Interested Experts: Do They Know More? Gorkem Celiky March 2003

Abstract A decisionmaker, whose optimal decision is state dependent, has to take an action in an uncertain environment. An expert, whose utility is monotonic in the action, has access to a veri…able evidence production technology. If the expert is fully informed about the state, then the intuitive criterion selects the equilibrium where the decisionmaker fully infers the state by observing the level of the created evidence. If the expert is uninformed at the beginning of her investigation, then the information is produced as a by-product of the evidence production technology. In that case, there exists no equilibrium where the generated information is fully revealed to the decisionmaker.

1

Introduction

In common law countries, courts are restricted to consider the evidence brought by the interested parties when making their decisions. Although the con‡ict of interest between the court and the source of the information is obvious, revelation of the entire information can be sustained in the following way. If the interested expert (the plainti¤ in civil cases or the prosecutor in criminal ones) fails to provide any evidence to the court, then the defendant is considered to be not guilty, which is the worst possible inference from the expert’s perspective. To avoid such a verdict, the expert reveals the evidence she has, even if the evidence falls short of supporting her most preferred inference.1 The discussion above is a justi…cation for the common law tradition as long as the expert has free excess to the evidence. However, the extent of economic resources directed to law practice is su¢ cient to argue against free availability of the evidence. An investigation to provide evidence is a costly and lengthy process that embeds many uncertainties concerning the level of evidence that will be obtained in the end. In this This is a revised version of a part of my dissertation submitted to Northwestern University. I am grateful to my advisors Je¤ Ely and Michael Whinston. y

Department of Economics, The University of British Columbia, #997-1873 East Mall, Vancouver, BC, V6T 1Z1 Canada. e-mail: [email protected] 1 Throughout the paper, we will use feminine pronouns for the “expert,”and masculine pronouns for the “decisionmaker.”

1

paper, we accommodate this observation by de…ning the expert as someone who has access to the technology to search for the evidence, rather than having excess to the free evidence. Then, we investigate whether full transmission of the expertise information is still possible. Once the evidence search stage is integrated to the “in court”behavior of the expert, we see that the extent of information transmission depends on a variable that is often overlooked by the existing literature. It is the quality of the expert’s initial “pre-search” information that determines the quality of “post-search”information transmission. If we assume that the expert is initially fully informed about the state, then the information transmission is perfect whenever there exists an a¤ordable evidence search technology. In this case, the above justi…cation of the common law tradition is intact. On the other hand, if the evidence search process is assumed to be the only source of the expert’s superior information, then we show that there exists no equilibrium where the expert’s information is fully transmitted to the court. The justi…cation for the full revelation of the “freely available” evidence is due to “persuasion game”analyses of Grossman (1981) and Milgrom (1981). A piece of evidence that is reported by the expert can be consistent with several states of nature. In the unique equilibrium, among these possible states, the decisionmaker makes the inference that is the least favorable for the expert. This “skeptical”behavior of the decisionmaker forces the expert to reveal the entire evidence she has. But again, for skepticism to be the decisionmaker’s equilibrium behavior, the expert should be known to have free access to the evidence that is perfectly correlated with the state. Farrell (1986) and Shin (1994a, 1994b, and 1996) consider a “modi…ed” version of the persuasion game where they allow for “not having free excess to the evidence” also be a possible state of nature. From the decisionmaker’s perspective, there is no way to distinguish such a state from another one where the expert is hiding the evidence that is unfavorable to her cause. Therefore, the full revelation result of the persuasion game is abandoned as a result of this modi…cation to the game. Dewatripont and Tirole (1999), on the other hand, consider lacking the evidence as a result of a conscious decision by the expert. In their model, the expert has to exert an e¤ort to obtain the evidence. If exerting the e¤ort is “su¢ cient” to obtain the evidence and moreover if it is an equilibrium behavior for the expert, then claiming not to have any evidence would not be credible, and therefore the analysis boils down to the original persuasion game analysis with a full revelation result. Dewatripont and Tirole also consider an alternative setup where exerting the e¤ort and still not getting the evidence are compatible. In this setup, the expert with the evidence that is unfavorable for her cause can pool with the no-evidence expert as in the modi…ed persuasion game environment and therefore the full revelation result is lost. Froeb and Kobayashi (1996) and Yilankaya (1998) share a similar approach with Dewatripont and Tirole. However, in their models, the expert can make many simultaneous draws from a costly random process that generates evidence. In other words, the expert determines the level of her e¤ort, rather than deciding whether to exert a speci…ed amount of e¤ort. As a result of this, when the court is making a decision, the relevant

