j),
from the perspective of player i, the distribution of m0i;j;l is determined only by player i’s action frequency for all j 2 fi
1; i + 1g and l 2
(i; j). Hence, if player i believes that
player j infers player i takes G (B, respectively) excluding l, then player i takes aG i often, 42
which means player i cannot believe that player j 0 infers player i takes B (G, respectively) excluding l0 , as desired. The rest of the proof is the same as in Section 6.2.1: we list the possible outcomes, verify the incentives, and check the coordination goes well with a high probability. List of possible outcomes (i) players
i mis-infer the state of some player j 2
(ii) players
i disagree on the inferences for player i’s state;
(iii) players
i coordinate on the same state pro…le and the inference of x
x(n) = x~ 2 X with x~
i
=x
i
for all n 2
i;
i
is correct:
i.
As we will verify in Section 6.3.2, if (iii) is the case, regardless of x~i and x(i), player i’s payo¤ is equal to wi (wi , respectively) if xi
1
= G (B, respectively).
Player i’s incentives Since (a), (b) and (c) are independent of player i’s strategy after the signaling round, player i conditions that none of (a), (b) and (c) is the case while players communicate about signal observations in the signaling round or they are in the con…rmation round. (This implies (iii).) Given this, player i’s payo¤ is fully determined by xi
1
and player
i is indi¤erent between any message to send. In addition, (a), (b) and (c) are almost independent of player i’s strategy in the signaling round, player i from the beginning of the block game conditions that none of them is the case after a small adjustment of player i’s incentive. Then, player i’s payo¤ is fully determined by xi 1 , which means player i has an incentive to follow the equilibrium path. Regular outcomes By the law of large numbers, with an ex ante high probability, for each j, l and n, we have m0i;j;l (n) = m0i;j;l = xi , no player is pivotal, and players coordinate on xi .
43
kth Main and Supplemental Rounds
6.3.2
With conditionally independent monitoring, each player j monitors players j creating mkj 2 fj 1; j +1; 0g from
x(j);k x(j);k j;t (faj 1 ; aj+1 g).
1 and j +1 by
As seen in Section 6.2.2, conditional
independence is used to show that, for each i 2 I, conditional on that players
i coordinate
on the same x~, regardless of player i’s signal observation, with a high probability, player i believes that 1. regardless of player i’s strategy, mkj = 0 for all j 2
(i
1; i; i + 1);
2. if player i takes ai 6= axi~ for non-negligible fraction of the kth main round, then mki mki+1 = i. Otherwise, mki
1
1
=
= mki+1 = 0;
With conditionally dependent monitoring, we modify the review in the kth main round to keep these properties endogenously, as mentioned in Section 5. That is, after the kth main round, players exchange their signal observations by cheap talk, so that each player j knows players
j’s signals. Based on this information, player j constructs mkj 2 fj
1; j + 1; 0g as
follows. First, player j uses yj; yj;
(j 1;j+1)
(j 1;j+1)
to monitor players j
1 and j + 1. By Lemma 1,
can statistically identify a j . Replacing yj with yj;
a random event
x(j);k x(j);k j;t (faj 1 ; aj+1 g)
x(j);k x(j);k j;t (faj 1 ; aj+1 g)
(j 1;j+1) ,
player j can create
2 f0; 1g such that (13) holds. If the time average of
is much lower than q2 , then player j “temporarily” infers player j + 1
has deviated: mkj;temp = j + 1. If it is much higher than q2 , then player j “temporarily”infers player j
1 has deviated: mkj;temp = j
1. Otherwise, player j infers neither player j
1
nor j + 1 has deviated: mkj;temp = mkj = 0 (in this case, this inference is not only temporary but also …nal). The di¤erence from Section 6.2.2 is that the inference of the deviation is temporary. By Lemma 1, players j
1 and j + 1 who are excluded from yj;
(j 1;j+1)
player j’s temporary review from their private signals. However, players may be able to infer player j’s review on players j the review as follows. 44
(j
cannot infer 1; j; j + 1)
1 and j + 1. Hence, we further adjust
If player j temporarily infers some player’s deviation, then player j proceeds to the second step. If player j temporarily infers player j + 1’s deviation, then player j “double-checks” player j + 1’s deviation by using the messages about signals from the other players 1; j; j + 1,24 since yj;
That is, for each l 6= j
E
for all a ~
(j+1) .
j;
can statistically infer player j + 1’s
x(j) j; (j+1;l);t (faj+1 g)
action, in each period t, player j can create h
(j+1;l)
(j + 1).
from yj;
(j+1;l);t
such that
8 x(j) < q if a ~j+1 = aj+1 ; 3 x(j) (fa g)j~ a = (j+1;l);t j+1 x(j) : q if a ~j+1 6= aj+1 2 i
If the time average of
x(j) j; (j+1;l);t (faj+1 g)
deviation is “double-checked” by players
is not close to q3 , then player j + 1’s
(j + 1; l): mkj;l = B. Otherwise, player j + 1’s (j + 1; l): mkj;l = G. If player j + 1’s deviation
deviation is not “double-checked”by players is double-checked by players
(15)
(j + 1; l) for all l 6= j
1; j; j + 1, then player j + 1’s deviation
is …nally inferred: mkj = j + 1. If player j + 1’s deviation is not double-checked by players (j + 1; l) for some l 6= j
1; j; j + 1, then player j + 1’s deviation is not inferred and mkj = 0.
Symmetrically, when player j temporarily infers player j 1; j; j + 1, player j can create
j
E
for all a ~
(j 1) .
h
j; (j
x(j) j; (j 1;l);t (faj 1 g)
from yj;
8 < q if a ~j 3 x(j) a = 1;l);t (faj 1 g)j~ : q if a ~j 2
If the time average of
i
x(j) j; (j+1;l);t (faj+1 g)
deviation is “double-checked” by players
(j
1’s deviation, for each l 6= (j 1;l);t
such that
x(j) 1;
1
= aj
1
6= aj
(16)
x(j) 1
is not close to q3 , then player j
1’s
1; l): mkj;l = B. Otherwise, mkj;l = G. If
player j
1’s deviation is double-checked by players
mkj = j
1. Otherwise, mkj = 0.
(j
1; l) for all l 6= j
1; j; j + 1, then
To make the notation well de…ned after all the histories, if no player’s deviation is temporarily inferred (mkj;temp = 0), then no player’s deviation is double-checked: mkj;l = G for all l 6= j 24
1; j; j + 1.
Since yj;
(j 1;j+1)
has been already used, we have l 6= j
45
1.
Now, we have de…ned mkj;temp , (mkj;l )l2
(j 1;j;j+1)
and mkj for each j. As in Section 6.2.2, in
the kth supplemental round, each player j 2 I sends the message about mkj and each player n creates the inference mkj (n). Based on (mkj (n))j2I , player n con…rms player i’s deviation if and only if player n infers that there is a unique player i whose two monitors i
1 and i + 1
agree that player i has deviated (mki 1 (n) = mki+1 (n) = i). In the report round, players to player i
i send (mkj;temp ; (mkj;l )l2
(j 1;j;j+1) )j2 i
and ((mkj (n))j2I )n2
1. If one of the following events happens, then player i
i
1 will change the
continuation play from the next block so that player i will be indi¤erent between any action pro…le sequence from the next main round: (a) in the main round, player i’s monitors temporarily infer the other player’s deviation, or player j double-checks some player’s deviation when player i is excluded from mkj;l : mki
1;temp
= i + 1, mki+1;temp = i + 2 or mkj;i = B for some j 2
(i
1; i; i + 1);
(b) in the supplemental, the communication is erroneous among players n2
i, mkj (n) 6= mkj for some j 2
i: for some
i;
As in Section 6.2.2, we verify that (a) and (b) do not a¤ect player i’s incentive, list the possible outcomes, verify players’ incentive, and make sure that with an ex ante high probability, the review goes well. (a) and (b) are independent of player i’s strategy while players communicate about signal observations in the main round or they are in the supplemental round Since player i is excluded, player i after the kth main round takes mki
1;temp ,
mki+1;temp and mkj;i as given. Hence, (a) is independent of player i’s strategy. Conditioning that (a) is not the case (otherwise player i has been indi¤erent), player j’s inference mkj is determined regardless of player i’s messages while players communicate about signal observations in the main round. Given mkj , the distribution of mkj (n) is independent of player i’s strategy.