2

pieces of information are the “level” of the generated evidence, as well as the intensity of the investigation that generated it. The …rst of these variables is directly observable for the court, and the second is deduced from the equilibrium strategy of the expert.2 The papers cited above have been successful to capture the cost that is associated with evidence gathering. However, since the decision of the expert is always modeled to be one shot, sequentiality of the investigation process is overlooked. Unlike the previous studies, in this paper we model evidence gathering as a lengthy “search process,”where we allow the expert make a continuation decision at any point in time. One complication with the suggested approach is about the decisionmaker’s inference of search intensity. In a search model, the total amount of the expert’s search e¤ort is partly determined by her observations along the search process. Therefore, the decisionmaker cannot deduce the investigation intensity from the expert’s equilibrium behavior. For the expert’s information to be fully transmitted to the decisionmaker, either the expert’s information should be independent of the search intensity, or the level of the …nal evidence should completely reveal the search intensity. When the expert is fully informed on the state of nature, her information is indeed independent of the e¤ort she exerts. In the …rst part of our analysis, we show that there exist separating equilibria such that the fully informed expert’s information is completely transmitted to the decisionmaker as long as the evidence search technology is a¤ordable. Moreover, the “intuitive criterion,” which is a re…nement of the perfect Bayesian equilibrium concept, selects such a separating equilibrium and rules out the equilibria where information transmission is incomplete. When the expert is not fully informed and therefore she is learning through the evidence search process, her information is a function of both the evidence level and the search intensity. In the second part of our analysis, we show that there exists no equilibrium where the search intensity is always accurately inferred by the decisionmaker. Therefore, we conclude that information transmission is incomplete with a learning expert. The alternative legal tradition to common law is the Roman law, where the evidence is investigated by the court, rather than by the advocates. Although the problems due to information transmission are avoided, the costs of evidence gathering are totally incurred by the court under the Roman law. In our analysis, we do not explicitly model the Roman law practice. Nevertheless, our results give the opportunity for a partial comparison of the two legal traditions. Whenever the expert is fully informed about the state of nature, the decisionmaker would prefer the common law to the Roman law, since the state is costlessly revealed from the expert’s behavior under the former legal tradition. However, in case that the expert’s information is imperfect, the …nal level of evidence does not reveal the expert’s entire information, and therefore our analysis is silent about the comparison. Our analysis is not exclusive for judicial processes. The model we suggest describes many situations where the decisionmaker’s source of information has a preset preference 2

However, both of these papers model the court as a “naive” player who does not take into account that the expert is strategic. Therefore, information transmission is exogenously constrained.

3

ordering over the decisionmaker’s choice set. A potential buyer taking the advise of the seller before buying a commodity and a legislature consulting to an administrative agency before setting the agency’s budget are two cases where our analysis can be useful.

2

The Model

We consider a setup with two players: The decisionmaker (D) and the expert (E). The utility level for D depends on the state as well as his own action. The realized state of nature is either 1 or 0 with probabilities p and 1 p respectively. D cannot observe the realized state, but knows the value for p. The action space for D is the closed interval [0; 1] : Let a and s denote the action and the state respectively. The function H (a; s) = (a s)2 gives the utility for D as a function of these parameters. Note that H (a; s) is strictly decreasing in the di¤erence between the state and the action. The optimal action for D is equal to the probability that D assigns to state 1. In the absence of any additional information, the optimal action is equal to p; the prior probability that the state is 1.3 The utility level for E is given by the function G (a) = a. Note that this function is independent of the state and strictly increasing in D’s action. Since E’s utility is increasing in the action, and D chooses the action that is identical to the posterior probability of being in state 1, E has an incentive to present this probability as high as possible. In the arbitration interpretation of our game, D is the court that is trying make the right decision and E is the advocate that is trying to secure a decision to serve her own interest. Until now, we did not specify our assumptions on what E can verify about the realized state. If she had access to veri…able evidence on the state of the nature, we would be in a persuasion game environment and D would infer the exact realization of the state by simply being skeptical. On the other hand, if there is a possibility that E did not observe any evidence at all, we would be in the modi…ed persuasion game environment. In that case, upon receiving no evidence from E, D would not know whether E did not observe the evidence or she is hiding the unfavorable evidence. We will make neither of the assumptions above. Instead, we will assume veri…able evidence is “searched”by E. The level of the evidence will be related to how much e¤ort E will allocate to the search technology.

2.1

Evidence Search Technology

E has access to a Poisson process that generates veri…able evidence. The parameter of the process, s , is state dependent and 1 > 0 > 0. Note that the level of evidence produced in t units of time will be a random natural number, which has a Poisson distribution with parameter s t, given the realized state of nature is s. Therefore, the 3

a = p uniquely maximizes D’s expected utility:

4

p (a

2

1)

(1

p) a2 .

probability of receiving exactly x units of evidence in t units of time is ( s t)x e s t =x! for x 2 N; where N is the set of natural numbers. The evidence can be interpreted as the output of a production function. To fully characterize the production technology, we should also specify the input. For our search technology, this input is the time that E allocates to search. For each unit of time she is spending to observe the process, E incurs a disutility at the amount c. Therefore, her net utility from action a, after observing the search process for t units of time is a tc. E’s search decision is sequential. At each point in time, E observes t and x, the amount of time passed till that point and the level of cumulated evidence respectively, and decides whether to “continue” searching for more evidence (and therefore continue incurring the disutility), or to “stop”the process.4 When E stops searching, D observes the …nal level of the evidence, but not the amount of time spent to produce that evidence. Since the …nal evidence level is the only observable variable for D, his updated belief on the state and his action choice are both functions of this variable. These functions will be denoted by ( ) and ( ) respectively. : N ! [0; 1] : N ! [0; 1] The pair f ( ) ; ( )g fully describes a strategy for D. The technology speci…es what E can report to D. What is still missing from our setup is what E knows. How to model the extent of E’s information certainly depends on the interpretation of the term “expert.” One possibility here is considering the expert as someone “who knows all the relevant information.” Alternatively, the expert can be thought as someone “who shares the same prior information with the other player, but who updates her information along the search process.” We will devote the following two sections to the analysis of our model under these two assumptions.

3

Fully Informed Expert

In this section, we will assume that E knows whether the realized state of nature is 0 or 1. However, she cannot directly convey her information to D. The only way of transmitting her knowledge to D is gathering evidence that is correlated with the state. Since E knows the state, she also knows the value for s , the parameter for the evidence search process. At each point in time, E observes x - the cumulated level of the evidence, t - the amount of time already spent in the search process, and s - the realized state of nature. After observing these, E decides whether to continue searching or not. Note that t cannot be reported to D, and since E is perfectly informed, it will not have any informative value to her. The variable t measures the sunk cost of the search E has already performed, but it is not strategically relevant. We will assume strategies not to be contingent on the non-strategic variables. Therefore, in this section, a state dependant strategy for E 4 E observes the exact timing of the previous occurances as well. Due to our Poisson assumption, this infomation of her is not strategically relevant.

5

is a function that maps each evidence level to a continuation decision. Let "s denote the strategy for E, given the state is s. "s : N ! f\continue"; \stop"g Note that the only relevant information that can be deduced from the function "s is the minimum evidence level that makes E stop searching. Therefore, in this section, xs = min fx : x 2 "s 1 (\stop")g is a reduced form for E’s strategy. Since the evidence is acquired through a Poisson process with parameter s , the searching time until the evidence level xs is attained is a random variable with gamma distribution with parameters xs and 1s . The expected value for this random variable is xss . Let Vs; (xs ) be the expected payo¤ for E associated with the speci…ed state and strategy choices for the players. Then, xs Vs; (xs ) = (xs ) c: s

De…nition 1 A Nash Equilibrium5 of the fully informed expert game is a collection n o fxs gs2f0;1g ; ( ) ; ( )

such that, i) xs 2 arg maxx fVs; (x)g for s 2 f0; 1g. ii) (x) is de…ned by the Bayes rule given fxs gs2f0;1g on the equilibrium path and arbitrary o¤ the equilibrium path. iii) (x) = (x) on the equilibrium path and arbitrary o¤ the equilibrium path. In this game, there are two types of Nash equilibria to consider. In a “pooling” equilibrium, E sends the same level of evidence in both states, so that D does not infer the state. In a “separating”equilibrium, the …nal evidence levels are di¤erent in di¤erent states and therefore reveal all the relevant information of E.

3.1

Pooling Equilibria

A pooling equilibrium is characterized by a uniform …nal evidence level. Both types of experts continue searching until that level of evidence is acquired, and they stop immediately after. We will let x denote this …nal evidence level. Note that x1 = x0 = x for a pooling equilibrium. Since D observes x as the evidence level in either state of nature, his equilibrium strategy is the same as in the absence of any evidence gathering process: (x ) = (x ) = p With the following proposition we will identify the evidence levels that can be sustained as equilibrium …nal evidence levels for pooling equilibria. 5

Throughout the paper, we restrict attention to the pure strategy Nash equilibria.

6

Proposition 1 Let x be a natural number. The equalities x1 = x0 = x can be induced by a pure strategy Nash equilibrium of the fully informed expert game if and only if 0 x p. c Proof. Necessity: Suppose x > c0 p. We need to show that there exists no equilibrium with the property x1 = x0 = x . To construct a contradiction, suppose n o fxs gs2f0;1g ; ( ) ; ( ) is such an equilibrium. The optimality of D’s decision requires (x ) = p. Therefore the expected equilibrium payo¤ for E in state 0 is V0; (x ) =

(x )

c

x

=p

c

x

0

< 0. 0

However, if E chooses not searching for the evidence at all in state 0, such a strategy would leave a higher expected payo¤ for her since V0; (0) =

(0)

c

0

0.

0

This is a contradiction to x being an optimal strategy for E in state 0. 0 Su¢ ciency: This part is proved by construction. Suppose x p. We will show c that x1 = x0 = x constitutes an equilibrium together with (x) and (x) such that (x) =

0 if x < x . p otherwise

(x) =

We know that the only requirement of a pooling equilibrium on D’s behavior is (x ) = 0 (x ) = p, which is satis…ed by our construction. And given x p, xs 2 arg maxx f (x) c x c s g for s = 1; 0. 0 0 is always an element of the set x 2 N : x p , whatever values the parameters c take. Therefore, there exists at least one pooling equilibrium for the fully informed expert game.

3.2

Separating Equilibria

In a separating equilibrium, x1 and x0 assume di¤erent values. As a result, the …nal level of the evidence is su¢ cient for D to infer the realized state. The equilibrium strategy for D requires (x1 ) = 1 and (x0 ) = 0. Provided that D is following this strategy, we can already identify the equilibrium strategy of E in state 0 as x0 = 0. That is, E will not invest any time in the search process in state 0. (Otherwise, her payo¤ would be strictly negative.) To determine the feasible values for x1 , we will follow an analysis that is similar to the one we conducted for the pooling equilibria. Proposition 2 Let x^ be a natural number other than 0. The equalities x1 = x^ and x0 = 0 can be induced by a pure strategy Nash equilibrium of the fully informed expert 1 game if and only if c0 x^ . c 7

n fxs gs=1;0 ;

Proof. Necessity: Suppose

( );

o ( ) is an equilibrium such that

1 . First, recall that the de…nition of x0 = 0 and x1 = x^ 6= 0. We need to show c0 x^ c the equilibrium requires (0) = 0 and (^ x) = 1. If x^ < c0 , then x0 = 0 is dominated by x0 = x^, since

V0; (^ x) =

(^ x)

c

x^

=1

c

0

Similarly, if x^ >

0

c

x^

> V0; (0) =

(0)

c

0

0

= 0.