46
(a) and (b) are almost independent of player i’s strategy Suppose players
i have
not con…rmed any player’s deviation yet. (In other words, focus on the main round where (a) or (b) has not happened before.) Player i conditions that players
i coordinate on the
same x~. For mki and yi+1; of yj;
1;temp
(i;i+2)
(j 1;i)
and mki+1;temp , the conditional independence property holds for yi
1; (i 2;i)
against player i. In addition, for mkj;i , player i is excluded from construction
and yj;
(j+1;i)
against player i. Hence, (a) and (b) do not a¤ect player i’s incentive
as in Section 6.2.2 with a small adjustment. List of possible outcomes As in Section 6.2.2, the list of possible outcomes is as follows: (i) players
i con…rm a deviation of some player j 2
(ii) players
i disagree on the con…rmation of player i’s deviation;
(iii) players
i do not con…rm a deviation of players
i;
i and coordinate on player i’s devi-
ation. As in Section 6.2.2, if (iii) is the case, regardless of players i’s inference of (mkj (i))j2I , player i’s payo¤ from the next main round will be constant. This implies that player i is indi¤erent between any inference (mkj (i))j2I . Player i’s incentives While players communicate about signal observations in the main round or they are in the supplemental round, since (a) and (b) are independent of player i’s strategy player i can condition that neither (a) nor (b) is the case, which implies that (iii) is the case: if mkj;i = G for all j 2
i, then mkj = 0 for all j 2
mkj (n) = mkj = G. Therefore, player n does not con…rm players
i. Since (b) is not the case, i’s deviations regardless of
player i’s messages and con…rms player i’s deviation if and only if (mki 1 (n); mki+1 (n)) = (i; i). Therefore, player i is indi¤erent between any messages. In the kth main round, since (a) and (b) are almost independent of player i’s strategy, player i from the beginning of the kth main round conditions that neither (a) nor (b) is the 47
case after a small adjustment of player i’s incentive. That is, player i will condition that (iii) is the case and that the distribution of mki and mki+1;i for all l 2
(i
1;temp ,
mki+1;temp , mki
1;l
for all l 2
(i
1; i; i + 1)
1; i; i + 1) is solely determined by player i’s action frequency by
the conditional independence property,25 and that (mki 1 (n); mki+1 (n)) = (mki 1 ; mki+1 ) for all n2
i with a high probability by the law of large numbers. Therefore, player i in the kth
main round believes that player i’s deviation will be con…rmed if and only if player i deviates for a non-negligible fraction of the kth main round. Therefore, the incentives are the same as with conditionally independent monitoring. By the law of large numbers, with an ex ante high probability, no player’s deviation is inferred, neither (a) nor (b) is the case, and players will take ax;k+1 in the next main round. 6.3.3
Four Players are Required
Before proceeding to the dispensability of the cheap talk, let us summarize where we use the assumption that there are no less than four players. After the Signaling Round When players coordinate on xi , player i should not be able to create a situation where she is pivotal. In the above construction, in such a case, we make player i indi¤erent between any action pro…le sequence. This might increase player i’s equilibrium payo¤. On the other hand, if we do not control player i’s incentive to tell the truth in such a way, player i would want to crease a situation where players
i have
di¤erent inferences of xi . Note that (9) and (11) do not control players’ payo¤ in such a mis-coordinated case. Hence, players
i should infer xi independently of player i’s messages about signal ob-
servations in the signaling round with an ex ante high probability, and the conditional independence property should hold while players are still in the signaling round. The latter implies that each player j needs to have another player n 2
(i; j) whose messages matter
for player j’s inference (see Section 5). This means that if it were not with another player 25
Note that unless mki respectively).
1;temp
=i
1 (mki+1;temp = i + 2 respectively), player i is excluded from mki
(mki+1;i
48
1;l
l2
(i; j; n), then player n would always matter for player j’s inference. As for player i in
the last paragraph, player n would want to create a situation where players i and j have different inferences of xi by telling a lie. Therefore, it is important to have player l 2
(i; j; n),
which requires at least four players. After the kth Main Round When players review player i in the kth main round, players i should infer player i’s deviation without relying on player i’s messages about signal observations in the kth main round (otherwise, player i would tell a lie). Since the conditional independence property should hold, each player j 2
i needs to have another player n 2
(i; j) whose messages matter for player j’s inference of player i’s deviation (see Section 5). This means unless there is another player l 2
(i; j; n), player n would always matter for
player j’s inference. If it is pro…table for player n to induce the situation to punish player ~
~
i by ai i (Ak ) or to reward player i by ai i (Ak ) (note that (9) does not control players payo¤s), then player n would tell a lie. Therefore, it is important to have player l 2
6.4
i’s
(i; j; n).
Conditionally Dependent Monitoring without Cheap Talk but with Public Randomization
In the previous section, players communicate about their signal observations in each period of signaling and main rounds by cheap talk. Now, we assume that the cheap talk communication devices are not available but public randomization devices are available. Instead of cheap talk, players communicate about signal observations by taking actions. Then, if players communicated about each period of the main rounds, then it would take too long to implement the targeted payo¤. To deal with this problem, consider the following protocol using public randomization: after each main round, players draw jIj public randomization devices, one for each player j independently. (After the signaling round, players communicate about all the periods since the length of the signaling round is much smaller than that of the main round.) For each player j, the public randomization device randomly selects a period tj;k from the set of 49
periods in the kth main round. If tj;k is selected for player j, then it means that player j uses periods tj;k , tj;k + 1, ..., tj;k + "KT
1 (mod "KT ) to review players
the length of the kth main round and " is a small number. Players their signal observations in periods tj;k , tj;k + 1, ..., tj;k + "KT
j. Here, KT is
j send messages about
1 by actions. That is, player
j picks "-fraction of the kth main round for the review by public randomization and players communicate only about these picked periods. As seen in Section 6.1.3, each player n 2
j attaches a sequence of log2 j j actions to
each of ! n and sends the messages about ! n;t with t = tj;k , tj;k + 1, ..., tj;k + "KT
1
by actions. To increase the precision of the message, player n repeats each action (the component of the message) for S times. (S will be …xed independently of T .) Given ! n;t , player j infers player n’s message correctly with probability 1
log2 j j exp( S). Although
there exists a positive probability of mis-inference, this high probability is su¢ cient for player j to construct statistics to monitor player i from player j’s own signal ! j;t and player j’s inference of ! n;t that satisfy the conditional independence property against player i. (See Lemma 6 in Section 8.3 for the formal proof.) This protocol is incentive compatible: each player is indi¤erent between any strategies while she sends the messages about the signal observations as seen in Sections 6.3.1 and 6.3.2. In addition, Assumption 2 guarantees that each player cannot manipulate the communication among the other players. While players are playing the kth main round, the realization of the public randomization device after the round is random and players think that each period will be used for the review with equal probability ". Therefore, the situation is as if the precision of the monitoring were multiplied by ", which does not a¤ect the precision of the review if we take the length of the round su¢ ciently long and rely on the central limit theorem. After the realization of the public randomization device, since players communicate about only "-fraction of the main round, if " is su¢ ciently small to make "S log2 j j su¢ ciently small, the communication does not take a long time compared to the length of the main round. Hence, it does not a¤ect the equilibrium payo¤.