0

, x1 = x^ is dominated by x1 = 0, since

V1; (^ x) =

(^ x)

c

x^ 1

=1

c

x^ 1

< V1; (0) =

(0)

c

0

= 0.

1

1 Su¢ ciency: This part is proved by construction. Suppose c0 x^ . We will c show that x1 = x^ and x0 = 0 constitute an equilibrium together with (x) and (x) where 0 if x < x^ (x) = (x) = . 1 otherwise

Note that (x0 ) = (x0 ) = 0, and (x1 ) = (x1 ) = 1 are both satis…ed by our construction. The condition c0 x^ guarantees 0 2 arg maxx f (x) c x0 g, whereas the 1 condition x^ guarantees x^ 2 arg maxx f (x) c x1 g. c If there is no natural number within the closed interval c0 ; c1 , then the previous proposition implies that there is no separating equilibrium. However, given any 0 and 1 ; provided that the search cost c is small enough, there exists such a natural number and therefore at least one separating equilibrium. We will postpone to state our assumption on the magnitude of c to the next subsection, where we will discuss the equilibrium re…nements. The two classes of equilibria of the informed expert game point to two dramatically di¤erent phenomena. In a pooling equilibrium, the created evidence is completely useless for D, since he cannot make any relevant inference based on the evidence. On the other hand, in a separating equilibrium, the …nal evidence level allows D to completely identify the state of nature and to make the ex post optimal decision. In the next subsection, we will use some well known re…nement concepts to contract the set of equilibria.

3.3

Re…nements

For a sequential game like the one we consider in this paper, the most common re…nement of the Nash equilibrium is the perfect Bayesian equilibrium concept. Unfortunately, in our case, this re…nement fails to contract the set of equilibrium outcomes. De…nition 2 A Perfect Bayesian Equilibrium (PBE) of the fully informed expert o n game is a collection fxs gs2f0;1g ; ( ) ; ( ) such that, i) xs 2 arg maxx fVs; (x)g for s 2 f0; 1g. 8

ii) (x) is de…ned by the Bayes rule given fxs g on the equilibrium path, and arbitrary o¤ the equilibrium path. iii0 ) (x) = (x) for all x. The only restriction that PBE imposes on a Nash equilibrium is about the out of equilibrium path actions. Any action should be rational given the corresponding belief. But this is not a material restriction for the game we consider, since there exists a belief on [0; 1] that rationalizes any action on the same interval. Although PBE does not succeed in eliminating some of the Nash equilibria, a further re…nement of this solution concept is more successful. n o De…nition 3 Let fxs gs=1;0 ; ( ) ; ( ) be a PBE. It satis…es the intuitive crite-

rion6 if

(x) =

(x) = 1 for all x 2 N such that (x0 )

c

x0

>1

c

x

0

and

0

(x1 )

c

x1

<1

1

c

x

.

1

The criterion above dictates the following on the belief of D: Suppose D can commit to taking the highest possible action (a = 1) after being reported a certain evidence level, x. Given E’s original state dependent expected equilibrium payo¤ and this promise, suppose acquisition of the evidence level x is not a pro…table deviation in state 0, but it is a pro…table in state 1 for E. Then, upon receiving the evidence level x; D should infer that the state is 1 and take the highest action. Proposition 3 Suppose c is small enough, such that there exists an integer in the interval (1

p)

0

c

; (1

p)

1

c

.

There exists a separating PBE such that x0 = 0 and x1 = x^ = min x > x2N

0

c

,

(1)

which satis…es the intuitive criterion. Moreover, any PBE satisfying the intuitive criterion should satisfy (1) as well. The proof for this proposition is in the appendix. Proposition 3 states that as long as the integer constraint is not material, the “reasonable”equilibrium of the fully informed expert game induces full information transmission from E to D. 6

For a textbook de…nition of intuitive criterion for two state games, see Mas-Colell, Whinston and Green (1995)

9

3.4

Persuasion or Signalling?

The equilibrium analysis above is closely related to the signalling model of Spence.7 The points that our approach di¤ers from the signalling paradigm are the integer constraint and the random characteristic of the signalling function. Neither of these has a qualitative e¤ect on our analysis. This close relationship between the signalling and the persuasion games was known to the economic researchers from the very beginning of the veri…able information literature.8 One may consider Grossman’s and Milgrom’s original persuasion game as a special case of the signalling game, where the cost of the signal is zero for the high-state expert, and in…nity for the low-state one. Since no signal can be sent by the low-state expert, provision of a signal would reveal all the relevant information for the decisionmaker. This is due to the consistency of the decisionmaker’s belief, without any further need to re…ne. Therefore the unique equilibrium is the fully revealing one. In our model, we have assumed a more general (yet, still state dependent) cost function for the signal. We have shown that there exist fully revealing equilibria with this general cost function. Our analysis so far implies that full revelation of the state through an interested party is a stronger phenomenon than it is presented in the framework of the persuasion game. However, there is one other “less realistic”aspect of the original persuasion game that we sustained in the previous analysis. We assumed that E has full information about the state. Since E is the “expert” that the decisionmaker is depending on, it is reasonable to model E as a player who has superior information than D, or who has an access to an information creation process that D does not have an access to. But it is not obvious why we should expect that the expert would know everything. Our next step will be relaxing this assumption. We will let E start the game with a non-degenerate prior about the state, as D also does. In that case, the Poisson process is not only a technology to produce veri…able evidence for D to observe, but also a technology to update E’s own belief on the state.

4

Learning Expert

In this section, we will assume that E does not know the realized state. At the beginning of the game what she knows is p, the prior probability of being in state 1, which is also known to D. As time passes and evidence is observed, she updates this prior via Bayes rule. Given the level of acquired evidence is x at time t; q (x; t) denotes the updated probability of being in state 1 for E. q (x; t) =

pe

1t

pe 1 t x1 x p)e 1 + (1

0t x 0

1

= 1+

(1 p) p

x 0 1

e[

1

0 ]t

7

For Spence Educational Choice model, see the source cited in the previous footnote.

8

See Leland (1981), which was written as a comment to Grossman (1981).

10

q (x; t) is increasing in x and decreasing in t. Let h be the amount of time that would pass until the acquisition of one more piece of evidence. This is a random variable for E with support (0; 1) and the following density function that depends on the acquired evidence level and the accumulated time: f(x;t) (h) = q (x; t)

1e

1h

+ [1

q (x; t)]

0e

0h

.

The higher q (x; t), the probability assigned to state 1, the higher the chances of obtaining an additional piece of evidence earlier. Since q (x; t) is decreasing in t, f(x;t0 ) …rst order 0 stochastically dominates f(x;t) as long as t < t. That is, Z k Z k f(x;t0 ) (h) dh f(x;t) (h) dh for all k 2 (0; 1) . 0

0

Also note that 1 e 1 h …rst order stochastically dominates f(x;t) for all (x; t) pairs. As before, D’s belief and action choice are two functions, (x) and (x), that map level of reported evidence to the interval [0; 1]. At each point in time, E observes the acquired evidence level and the time that has already passed, but not the state of nature. Her strategy is a mapping from her observation to her continuation decision. ":N

R+ ! f\continue"; \stop"g

where R+ is the set of non-negative real numbers. The …nal evidence level and total search time are random variables that depend on (but not fully identi…ed by) the state and E’s strategy. Given the strategies of both players, each decision node (x; t) is mapped into a continuation payo¤ for E.9 V ;" (x; t) denotes this continuation payo¤. Now we are ready to introduce the equilibrium concept that we will employ for the analysis of this game. De…nition 4 A Nash Equilibrium of the learning expert game is a collection f" ;

( );

( )g

such that, i) " 2 arg max" fV ;" (x; t)g for all (x; t). ii) (x) is de…ned by the Bayes Rule on the equilibrium path given " and arbitrary o¤ the equilibrium path. iii) (x) = (x) on the equilibrium path and arbitrary o¤ the equilibrium path. The de…nition above imposes sequentially rational choices for E by asking her equilibrium strategy to be optimal not only on the equilibrium path, but on any decision node for her.10 Since ( ) takes values within the interval [0; 1], the function V ;" (x; t) 9

We will consider the continuation payo¤ net of the sunk cost of the search that has already been conducted. 10 Alternatively, we could de…ne an equilibrium concept where E’s equilibrium strategy maximizes E’s payo¤ at the starting point of the evidence search, where (x; t) = (0; 0). This would complicate the analysis but not change the central result of this section.

11

is bounded by 0 and 1 from below and above respectively: Stopping the search process immediately gives a payo¤ of at least 0 to E, and there is no evidence level that would provide her with a terminal payo¤ larger than 1. The boundedness of the expected payo¤ will be crucial for our central result regarding the learning expert game. At decision node (x; t), E’s strategy determines a stopping time, t" , such that t" (x; t) = inf k

t : (x; k) 2 "

1

(\stop")

If E does not receive one more piece of evidence until time t" , her strategy instructs her to stop and report x as the …nal level of evidence. This would give her the continuation payo¤ (x) c (t" t) at node (x; t), provided that (x) is the action choice for D. On the other hand, if E acquires an additional unit of evidence by time t" , then her expected continuation payo¤ will be determined by the function V ;" (x + 1; ). Therefore V ;" (x + 1; ) and (x) are su¢ cient to identify V ;" (x; t) as follows:

V

;"

(x; t) =

Z

t" t

[V

;"

(x + 1; t + h)

ch] f(x;t) (h) dh +

Z

1

[ (x)

c (t"

t)] f(x;t) (h) dh

t" t

0

(2) where the arguments of the function t" (x; t) are dropped for notational simplicity. Since the de…nition of the equilibrium requires " to maximize V ;" (x; t), t" should maximize the right hand side of the above equation for = and " = " .