50
6.5
Conditionally Dependent Monitoring without Cheap Talk or Public Randomization
Finally, we consider the situation where players have neither cheap talk nor public randomization device. Instead of picking period tj;k by public randomization, player j picks period tj;k randomly after the kth main round by herself. After that, each player j sends the message about which period tj;k player j picks by actions. Each player n 2
j infers player j’s
message from her own private signals and sends the messages about what signal she observed in periods tj;k (n), tj;k (n) + 1, ..., tj;k (n) + "KT
1, where tj;k (n) is player n’s inference of
player j’s message tj;k . As explained in Section 6.1.3, to communicate tj;k 2 f1; :::; KT g, player j needs to convey the sequence of actions whose length is log2 KT . If player j repeats each action (the 1
component of the message) T 2 times to increase the precision, player j’s message transmits correctly with an ex ante probability no less than of order 1
1
(log2 KT ) exp( T 2 ). Thus,
player j’s message transmits correctly with an ex ante high probability for su¢ ciently large 1
T . In addition, player j spends (log2 KT )T 2 periods to send the message about tj;k , which is much smaller than KT , the length of the main round. Hence, this does not a¤ect the equilibrium payo¤. Consider the incentives of players. While they are communicating about tj;k and signals, as explained in Sections 6.3.1 and 6.3.2, players are indi¤erent between all the strategies. In addition, Assumption 2 ensures that players do not have an incentive to manipulate the communication among the other players. Next, let us consider the incentives in the kth main round. Suppose player j reviews player i. If the message tj;k transmits correctly, then player i’s signal observation does not reveal anything about player j’s review on player i as explained in Section 6.4, as desired. We are left to verify that a small probability that tj;k will not transmit properly does not a¤ect player i’s incentive. After such a mistake, the statistics that player j creates do not satisfy the conditional independence property, which means that player i’s signal observation in the kth main round may have information about how player j reviews player i in such a 51
1
case. However, such an event happens with a small probability (log2 KT ) exp( T 2 ) from the perspective in the kth main round. Hence, player i in the kth main round believes that, after any history, the signal observation in the kth main round does not have information about player j’s review with a high probability. In other words, “almost conditional independence property”is satis…ed. Since the original equilibrium of Yamamoto (2012) is strict except for the initial period of each phase, it is su¢ cient to have the almost conditional independence property, as Proposition 5 of Yamamoto (2012). 6.5.1
Structure of the Block
Given the equilibrium construction without cheap talk or public randomization, the structure of the block is determined as below. Again, since De…nition 1 requires that players take a constant action within a phase, we need to divide a round into several phases. Especially, when players communicate about signal observations to recover the conditional independence property, they need to change an action frequently and the communication consists of a lot of phases. Signaling Round This round is one phase which consists of N T periods and players communicate x as in Section 6.2. Recovering Round for the Signaling Round This is the name of the round where players communicate about signal observations in the signaling round to recover the conditional independence property. This round consists of N (N
2)T log2 j j phases, (N
2)T log2 j j
phases for each i. Players communicate sequentially. That is, for the …rst (N
2)T log2 j j
phases, players communicate about the signaling round in order to infer player 1’s message x1 , and so on, until players communicate about the signaling round in order to infer player N ’s message xN . When players communicate in order to infer xi , each player n 2 sends the signal observations in periods (i
1)T + 1, (i
(i; i
1) sequentially
1)T + 2, ..., iT to player i
1.
(Remember that the signaling round is from period 1 to period N T and the ith T periods 52
are used to infer player i’s state xi .) Player n spends log2 j j phases to send a message about a signal observation in each period.26 Repeating these phases for each xi , in total, we have (N
2)T log2 j j phases. Since we repeat these phases for each xi , we have N (N
Con…rmation Round This round is divided into N (2N
2)T log2 j j phases. 3) phases. The basic structure
is the same as in Section 6.2 but now, for each i, instead of (m0i;j )j2fi to communicate m0i;i , (m0i;i
1;l )l2 (i;i 1)
and (m0i;i+1;l )l2
(i;i+1) .
1;i;i+1g ,
players need
Note that the conditional
independence property is not necessary for the con…rmation round. kth Main Round This round is one phase which consists of KT periods and players take actions as in Section 6.2. kth Recovering Round This is the name of the round where players communicate about signal observations in the main round to recover the conditional independence property. This round consists of N (log2 KT + (N
1)"KT log2 j j) phases. We repeat the following phases
sequentially for each j: player j spends log2 KT phases to send the message about tj;k .27 After than, each player n 2
j sequentially sends the messages about the signal observations
in tj;k (n), tj;k (n) + 1, ..., tj;k (n) + "KT
1 to player j. Player n spends log2 j j phases to
send a message about a signal observation in each period.28 Since there are log2 KT + (N 1)"KT log2 j j phases for each j, in total, we have N (log2 KT + (N
1)"KT log2 j j) phases
for the kth recovering round. kth Supplemental Round This round is divided into 2N phases and each player i sends mki as in Section 6.2. Note that the conditional independence property is not necessary for 26
Each phase consists of S periods, that is, player n repeats an action (the component of the message) for S times. See Section 8.2 for the formal description of the communication. 1 27 As explained in Section 6.5, player j repeats an action (the component of the message) for T 2 times, 1 that is, each phase consists of T 2 periods. 28 As in footnote 26, each phase consists of S periods, that is, player n repeats an action (the component of the message) for S times.
53
the kth supplemental round. Report Round The structure of the round is the same as in Section 6.2 but since player i needs to send more messages (precisely, instead of xi , (m0j;i )j2fi mki and (mkj (i))j2I , player i sends xi , m0i;i , (m0j;i;l )j2fi (m0j;j+1;l [1](i))l2
(j;j+1);j2I ,
mki;temp , (mki;l )l6=i
1;i;i+1
1;i;i+1g ,
1;i+1g;l2 (j;i) ,
(m0j;j 0 (i))j2I;j 0 2fj
(m0j;j (i))j2I , (m0j;j
1;j;j+1g ,
1;l (i))l2 (j;j 1);j2I ,
and (mkj (i))j2I ), the number of phases is
larger.
7
E¢ ciency Result and Folk Theorem
Since we construct BFRSE to attain the same equilibrium payo¤ set as Yamamoto (2012), a su¢ cient condition for the e¢ ciency result in Yamamoto (2012) is also valid for our equilibrium: Proposition 2 Suppose that the feasible payo¤ set is full dimensional and that there are pro…les a 2 A and a
2 A such that maxai 2Ai
i
ai ; a
i 2 I. If Assumptions 1, 2 and 3 are satis…ed, then
i
<
i
(a ) 2 lim
(a ) <
!1
i
ai ; a
i
for all
E ( ).
Proof. Proposition 3 of Yamamoto (2012) shows that the existence of such a and a (a ) 2 F BFRSE . Hence, Theorem 1 derives
implies that Assumptions 4 is satis…ed and that the result.
In addition, the next proposition assures that the folk theorem holds for the games with prisoners’-dilemma structure. De…nition 3 The stage game is an N -player prisoners’ dilemma if Ai = fCi ; Di g for all i 2 I,
i
(Di ; a i )
all j 6= i and a
j
2A
i j
(Ci ; a i ) for all i 2 I and a and
i
(C1 ; : : : ; CN ) >
i
i
2 A i,
!1
(Cj ; a j )
i
(Dj ; a j ) for
(D1 ; : : : ; DN ) for all i 2 I.
Proposition 3 For the N -player prisoners’dilemma with N 3 are satis…ed, then lim
i
4, if Assumptions 1, 2 and
E ( ) is equal to the feasible and individually rational payo¤ set.
54
Proof. Proposition 4 of Yamamoto (2012) implies that for the N -player prisoners’dilemma, F BFRSE is the feasible and individually rational payo¤ set with full dimension. Hence, Theorem 1 derives the result.
8
Appendix
8.1
Identi…ability
Assumption 2 implies that each player i can identify player j’s action regardless of players (i; j)’s deviation. Speci…cally, we can show the following lemma: Lemma 2 If Assumption 2 is satis…ed, then for some q3 and q2 with 1 > q3 > q2 > 0, there are random events
i (Aj )
2 f0; 1g for all i; j 2 I and Aj
Aj such that, for all a ~ 2 A,
8 < q if a ~ j 2 Aj ; 3 Pr(f i (Aj ) = 1g j a ~) = : q if a ~j 2 = Aj : 2
Proof. The same as Lemma 1 of Yamamoto (2007).
Important implications are that (i) player i can statistically identify player j’s action by the realization of i (Aj ).
i (Aj )
and that (ii) players
(i; j) cannot change the distribution of
Note that these properties hold whether or not the monitoring is conditionally
independent since Lemma 2 considers ex ante distributions (before conditioning on signal observations). In addition, since player i
1 can identify players
(i
1)’s actions, player i
1 can
cancel out the di¤erence in player i’s instantaneous utilities with respect to action pro…les by changing the continuation payo¤: Lemma 3 If Assumption 2 is satis…ed, then for all i, there exists ui : Ai
1
i 1
! [ ui ; ui ]
such that i
(a) + E [ui (ai 1 ; ! i 1 ) j a] = 0 for all a. 55
(17)
Proof. The same as the construction of ui and ui in Yamamoto (2012). Intuitively, ui (ai
1;t ; ! i 1;t )
denote the e¤ect of the marginal movement of the continua-
tion play with respect to player i
8.2
1’s history in period t on player i’s payo¤.
Message Exchange
As seen in Section 6.1.3, when player i sends a message m to player j from a …nite message B space M , we encode the messages in M into sequences of two di¤erent actions faG i ; ai g and B choose a one-to-one map from M to the set of …nite sequences of faG i ; ai g. To increase the
precision of the message, player i repeats each action (the component of the message) for su¢ ciently many times. Since player j infers player i’s action by
G j (fai g),
players
(j; i)
cannot change player j’s inference. Below, we consider player i’s strategy to send m and player j’s inference of m more speci…cally. Message Protocol for t 2 f1; :::; KT g As seen in Section 6.5.1, player i sends the message about t 2 f1; :::; KT g in the kth recovering round. Let us attach a sequence of actions B log2 KT ai (t) = (ai;1 (t); : : : ; ai;log2 KT (t)) 2 faG to each t 2 f1; : : : ; KT g as follows: i ; ai g B ai;1 (t) = aG i if t 2 f1; : : : ; KT =2g and ai;1 (t) = ai otherwise. That is, we attach B ai;1 (t) = aG i if t is in the …rst half of KT and ai;1 (t) = ai otherwise;
G B – for t with ai;1 (t) = aG i , ai;2 (t) = ai if t 2 f1 : : : ; KT =4g and ai;2 (t) = ai otherwise.