V

;"

(x; t) = max t t

(Z

t t

[V

;"

(x + 1; t + h)

ch] f(x;t) (h) dh +

Z

1

[ (x)

c (t

t)] f(x;t) (h) dh

t t

0

(3) Since t = t, stopping immediately at time t is an option for E, her expected equilibrium continuation payo¤ is at least as large as her stopping payo¤: V

;"

(x; t)

(x) for all (x; t) :

With the following proposition we will state our …rst result on the equilibrium behavior of E. Proposition 4 Suppose f" ; ( ) ; ( )g is a pure strategy Nash equilibrium of the 1 learning expert game. There exists x such that t" (x; t) = t for all t. (Or equivac lently, " (x; t) = \stop" almost everywhere with respect to t.) The proof is in the appendix. The proposition above states the existence of an evidence level x such that E ends the search process immediately after obtaining that level regardless of the accumulated amount of time. Any evidence level higher than x will not be observed on the equilibrium path. Therefore, we can restrict attention to evidence levels lower than x. Another corollary is that V ;" (x; t) = (x) for all t. Now that we have the value for the function V ;" (x; ) ; we can identify V ;" (x; ) iteratively for x x, using equation (2). 12

)

We know that q (x; t) is declining in t. Unless she receives a new piece of evidence, E becomes more pessimist about the state as time passes. Our next proposition will prove that E’s continuation payo¤ is declining in time as well. Proposition 5 Suppose f" ; ( ) ; ( )g is a pure strategy Nash equilibrium of the learning expert game. V ;" (x; t) is decreasing in t. The proof is in the appendix. For the fully informed expert game, the evidence level that would make E end the search process was su¢ cient to identify her equilibrium path behavior. There exists a reduced form “equilibrium” strategy for the learning expert as well. The proposition above implies the existence of a stopping time, t~ (x), for each level of the evidence, x. As long as t < t~ (x), the value for V ;" (x; t) is greater than (x), and E continues to search for the evidence. And for t t~ (x), V ;" (x; t) is equal to (x), and E stops searching and reports x to D. With the function t~ ( ), we can also accommodate the evidence levels where E stops searching regardless of the accumulated time, or where E does not stop searching unless she acquires another piece of evidence. For the …rst case t~ (x) takes the value 0, and 1 for the second one. With Proposition 4 we already established the existence of the evidence level x where t~ (x) = 0. The function t~ (x) de…ned for x x entirely explains the part of E’s strategy that is relevant for the equilibrium outcome. Now we are ready to identify the …rst class of Nash equilibria for the learning expert game. n Proposition 6 Let x be an integer no larger than c0 p. ( Or equivalently, let fxs gs2f0;1g ; be a Nash equilibrium such that x1 = x0 = x .) Then x can be supported as the unique …nal evidence level for a learning expert game Nash equilibrium, where t~ (x) =

1; for all x < x 0; otherwise

and

(x) =

(x) =

0; for all x < x p; otherwise

. Proof. Given t~ ( ), the only …nal level of evidence that would be observed in equilibrium is x . Note that (x ) = p is consistent with this fact, and (x ) = p is rational given this belief. In order to conclude that the above functions constitute an equilibrium, we should also ensure the optimality of E’s choices. The strategy requires E to continue searching as long as the evidence level is below x . Given ( ), such a strategy would provide E with the following expected continuation payo¤s: ( p c q(x;t) + 1 q(x;t) (x x) ; for x < x 1 0 V ; (x; t) = : p; for x x Recall that q (x; t) is the posterior probability that E assigns to state 1 at her decision node (x; t). Therefore the expression q(x;t) + 1 q(x;t) is the expected amount of time 1 0 13

( );

o ()

until an additional piece of evidence is acquired by E. For evidence levels smaller than x , this expectation is relevant for determination of the search cost. For x x , the strategy is optimal since it provides E with a continuation payo¤ equal to maxy f (y)g. To argue that it is optimal for x < x as well, we need to show E prefers to continue searching and receive expected payo¤ V ; (x; t), rather than stopping and receiving payo¤ 0. Note that

q(x;t) 1

+

1 q(x;t) 0

is smaller than

1 0

. This

implies V ; (x; t) p c (x 0 x) . The proof follows from the observation that the right hand side of this last inequality is larger than 0 due to the supposition of the proposition. For any integer that can be supported as the …nal evidence level of a pooling equilibrium of the informed expert game, the proposition above states the existence of an equilibrium of the learning expert game which induces that integer as its own …nal evidence level. As long as this integer is larger than 0, E obtains some additional information through the search process regarding the realized state. However, this information is not transmitted to D. Therefore, he cannot update his prior belief even after his interaction with E, and implements the ex ante optimal action, p. As in the informed expert game, x = 0 will constitute an equilibrium together with the uniform belief and action functions (x) = (x) = p regardless of the parametrization of the model. If an equilibrium such as the one above de…nes one end of the spectrum of all the possible equilibria, on the other end would be an equilibrium where the entire additional information of E is transmitted to D. The following subsection is devoted to the analysis of such equilibria.

4.1

Fully Revealing Equilibria

In the informed expert game, separating equilibrium was de…ned to be an equilibrium where D infers the state of nature from the reported level of the evidence. In the learning expert game, E does not know the realized state herself. Therefore inferring the state from the interaction with E is not a possibility. The analogue of a separating equilibrium in this case would be an equilibrium where D infers all the relevant information E has. By the relevant information, we mean q (x; t), the most recently updated posterior belief for E. Since x, the …nal evidence level is directly observed by D, and q (x; t) is a continuous and strictly declining function of t for any x, inferring q (x; t) is equivalent to inferring t, the total amount of time spent to observe the search process. De…nition 5 Suppose x is the …nal evidence level reported to D while " is the equilibrium strategy for E. If x and " are su¢ cient to identify the accumulated time until the end of the search process, then x is called a revealing evidence level for " . With the following proposition, we will establish that x is a revealing evidence level for " if and only if " assigns x a stopping time t~ (x) that is larger than the stopping time for x 1.

14

Proposition 7 Let " be an equilibrium strategy for a learning expert. x > 0 is a revealing evidence level for " if and only if t~ (x 1) t~ (x). In that case x reveals t = t~ (x). For any " , the evidence level 0 is a revealing evidence level and it reveals t = t~ (0). Proof. The second part of the proposition is trivial. t~ (0) is the time that would make E stop searching as long as she did not receive any evidence by then. For the …rst part, we should prove su¢ ciency and necessity separately. Suppose t~ (x 1) t~ (x), and x is the level of the evidence reported to D. Consider D’s inference of t, the amount of time spent searching for the evidence. t cannot be lower than t~ (x), since requires E to “continue”in that case. And if t is greater than t~ (x), then E must have ended the search process as soon as the last piece of evidence is obtained. In that case E must have continued searching with evidence level x 1 while t was larger than t~ (x). However, this is a contradiction to t~ (x 1) t~ (x). Therefore t = t~ (x) with probability 1. Suppose x is a revealing evidence level, but t~ (x 1) > t~ (x). If x is the level of …nal evidence reported to D, t could assume any value between t~ (x) and t~ (x 1), which is a contradiction to x being a revealing evidence level. Now we are ready to introduce a property for the equilibrium behavior of E that depends on the extent of information transmission from her to D. For the equilibrium strategy " , there exist evidence levels that will not be observed on the equilibrium path. The property we will de…ne shortly does not impose any requirement for those evidence levels. However for the strategy " to reveal the relevant information to D, all the equilibrium path evidence levels need to be revealing. De…nition 6 Suppose f" ; ( ) ; ( )g is a Nash equilibrium of the learning expert game. It is called fully revealing if all the …nal evidence levels that can be observed on the equilibrium are revealing for " . Identically, a fully revealing equilibrium is de…ned as t~ (x 1) t~ (x) for all x which can be observed as …nal evidence levels on the equilibrium path. Proposition 6 implies the existence of one family of fully revealing equilibria where ~ t (0) = 0 regardless of the parametrization of the model. Such an equilibrium is fully revealing since the only evidence level observed on the equilibrium path is 0, which is always a revealing level. In this equilibrium, E will not search for any evidence and therefore she will not get any superior information about the state. Although there is no additional information created by E’s behavior, since there is no ex post informational asymmetry between D and E, such an equilibrium would satisfy the criteria for full revelation. We will call this type of equilibria as “unproductive equilibria.” Proposition 8 There is no fully revealing equilibrium of the learning expert game other than the unproductive ones. Proof. Suppose there exists an equilibrium which is fully revealing but not unproductive. Since the equilibrium is not unproductive, it must be that t~ (0) > 0. And 15

since it is fully revealing, t~ (x) t~ (0) for the levels of x that can be observed on the equilibrium path. Therefore, the search process will continue at least until time t~ (0), whatever level of the evidence is acquired until that time. Given t~ (0) is strictly positive, any natural number could be the level of the acquired evidence level by the time t~ (0). This implies that any evidence level can be observed on the equilibrium path. Therefore, t~ (x) t~ (0) > 0 for all x. But this contradicts to the existence of x such that t~ (x) = 0. The proposition above is the non-existence result of this section. Some properties of this result are worth mentioning here. First, there might be fully revealing levels of evidence in an equilibrium. The proposition does not rule out this possibility. In fact, we already showed that 0 is always a revealing level of evidence for any equilibria. The proposition states that this cannot be true for all levels that can be observed on the equilibrium. Second, the proof for the proposition makes use of the Poisson evidence technology, which dictates that the level of evidence that can be observed till time t > 0 is unbounded. If sequential Bernoulli trials technology was used rather than the Poisson technology, the cumulated time would be an integer representing the total number of trials so far. And the level of the evidence to be observed till time t would be bounded by t itself.11

5

Conclusion

The analysis we conduct yielded two di¤erent conclusions on two di¤erent cases of interested expertise. First, we showed that there exists an equilibrium where a fully informed expert’s information is fully transmitted to the decisionmaker, given there exists a hard evidence gathering technology which is “a¤ordable” enough for the expert. We also proved that such an equilibrium is selected by the intuitive criterion. Second, if the expert is not fully informed, then we showed that there is no equilibrium where the expert’s information is completely revealed through the evidence she reports. The success of the common law system depends on the extent that the interested party’s information is transmitted to the court. Since transmission is complete with a fully informed expert, the court achieves its …rst best solution for just relying on the expert’s report of the evidence. Even if the court had a technology to conduct the evidence search itself, it would not choose to do so, regardless of the associated cost. The court’s conducting its own evidence search can be regarded as a Roman law practice as opposed to common law. Our …rst result shows the court would prefer common law to Roman law, if there exist a fully informed interested expert and an a¤ordable evidence gathering technology for her. However, this conclusion does not hold in the case where the expert is learning. Information transmission from a learning expert is 11