That is, among t in the …rst half of KT , ai;2 (t) = aG i if t is in the …rst quarter and ai;2 (t) = aB i otherwise; G – similarly, for t with ai;1 (t) = aB i , ai;2 (t) = ai if t 2 fKT =2 + 1; :::; 3KT =4g and
ai;2 (t) = aB i otherwise; – keep this procedure until we can identify t uniquely from ai (t).
56
1
When player i wants to send the message about t, she takes ai;1 (t) for T 2 periods, then 1
1
ai;2 (t) for T 2 periods and so forth. Hence, to send the message about t, it takes (log2 KT ) T 2 periods. Player j 6= i infers ai;m (t) = aG i if
G j (fai g)
occurs
q2 +q3 T 2
1 2
times or more while player i
sends the component of the message ai;m (t). Otherwise, she infers the action (component of log2 KT 29 . the message) is aB i . Player j infers t combining fai;m (t)gm=1
Lemma 2 implies that player j can infer player i’s message t correctly with probability of order 1
log2 KT exp(T
of players
1 2
) and that the distribution of player j’s inference is independent
(i; j)’s strategy.
Message Protocol for ! i
As seen in Section 6.5.1, player i sends the message about ! i
B log2 j in the recovering round. We can attach a sequence of actions ai (! i ) 2 faG i ; ai g
!i 2
i
j
to each
B log2 KT as we attach ai (t) 2 faG to each t 2 f1; : : : ; KT g.30 i ; ai g
When player i wants to send the message about ! i , she takes ai;1 (! i ) for S periods, then ai;2 (! i ) for S periods and so forth. Hence, to send message about ! i , it takes S log2 j j periods. Player j 6= i infers ai;m (! i ) = aG i if
G j (fai g)
occurs
q2 +q3 S 2
times or more while player i
sends the component of the message ai;m (t). Otherwise, she infers the action (component of log2 j j 31 the message) is aB i . Player j infers ! i combining fai;m (! i )gm=1 .
Lemma 2 implies that player j can infer player i’s message ! i correctly with probability of order 1
log2 j j exp( S) and that the distribution of player j’s inference is independent
of players
(i; j)’s strategy.
29
If log2 KT 2 = N, then jM j = minn2N;n log2 KT . In such a case, there may not exist t 2 f1; :::; KT g jM j corresponding to fai;m (t)gm=1 . If so, player j randomly infers t. 30 We use j j instead of j i j for simple notation. This is without loss since we can attach “an uninformative G G B log2 j i j sequence aG . i ; :::; ai ” after the informative sequence of fai ; ai g log2 j j 31 Again, if there is no ! i corresponding to fai;m (! i )gm=1 , then player j randomly infers ! i .
57
8.3
Endogenously Conditionally Independent Monitoring
Given the message exchange protocol above, players construct the following statistics based on the recovering round, which satisfy the conditional independence property endogenously. Signaling Round In Lemma 1, we construct yj; Suppose player j receives the messages about !
(i;l) ,
(i;l);t
assuming that player j knew !
(i;l) .
in period t of the signaling round as
explained above. For su¢ ciently large S, since the messages transmit correctly with a high probability, we can construct y^j;
(i;l)
with the same properties.
Lemma 4 If Assumption 3 is satis…ed, then there exists S such that, for all j; i; l 2 I with j 6= i 6= l 6= j and aj 2 Aj , player j taking aj can construct a new signal y^j; from player j’s inference of ! 1. y^j;
(i;l)
(i;l)
j 2A j
of Yj;
(i;l)
is the distribution of y^j; player j constructs y^j;
j’s action: the collection (pj;
(i;l)
yj; (i;l) (i;l) (^
-dimensional vector is linearly independent. Here, pj;
(i;l) ,
(i;l)
2 Yj;
explained in Section 8.2 such that
can statistically identify players
aj ; a j ))a
(i;l)
determined by q, how player j infers !
from the inference. In addition, pj;
(i;l)
(i;l)
j
(i;l)
and how
is the distribution
measured in the signaling round. (That is, players i and l do not condition their signal observation in the recovering round.) 2. y^j;
(i;l)
satis…es the conditional independence property for players i and l: pj;
for all a, yj;
yj; (i;l) (i;l) (^ (i;l) ,
j a; ! i ) = pj;
yj; (i;l) (i;l) (^
j a; ! l ) = pj;
(i; l)’s messages correctly with
a high probability. Hence, Assumption 3 is satis…ed with !
Using y^j;
(i;l) .
(i;l) ,
j a)
! i and ! l .
Proof. For su¢ ciently large S, player j can infer players
inference of !
yj; (i;l) (i;l) (^
(i;l)
replaced with player j’s
The rest of the proof is the same as Lemma 1.
player j can constructs random events with the conditional independence
property: 58
Lemma 5 Fix S such that Lemma 4 holds. For some q3 and q2 with 1 > q3 > q2 > 0, for all j; i; l 2 I with j 6= i 6= l 6= j and aj 2 Aj , player j taking aj can construct a random event
j; (i;l) (Ai )
Pr(f
2 f0; 1g for all Ai
j; (i;l) (Ai )
Ai such that, for all a ~ 2 A, ! i 2
= 1g j ! i ; a ~) = Pr(f
i
and ! l 2
l,
8 < q if a ~ i 2 Ai ; 3 ~) = (i;l) (Ai ) = 1g j ! l ; a : q if a ~i 2 = Ai : 2
j;
Here, Pr is the probability measure in the signaling round. (That is, players i and l do not condition their signal observation in the recovering round.) Proof. As we construct
we can construct
j (Ai )
from ! j in Lemma 2 with
8 < q if a ~ i 2 Ai ; 3 Pr(f j (Ai ) = 1g j a ~) = : q if a ~i 2 = Ai ; 2
j; (i;l) (Ai )
that Pr(f
j;
from y^j;
(i;l) .
By Condition 1 of Lemma 4, we can make sure
8 < q if a ~ i 2 Ai ; 3 (A ) = 1g j a ~ ) = i (i;l) : q if a ~i 2 = Ai : 2
From Condition 2 of Lemma 4, the conditional independence property holds for ! i and ! l , as desired. Without loss of generality, we can assume that q3 and q2 in Lemmas 2 and 5 are the same by applying the a¢ ne transformation properly for the probability with which player j constructs the random variables. See Lemma 1 of Yamamoto (2007) for the details. kth Main Round Contrary to the signaling round, player j yields the information about !
j
only for an "-fraction of the kth main round.
With public randomization, on equilibrium path in the recovering round, it is common knowledge that what periods players are communicating about (that is, which is tj;k ). In this case, player j can construct y^j;
(i;l);t
add one more possible realization of y^j;
as follows: compared to Yj;
(i;l);t :
Yj;main (i;l) = Yj;
59
(i;l)
(i;l)
in Lemma 4, we
[ f0g. If t is not included in
tj;k , tj;k + 1, ..., tj;k + "KT y^j;
(i;l);t
1, then y^j;
(i;l);t
= 0. If t is included, then player j constructs
as in the signaling round. Then, for all " > 0, y^j;
(i;l);t
satis…es Conditions 1 and 2
of Lemma 4. Therefore, from y^j;
(i;l);t ,
player j can construct the random events to monitor player i:
Lemma 6 Fix S such that Lemma 4 holds. With public randomization device, for some q3 , q2 and q1 with 1 > q3 > q2 > q1 > 0, for all j; i; l 2 I with j 6= i 6= l 6= j and aj 2 Aj , player j taking aj can construct random events j; (i;l) (faj 1 ; aj+1 g)
Pr(f
=
j; (i;l) (Ai )
j; (i;l) (Ai )
2 f0; 1g such that, for all a ~ 2 A, ! i 2 = 1g j ! i ; a ~) = Pr(f
j;
1
6= aj
1
i
and ! l 2
Ai and
l,
8 < q if a ~ i 2 Ai ; 3 ~) = (i;l) (Ai ) = 1g j ! l ; a : q if a ~i 2 = Ai ; 2
Pr f j; (i;l) (faj 1 ; aj+1 g) = 1g j ! i ; a ~ = Pr f 8 > > q if a ~j 1 = aj 1 and a ~j+1 6= aj+1 ; > < 1 q3 if a ~j > > > : q 2
2 f0; 1g for all Ai
j; (i;l) (faj 1 ; aj+1 g)
= 1g j ! l ; a ~
and a ~j+1 = aj+1 ;
otherwise.
Here, Pr is the probability measure in the main round. Since player j’s signal y^j;
(i:l)
is informative only if t 2 ftj;k ; :::; tj;k + "KT
1g, q3 and q2
becomes smaller than that of Lemmas 2 and 5. However, applying the a¢ ne transformation again, we can assume that q3 and q2 in Lemmas 2, 5 and 6 are the same. Without public randomization device, there is a small probability that player n makes a mistake to infer tj;k : tj;k (n) 6= tj;k . However, the probability of such an event is bounded by 1
log2 KT exp( T 2 ), which is very small for su¢ ciently large T . Hence, the almost conditionally independent property holds: Lemma 7 Without public randomization device, for all j; i; l 2 I with j 6= i 6= l 6= j, the random events created in Lemma 6 is almost conditionally independent with respect to player 60
i’s history and player l’s history: for any history of player i in the kth main round, hi , and that of player l, hl , all of
Pr(f Pr(f
Pr(f
j; (i;l) (Ai )
= 1g j hi ; a ~)
Pr(f
j; (i;l) (Ai )
= 1g j a ~) ;
Pr(f
j; (i;l) (Ai )
= 1g j hl ; a ~)
Pr(f
j; (i;l) (Ai )
= 1g j a ~) ;
j; (i;l) (faj 1 ; aj+1 g)
= 1g j hi ; a ~)
Pr(f
j; (i;l) (faj 1 ; aj+1 g
= 1g j a ~) ;
j; (i;l) (faj 1 ; aj+1 g
= 1g j hl ; a ~)
Pr(f
j; (i;l) (faj 1 ; aj+1 g
= 1g j a ~)
1
are less than log2 KT exp( O(T 2 )).
8.4 8.4.1
Equilibrium Construction Structure of the Block
Given the structure of the block in Section 6.5.1 and the message exchange protocol in Section 8.2, we can fully specify the structure of the block. Signaling Round This round consists of N T periods and players communicate x as in Section 6.2. Recovering Round for the Signaling Round This round consists of N (N
2)T log2 j j
phases, where players communicate about signal observations as explained in Section 6.5.1. As seen in Section 8.2, players repeat each message S times, each phase takes S periods. Hence, this round consists of N (N
2)ST log2 j j periods.
Con…rmation Round As explained in Section 6.5.1, this round is divided into N (2N
3)
phases, where players communicate the inferences in the signaling round. Since each phase consists of T periods, this round takes N (2N
3)T periods.
kth Main Round This round consists of KT periods and players take actions as in Section 6.5.1. 61
kth Recovering Round This round consists of N (log2 KT +(N
1)"KT log2 j j) phases,
where players communicate about signal observations as explained in Section 6.5.1. As seen 1
in Section 8.2, each of N log2 KT phases where players communicate about tj;k takes T 2 periods while each of N (N
1)"KT log2 j j phases where players communicate about signals 1
takes S periods. In total, this round takes N (T 2 log2 KT + (N
1)"KST log2 j j) periods.
kth Supplemental Round As explained in Section 6.5.1, this round is divided into 2N phases, where players communicate the inferences in the kth main round. Since each phase consists of T periods, this round takes 2N T periods. Report Round Since each player i sequentially sends xi , m0i;i , (m0j;i;l )l2 (m0j;j (i); (m0j;j
0 1;l (i))l2 (j;j 1) ; (mj;j+1;l (i))l2 (j;j+1) )j2I ,
mki;temp , (mki;l )l6=i
1;i;i+1
(i;j);j2fi 1;i+1g; ,
and (mkj (i))j2I ,
we need N times 1 |{z} xi
+
one phase is su¢ cient to send xi 2fG;Bg
2(N 2) | {z }
(m0j;i;l )l2
+
1 |{z}
mki;temp
+ |{z} NK
(mkj (i))j2I
m0i;i
N (2N | {z
1;l (i))l2 (j;j
2 |{z}
1 |{z}
one phase is su¢ cient to send m0i;i 2fG;Bg
2 |{z}
two phases are su¢ cient to send a message in fG;B;Eg
(i;j);j2fi 1;i+1g;
(m0j;j (i);(m0j;j
+ |{z} K
+ |{z} 1
3) }
0 1) ;(mj;j+1;l (i))l2 (j;j+1) )j2I
+
two phases are su¢ cient to send mki;temp 2fi 1;i+1;0g
2 |{z}
(N 3) | {z }
(mki;l )l6=i
1;i;i+1
2 |{z}
two phases are su¢ cient to send a message in fG;B;Eg
2 |{z}
two phases are su¢ cient to send mki;l 2fi 1;i+1;0g
two phases are su¢ cient to send mkj (i)2fG;B;Eg
= 2(K + KN + 2N 2
6)
phases, each of which consists of T periods. In total, this round takes 2N (K + KN + 2N 2 periods.
62
6) T
The Total Length Tb
In total, the total length of the block is
Tb = N T + N (N
2)ST log2 j j + N (2N 1
+K(KT + N (T 2 log2 KT + (N +2N K + KN + 2N 2
3)T
1)"SKT log2 j j) + 2N T ) (18)
6 T:
It will be important that the length of the main rounds is K 2 T . 8.4.2
Variables
So far, we have introduced variables S, ui , K, ", q3 , q2 , q1 , T and . We pin down these variables in the sequel. First, …x S so that Lemma 4 holds and ui so that Lemma 3 holds. Second, let us …x K and ". To de…ne them, for a moment, consider a Tb -period …nitely repeated game where the monitoring is perfect. Since the length of the …nitely repeated game and the length of the block in the in…nitely repeated game are the same, we use the …nitely repeated game and the block game interchangeably. Player i’s strategy in the block TS b 1 Tb t game is sTi b : Hit ! 4(Ai ) with Hit = (Ai i ) . Let Si be the set of strategies in the t=0
block game. The strategy pro…le is sTb = (sT1 b ; :::; sTNb ).
The structure of the block game is as explained in 8.4.1. Let wiP (sTb ) be the summation of player i’s instantaneous utilities from the main rounds when players take Tb -period block game strategy pro…le sTb : wiP (sTb )
1 1
Tb
E
"
X
t2main round
t 1
i (at )
#
j sTb :
In other words, wiP (sTb ) is the average payo¤ in the block game with perfect monitoring where payo¤s in the periods other than the main rounds are replaced with zero. Consider the following strategy as in Section 3.3.4 of Yamamoto (2012), where he also considers perfect monitoring. His strategy is valid in our structure of the block except that we have extra rounds to recover the conditional independence property in the signaling and
63
main rounds. Since the monitoring is perfect, the review does not play a role. In addition, we replace payo¤s in these extra rounds with zero. Hence, we leave the strategies in these rounds arbitrary and make the continuation play independent of the histories in these rounds. For concreteness, we explain the strategy in Section 3.3.4 of Yamamoto (2012):32 let SiTb
be the set of all strategies sTi b 2 SiTb such that player i plays a constant action in each review phase and such that for each k 2 f1; :::; Kg, player i chooses an action from the set Aki in the kth main round. Tb The following two block strategies are important: a good strategy sG i 2 Si and a bad
Tb xi strategy sB i 2 Si . Let us explain player i’s strategy si with xi 2 fG; Bg. In the signaling
round, each player i tells the truth about xi , that is, takes axi i constantly. Player i constructs an inference of xj , xj (i) as follows: xj (i) = G if the initial action of the signaling round is
aG j and xj (i) = B otherwise. Since the monitoring is perfect, the review is not necessary and regardless of player i’s deviation, players for all n 2
i infer x
i and xi (n) = xi (n0 ) for all n; n0 2
i
correctly and agree on xi : x i (n) = x
i
i. The recovering round for the signaling
round and the con…rmation round are irrelevant for the continuation play. x(i);k
In the kth main round, if no player’s deviation has been con…rmed, player i takes ai
.
If player j’s deviation has been con…rmed, then if xj 1 (i) = B, then player i takes aji (Ak ) and if xj 1 (i) = G, then player i takes aji (Ak ). Since the monitoring is perfect, the review is not necessary. Player i con…rms player j’s deviation if and only if “there has not been any player whose deviation is con…rmed and there is a unique player i who does not take x(i);k
aj
for at least one period in the kth main round.” Hence, the kth recovering round and
the supplemental round are irrelevant for the continuation play. Further, since the history is common knowledge, the report round is not necessary either. Since players
i infer x
i
correctly and agree on xi , player i’s deviation is con…rmed if
and only if player i does not take axi~;k with x~ = x(n) for all n 2 value from the main rounds is less than wi if xi 32
1
i. Therefore, player i’s
= B and more than wi if xi
1
= G for
Our inference of the states and con…rmation of a player’s deviation are slightly di¤erent from those in Yamamoto (2012) to recover conditional independence with imperfect monitoring. However, with perfect monitoring, it does not change player i’s payo¤ regardless of player i’s deviation.
64
su¢ ciently large K from (9) and (11). In addition, given S, for su¢ ciently small ", large K and large T , the length of the main rounds is much larger than that of the other rounds. Hence, Lemma 2 of Yamamoto (2012) holds: Lemma 8 If Assumption 4 is satis…ed, then there exist " > 0 and K such that for all " and K
K, there exists T such that for all T
2 ( ; 1), i 2 I, x
i
s~Ti b 2 SiTb ,
2 X
i
with xi x
wiP (sTi b ; s
i
i
1
= B, x~
T , there exists i
2 X
i
with x~i x ~
) < wi < wi < wiP (~ sTi b ; s
i
i
"
< 1 such that for all 1
= G, sTi b 2 SiTb and
):
Proof. As Yamamoto (2012), it su¢ ces to show that the fraction of the main rounds is su¢ ciently large: for any , there exist " > 0 and K such that for all " there exists T such that for all T Tb K 2T
=
T , T b =K 2 T
N (1 + (N K2 +1 + (N +
" and K
K,
1 + . From (18), we have
2)S log2 j j + (2N 1)"S log2 j j +
2 2 (1 + N ) + 2 (2N 2 K K
N K
3))
log2 KT 1
T2
+2
6):
Since Tb = 1; "!0;K!1 T !1 K 2 T lim
lim
we are done. Note that S has been …xed. Fix K and " so that Lemma 8 holds. Given S, K and ", we can take q3 , q2 and q1 so that Lemmas 5 and 6 are satis…ed. Fix T so that Lemma 8 holds. T
T and
will be
determined later. 8.4.3
Finitely Repeated Game (Block Game) with a “Reward Function”
As Fudenberg and Levine (1994), Hörner and Olszewski (2006) and Yamamoto (2012), instead of the in…nitely repeated game, we consider Tb -period …nitely repeated game (block 65
game) with private monitoring. As in perfect monitoring, player i’s generic strategy is deTS b 1 t noted by siTb : Hit ! 4(Ai ) with Hit = (Ai i ) . Player i’s payo¤ is modi…ed by the t=0
“reward function” Ui : HiTb1 ! R, that is, player i’s payo¤ from a strategy pro…le sTb 2 S Tb is given by wiA (sTb ; Ui )
1 1
Tb
E
"
Tb X
t 1
i (at )
+
Tb
U (hTi b 1 )
t=1
Intuitively, the reward function maps player i
js
Tb
#
:
1’s history into the real number that cor-
responds to the movement of player i’s continuation payo¤ in the in…nitely repeated game after player i
1 observes hTi b 1 .
Let sTi b j hti denote player i’s continuation strategy after history hti 2 Hit induced by
sTi b 2 SiTb . Also, let BRA (sTbi j ht i ; Ui ) be the set of player i’s best replies in the …nitely repeated game after history, given that the payo¤ is augmented by Ui , that the opponents play sTbi j ht i in the …nitely repeated game and that their past history was ht i . As in Section 3.3.7 of Yamamoto (2012), the following lemma is su¢ cient to support v in the equilibrium: Lemma 9 (Lemmas 4 and 5 of Yamamoto (2012)) Suppose Assumptions 1, 2, 3 and 4 are satis…ed. Then, there is T such that for all T > T , there is
2 (0; 1) such that for all
2 ( ; 1) and for all i 2 I, 1. there is UiB : HiTb1 ! R such that for all l xi
1
0, htl 2 H tl , hTi b 1 2 HiTb1 and x 2 X with
= B,
(a) player i’s strategy sxi i is optimal and the equilibrium is belief-free at the beginning of each phase l: sxi i j htil 2 BRA (s
xi i
1
j htl i ; UiB );
(b) player i’s payo¤ is equal to wi : wiA (sx ; UiB ) = wi ; (c) the reward is feasible: 0
UiB (hTi b 1 )
2. there is UiG : HiTb1 ! R such that for all l xi
1
= G, 66
wi w i ; 1
0, htl 2 H tl , hTi b 1 2 HiTb1 and x 2 X with
(a) player i’s strategy sxi i is optimal and the equilibrium is belief-free at the beginning of each phase l: sxi i j htil 2 BRA (s
xi i
1
j htl i ; UiG );
(b) player i’s payo¤ is equal to wi : wiA (sx ; UiG ) = wi ; wi w i 1
(c) the reward is feasible:
UiG (hTi b 1 )
0.
To see why this is su¢ cient,33 consider Conditions 1-(a) and 2-(a). This means both sG i and sB i are best response to s
x
i
i
regardless of x
i
and the belief-free property is satis…ed at
the beginning of each phase. Hence, it is a belief-free review-strategy equilibrium that each B player i takes either sG i or si in each block if player i
1’s state transition between blocks
implements the reward function through the movement of the continuation play. Conditions 1-(b) and 2-(b) say that player i’s value is wi if player i wi if player i
1’s state is G, regardless of x
i’s value by unilaterally switching between xi
(i 1) . 1
That is, player i
can implement
1 (hTi b 1 )
generous state xi
1
1 with current state xi
1
= B, then player i
1 is in the worst state for player
1 can only increase player i’s continuation payo¤ by switching to the = G in the next block: 0
UiB (hTi b 1 ). In addition,
wi w i 1
increase of player i’s (non-normalized) continuation payo¤ that player i switching to xi
1
1
by determining the probability to transit to the next state after
observing hTi b 1 property. If xi i. Hence, player i
1 can control player
= B and G.
Finally, Conditions 1-(c) and 2-(c) guarantee that player i x Ui i
1’s state is B and
wi w i . 1
= G with probability one. Hence, UiB (hTi b 1 )
is the maximum 1 can create by
In total, 1-(c) implies
that it is feasible to implement UiB (hTi b 1 ). Symmetrically, 2-(c) implies that it is feasible to implement UiG (hTi b 1 ).34 x
The rest of the paper is devoted to the construction of Ui i 1 (hTi b 1 ) satisfying the conditions in Lemma 9. 33
See Section 3.3.8 of Yamamoto (2012) for the formal proof. If xi 1 = G, then player i 1 is in the best state for player i. Hence, player i 1 can only decrease player i’s continuation payo¤ by switching to the harsh state xi 1 = B in the next block: UiG (hTi b 1 ) 0. wi w i In addition, 1 is the maximum decrease of player i’s (non-normalized) continuation payo¤ that player wi w i i 1 can create by switching to xi 1 = B with probability one. Hence, UiG (hTi b 1 ). 1 34
67
8.5
Proof of Lemma 9
8.5.1
Histories
To de…ne the reward function, the following notation is useful: for each k 2 f1; : : : ; Kg, let [k]
hi denote player i’s private history up to the end of the kth supplemental round. Also, let [k;r]
hi
[k;m]
be player i’s history up to the end of the kth recovering round, and hi
be player i’s
[0]
history up to the end of the kth main round. Similarly, let hi be player i’s history up to [ 1;r]
be player i’s history up to the end
[ 1]
be player i’s history up to the end
the end of the con…rmation round. In addition, let hi
of the recovering round for the signaling round, and hi [k;r]
[k]
of the signaling round. Let Hi , Hi [k;r]
[k]
hi , hi
[k;m]
, hi
[0]
[ 1;r]
, hi , hi
[ 1]
and hi
Let Mj0 3 (m0n;n (j); (m0n;n (mkn;l (j))l2
(n 1;n;n+1) )n2I
[k;m]
, Hi
[0]
[ 1;r]
, Hi , Hi
[ 1]
and Hi
represent the set of all
, respectively.
0 1;l (j))l2 (n;n 1) ; (mn;n+1;l (j))l2 (n;n+1) )n2I
and Mjk 3 (mkn;temp (j);
be the set of player j’s inferences of players’messages in the con-
…rmation and kth supplemental rounds. (If player j is the sender of a message, then Mj0 is the true message: for example, m0j;j (j) = m0j;j . Also, since m0j;j = xj by de…nition, x
i
(the messages in the signaling round) is also included in Mj0 . ) Since each player j sends these inferences in the report round, given hTi b 1 2 HiTb1 , let I represent player i
i
1’s inference on the messages from players
I k i be the projection of I
i
onto (Mj0
Mj1
MjK )j6=i
i in the report round. Let
Mjk )j6=i for k 2 f0; : : : ; Kg. As explained
Mj1
in Section 6.2.4, we can assume that player i
2 (Mj0
1’s inference in the report round I
i
is correct
with probability one regardless of player i’s strategies. 8.5.2
Pivotal [0]
[0] i
Suppose the true state pro…le is x. Let H i (x) be the set of h
2H
[0] i
such that neither (a),
(b) nor (c) in Section 6.3.1 is satis…ed: (a) in the signaling round, there exist j 2
i and j 0 2 (j
j 0 infers m0j;j 0 ;i 6= xj ;
68
1; j + 1) \ i such that player
(b) in the con…rmation round, there exist j 2 I, n 2 that m0j;j 0 ;i (n) 6= m0j;j 0 ;i or there exist j 0 ; n 2 (c) there exist j 2 I and n 2
i and j 0 2 (j
1; j + 1) \
i such
i such that m0j 0 ;j 0 (n) 6= m0j 0 ;j 0 ;
i such that xj (n) depends on player i’s messages in the
con…rmation round. [0] i
As explained in Section 6.3.1, if h
[0]
2 H i (x), then regardless of player i’s strategy in the
recovering round for the signaling round and in the con…rmation round, players on the same x~ with x~
i
= x i . This is comparable to
that Yamamoto (2012) includes the case where x~ the distribution of x~
i
i
[0] H i (x)
i coordinate
in Yamamoto (2012) except
6= x i . Since player i cannot manipulate
without being pivotal and x~
i
6= x
i
happens only with a negligible
probability, this di¤erence does not a¤ect the rest of the proof. [k]
[k] i
In addition, let H i (x) be the set of h
[k]
2 H i (x) such that none of the following
conditions is satis…ed: [0] i
1. h
[0]
(history up to the end of the con…rmation round) is not included in H i (x);
2. there exists player n 2
~ i who infers some player j’s deviation with j 2 I in the kth
supplemental round with k~
k
1;
3. (a) or (b) in Section 6.3.2 is satis…ed: (a) in the main round, mki j2
(i
1;temp
= i + 1, mki+1;temp = i + 2 or mkj;i = B for some
1; i; i + 1);
(b) in the supplemental, there exists n 2
i such that mkj (n) 6= mkj for some j 2
4. both of player i’s monitors infer player i’s deviation: mki
1
i;
= mki+1 = i.
Then, regardless of player i’s strategy in the kth recovering and supplemental rounds, no player’s deviation is con…rmed by any player in [k]
comparable to H i (x) in Yamamoto (2012).
69
i in the kth supplemental round. This is
[k]
[k] i
On the other hand, let H i (x; i) be the set of h [k]
[k]
2 H i (x) such that none of 1, 2
and 3 for H i (x) is satis…ed and that both of player i’s monitors infer player i’s deviation: mki
1
= mki+1 = i (4 is satis…ed). Then, regardless of player i’s strategy in the kth recovering
and supplemental rounds, player i’s deviation is con…rmed by all the players
i in the kth
[k]
supplemental round. This is comparable to H i (x; i) in Yamamoto (2012). S S [0] [0] [k] [k] [k] De…ne H i (x i ) xi H i (x) and H i (x i ) xi H i (x) [ H i (x; i) . Compared to
H
[0] i
and H
[k] i
in Yamamoto (2012), we do not take the union over x i . This di¤erence is
irrelevant since, as seen above, if players
i cannot coordinate on the true x i , then player
1 makes player i indi¤erent between any action pro…le sequence and the history is not
i
[0]
included in H i (x). 8.5.3
Reward Function x
The reward function Ui i 1 (hTi b 1 ) is decomposable into real-valued functions
1;r
;
0
;:::;
K+1
such that 1 4
x
Ui i 1 (hTi b 1 ) =
where T
1;r
2
Tb
T
+
PK
1;r
1;r
Tk k
k=1
[ 1;r]
(hi
T0 0
)+
[k] (hi 1 ; I k i )
+
[0]
(hi 1 )
Tb K+1
(hTi b 1 )
3
5;
is the last period of the recovering round for the signaling round, T0 is the last
period of the con…rmation round, and Tk is the last period of the kth supplemental round. Intuitively, player i receives a transfer 0
k
after the con…rmation round,
and
K+1
1;r
after the recovering round for the signaling round,
after the kth supplemental round for each k 2 f1; : : : ; Kg,
after the report round.
We will determine
1;r
,
0
,
k
and
K+1
by backward induction. Note that, by the
formulation of the transfer scheme, the transfers for the past rounds are irrelevant to player i’s incentive compatibility. For example, since the transfers
1;r
;
0
;:::;
k 1
are sunk at
the beginning of the kth main round, they are irrelevant to player i’s incentive compatibility in the continuation game from the kth main round.
70
8.5.4
With Public Randomization
First, let us consider the case with public randomization (but without cheap talk), by backward induction. K+1
Construction of
As Appendix C.1 in Yamamoto (2012), we de…ne Treport K+1
(hTi b 1 )
=
X ui (ai
1;t ; ! i 1;t ) ; Treport t+1
t=1
where Treport is the length of the report round and ai
1;t ; ! i 1;t
is player i
1’s action and
signal in the tth period from the beginning of the report round. By Lemma 3, player i is indi¤erent between any action pro…le sequence in the report round. Since player i cannot manipulate the communication among the other players and player i’s own messages are used only for the rewards on players k
Construction of de…ne
1
;:::;
K
i, player i’s incentive is satis…ed.
with k 2 f1; : : : ; Kg As in Appendix C.2 in Yamamoto (2012), we
by backward induction. To de…ne
k
, assume that (
k+1
K+1
;:::;
[k] i
already been determined so that player i’s continuation payo¤ after history h k+1
augmented by
;:::;
K+1
[k]
) have [k]
2 H i,
, is equal to Vi (h i ) and 1-(a) and 2-(a) in Lemma 9 hold for
the initial period of the k + 1th main round and the subsequent phases (induction hypothesis for k + 1. Note that by de…nition of
K+1
, this hypothesis is true for K + 1). Here, the value
[k]
Vi (h i ) denotes as follows: [k] i
1. for each k 2 f0; 1; : : : ; Kg and h (a) if xi
[k]
2 H i (x i ),
[k]
1
= B, then Vi (h i ) is the maximum of player i’s actual continuation payo¤ [k] i
(i.e., the discounted sum of stage game payo¤s) after history h
over all her
continuation strategies, subject to the constraints that the monitoring is perfect and that payo¤s in the communication stages are replaced with zero;
71
[k]
(b) if xi
= G, then Vi (h i ) is the minimum of player i’s actual continuation payo¤
1
[k] i
after history h
over all her continuation strategies compatible to SiTb , subject to
the same constraints above; [k] i
2. for each k 2 f0; 1; : : : ; Kg and h
[k]
2 = H i (x i ),
[k]
(a) if xi
= B, then Vi (h i ) is player i’s actual continuation payo¤ when she earns
1
maxa2A
i (a)
in periods of the main rounds and zero in other periods; [k]
(b) if xi
= G, then Vi (h i ) is player i’s actual continuation payo¤ when she earns
1
mina2A
i (a)
in periods of the main rounds and zero in other periods.
Notice that the transfers ( 1 ; : : : ;
K
) are speci…ed in such a way that player i’s continua-
[k]
tion payo¤ Vi (h i ) after the kth supplemental round is high for xi [k] i
and the same for all histories h
1
= B, low for xi
1
= G,
[k]
2 = H i (x i ). As in Yamamoto (2012), this “constant con-
tinuation payo¤”property is used to show that player i has a truthtelling incentive in the kth [k] i
recovering and supplemental round, even when h
[k]
2 = H i (x i ) so that player i’s messages
in that round can a¤ect the opponents’continuation play. In what follows, it is shown that there is history
[k 1] h i
2
[k 1] H i ,
k
such that player i’s continuation payo¤ after
augmented by ( k ; : : : ;
K+1
[k 1] i )
), is equal to Vi (h
and 1-(a) and
2-(a) in Lemma 9 hold for the initial period of the kth main round and subsequent phases (induction hypothesis for k). We consider
k
which has the following form: Tk;non
k
[k] (hi 1 ; I k i )
=
~k
[k;m] (hi 1 ;
^
k 1 (i 1;i;i+1) ; I i )
+
Xmain ui (ai
1;t ; ! i 1;t ) ; Tk;non main +1 t
(19)
t=1
where ^
(i 1;i;i+1)
is the realized messages from players
the kth recovering round; Tk;non rounds; ai
1;t ; ! i 1;t
is player i
main
(i
1; i; i + 1) to player i 1 during
is the total length of the kth recovering and supplement
1’s action and signal in the tth period from the beginning
of the kth recovering round.
72
The di¤erence from [k;m] 1 ,
hi
^
(i 1;i;i+1)
k
in Yamamoto (2012) is that [k;m] 1
and I k i 1 , rather than hi
k
here is a real-valued function of
and I k i 1 .
As Lemma 8 in Yamamoto (2012), since the second term of (19) cancels out the di¤erence in the instantaneous utilities by Lemma 3 and if player i’s messages in the kth recovering and supplemental rounds have an impact on the future equilibrium path, player i’s value from the subsequent rounds is constant (inductive hypothesis for k +1), player i is indi¤erent between any action sequence in the kth recovering and supplemental rounds: Lemma 10 Player i is indi¤erent over all the actions in every period of the kth recovering and supplemental rounds regardless of the past history, and hence 1-(a) and 2-(a) in Lemma 9 hold from the beginning of the kth recovering round. Proof. The same as Lemma 8 in Yamamoto (2012). Next, we specify ~k . In Yamamoto (2012), player i
1 constructs the following variables
[k;m] 1 :
from I k i 1 and hi [k;m] 1
1. 1ai : Hi
[k;m] 1 )
! f0; 1g, which is the indicator function such that 1ai (hi
only if player i is constructed by
1 thinks that player i took ai in the kth main round. This indicator i 1 (fai g); jAi j
2. for each I k i 1 , for some speci…c ordering of player i’s action a1i ; :::; ai and each l 2 f1; :::; jAi jg, 1[I k in
1 i
;l]
[k;m] 1
: Hi
[k;m] ;l] (hi 1 ) = 1 if and only jA j fali (I k i 1 ); :::; ai i (I k i 1 )g in the
that 1[I k
= 1 if and
1
i
based on I k i 1 ,
! f0; 1g, which is the indicator function such
if player i
1 thinks that player i took ai included
kth main round. This indicator is constructed by
jAi j k 1 l k 1 i 1 (fai (I i ); :::; ai (I i )g);
Since the conditional independence property is important for 1ai and 1[I k i 1 (fai g)
and
jAi j k 1 l k 1 i 1 (fai (I i ); :::; ai (I i )g)
jAi j k 1 l k 1 i 1; (i;i+1) (fai (I i ); :::; ai (I i )g) [k;m] 1 ,
from hi
^
(i 1;i;i+1)
with
i 1; (i;i+1) (fai g)
by Lemma 6. Then, 1ai and 1[I k
1 i
;l] ,
we replace
and 1 i
;l]
are now functions
and I k i 1 to f0; 1g. We have the following counterpart of Lemmas 9
and 25 of Yamamoto (2012): 73
Lemma 11 There exists T such that for all T > T , there exists [k;m] 2 ( ; 1), there exists ~k (hi 1 ; ^
1. if xi
1
= B, then for
1
= G, then for
[k]
such that, for all (hi 1 ; I k i 1 ),
de…ned in Equation (34) in Yamamoto (2012),
Tk;non
2. if xi
k 1 (i 1;i;i+1) ; I i )
2 (0; 1) such that for all
main u
k
2KT <
[k]
(hi 1 ; I k i 1 ) <
wi wi (K + 3)(1
)
;
de…ned in page 65 in Yamamoto (2012),
wi wi (K + 3)(1
)
<
k
[k]
(hi 1 ; I k i 1 ) < Tk;non
main u
+ Aki KT :
Also, using this reward function, 1-(a) and 2-(a) in Lemma 9 hold from the beginning [k 1] 1
of the kth main round and player i’s continuation payo¤ after history hi
is equal to
[k 1] i ).
Vi (h
Proof. The same as Lemmas 9 and 25 in Yamamoto (2012) with i 1; (i;i+1) (Ai )
for all Ai
i 1 (Ai )
replaced with
Ai .
Con…rmation Round As Appendix C.3 in Yamamoto (2012), we de…ne
0
[0]
(hi 1 ) =
Tcon Xrm
ui (ai
1;t ; ! i 1;t ) ; Tcon rm t+1
t=1
where Tcon
rm
is the length of the con…rmation round and ai
1;t ; ! i 1;t
is player i
1’s action
and signal in the tth period from the beginning of the con…rmation round. As C.3 in Yamamoto (2012), since
0
cancels out the di¤erence in the instantaneous
utilities by Lemma 3 and if player i’s messages in the con…rmation round have an impact on the future equilibrium path, player i’s value from the subsequent rounds is constant by construction of
1
; :::;
K+1
, player i is indi¤erent between any action sequence in the
con…rmation round.
74
Signaling Round and Recovering Round for the Signaling Round As in the kth 1;r
main, recovering and supplemental rounds, we consider T 1;r
[ 1;r] (hi )
[ = ~ 1 (hi
1;r]
)+
1;r X ui (ai
T
which has the following form:
1;t ; ! i 1;t ) : 1;r +1 t
t=1
Remember that T
1;r
which means T
is the total length of the signaling round and recovering round for the
1;r
is the last period of the recovering round for the signaling round,
signaling round. Here, ai
1;t ; ! i 1;t
is player i
1’s action and signal in the tth period from
the beginning of the signaling round. As in C.4 of Yamamoto (2012), the second term cancels out the di¤erence in player i’s instantaneous utilities. Next, we specify ~ 1 . In Yamamoto (2012), player i
1 constructs the following variables
[ 1] 1:
from hi
1. for each j 2
[ 1] 1
(i
1; i), x^j : Hi
! fG; E; Bg, which is player i
player j’s state. This indicator is constructed by from player i
G i 1 (faj g).
1’s interpretation of
Note that this is di¤erent
1’s inference of xj at the end of the con…rmation round xj (i
1). See
[ 1] 1 );
page 29 of Yamamoto (2012) for the de…nition of x^j (hi [ 1] 1
2. x^i : Hi
! fG; Bg, which is player i G i 1 (fai g).
indicator is constructed by
1’s interpretation of player i’s state. This Again, this is di¤erent from player i
inference of xi at the end of the con…rmation round xi (i
1’s
1). See C.4 in Yamamoto
[ 1] 1 );
(2012) for the de…nition of x^i (hi 3. x^i
1
[ 1] 1
: Hi
! fG; Bg, which is the straightforward interpretation of player i [ 1] 1)
state: x^i 1 (hi
= xi
1
2 fG; Bg; [ 1] 1
4. for each x~ 2 X, 1x~ : Hi [ 1] 1)
1x~ (hi
! f0; 1g, which is the indicator function such that [ 1] 1)
= 1 if and only if x^j (hi
= x~j for all j 2 I; [ 1] 1)
Since the conditional independence property is important for x^j (hi replace
G i 1 (faj g)
and
1’s own
G i 1 (fai g)
with
G i 1; (i;i+1) (faj g)
75
and
[ 1] 1 ),
and x^i (hi
G i 1; (i;i+1) (fai g)
we
by Lemma
[ 1;r]
5. Then, 1x~ is now functions from hi
to f0; 1g. We have the following counterpart of
Lemmas 10 and 26 of Yamamoto (2012): Lemma 12 There exists T such that for all T > T , there exists [ 2 ( ; 1), there exists ~ 1 (hi
1. if xi
1
1
[ 1;r] 35
) such that, for all hi
,
= B, then 3 Tb
2. if xi
1;r]
2 (0; 1) such that for all
T
1;r ui
1
<
[ 1;r]
(hi
)<
wi wi (K + 3)(1
)
;
= G, then wi wi (K + 3)(1
)
1
<
[ 1;r]
(hi
)
1;r ui
(jAi j + 1) Tb :
Also, using this reward function, 1-(a) and 2-(a) in Lemma 9 hold from the beginning of the signaling round. Given these de…nitions of
1;r
,
0
,
k
and
K+1
, the rest of the proof is the same as in
Yamamoto (2012). 8.5.5
Without Public Randomization
By Lemma 7, the almost conditionally independent property holds. Hence, the result follows from Lemma 9 and Proposition 5 of Yamamoto (2012).
References Abreu, D., P. Milgrom and D. Pearce (1991) ‘Information and timing in repeated partnerships.’Econometrica 56(2), 1713–1733 35
The de…nition of
is the same as in Lemma 11.
76
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