To illustrate this last point, consider the following equilibrium of the Bernoulli model as an example. t(0) = 1; t(1) = 2; and t(y) = y. If such an equilibrium exists, it is a fully revealing equilibrium where the information transmission is perfect. If a failure is received in the …rst trial, E stops and reports 0 evidence. If a success is received, she tries one more time and then stops whatever the result of the second draw. The proposition above does not rule out this equilibrium.

16

incomplete. The court would certainly get better information if it conducted the search itself. However, comparison of the two systems would depend on the magnitude of the cost for self search relative to the value of making a more accurate decision. Throughout the analysis, we assumed that there is no commitment technology for the decisionmaker. Given the …nal evidence level, he updates his prior distribution on the state and chooses the action that maximizes his expected payo¤, where expectation is taken with respect to his updated belief. Alternatively, we could have modeled the decisionmaker as a mechanism designer, where he commits to an action function even if the suggested action by the function does not always maximize his expected payo¤. It is trivial that a decisionmaker cannot be worse o¤ under a commitment technology, since committing to the “no commitment” choices is still available for him. For the informed expert game, a separating equilibrium nulli…es the e¤ect of uncertainty for D. Therefore, introduction of the commitment technology would not change D’s payo¤ at all. However, for the learning expert game, commitment has the potential to sustain a strictly higher payo¤ for D. Even with the commitment technology, non-existence of fully revealing equilibrium survives for the learning expert game. For our equilibrium analysis, we only made use of the fact that E’s strategy is a best response to D’s. Any action choice function for D would not make a qualitative di¤erence in our analysis.

6 6.1

Appendix Proof of Proposition 3

Under the supposition of the proposition, a separating equilibrium always exists. To see this, note that x^ < c1 . Otherwise, there would be no natural number in the interval (1 p) c0 ; (1 p) c1 . Since c0 < x^ < c1 , it follows from Proposition 2 that there exits a separating equilibrium where x1 = x^. Together with x0 = x^ and (x) = (x) = 0 if x < x^ : 1 otherwise The constructed equilibrium satis…es the intuitive criterion. The rest of the proof will follow from the two claims below: Claim: There exists no pooling equilibrium that satis…es the intuitive criterion. Let x be the …nal evidence level for a pooling equilibrium. Consider x0 such that, (x0 )

c

(x1 )

c

x0

= p

c

x

0

x1

<1

c

0

= p

1

c

x

>1 1

c

x0 0 0

x

(4) (5)

1

Existence of x0 is guaranteed by the supposition we make in the proposition. If this equilibrium satis…es the intuitive criterion, it must be that (x0 ) = (x0 ) = 1. Then x1 = x cannot be an equilibrium strategy for E in state 1, since x1 = x0 dominates. 17

n o Claim: Suppose fxs gs=1;0 ; ( ) ; ( ) is a separating equilibrium that satis…es the intuitive criterion. Then x1 = x^. From Proposition 2, we know that x0 = 0 and x1 x^ : Consider a separating equilibrium where x1 > x^. Note the following inequalities: (x0 )

c

(x1 )

c

x0

= 0>1

c

x^

0

(6)

0

x1

= 1

c

x1

1

<1

c

x^

1

(7)

1

If this equilibrium satis…es the intuitive criterion, it must be that (^ x) = (^ x) = 1. But then, x1 > x^ cannot be induced by an equilibrium strategy of E, since x1 = x^ dominates.

6.2

Proof of Proposition 4

Let x be an evidence level such that there exists a point in time, t, where t" (x; t) > t. Rewrite equation (2) for the equilibrium strategies ; " , and the observation pair (x; t) : Z t" t Z 1 V ;" (x; t) = [V ;" (x + 1; t + h) ch] f(x;t) (h) dh+ [ (x) c (t" t)] f(x;t) (h) dh t"

0

De…ne V ;" (x) = supt V ;" (x; t). Also recall that V implies the following inequality. Z t" t Z V ;" (x; t) V ;" (x + 1) ch f(x;t) (h) dh+

(8) (x). Equation (8)

(x; t)

;"

1

t"

0

t

[V

;"

(x; t)

c (t"

t)] f(x;t) (h) dh

t

(9) It follows from inequality (9) that V ;" (x + 1) V ;" (x; t), which in turn implies that inequality (9) would still hold if f(x;t) (h) is replaced by a …rst order stochastically dominating density function, such as 1 e 1 h . Z t" t Z 1 1h V ;" (x; t) V ;" (x + 1) ch 1 e dh+ [V ;" (x; t) c (t" t)] 1 e 1 h dh t"

0

t

(10)

After rearranging, this last inequality can be written as follows: " R t" t Z t" t V ;" (x + 1) 1 e 1h R t" t 0 R1 V ;" (x; t) 1 e dh 1h ch e dh c (t" 1 0 0 t" t 1

e

1 (t"

t)

V

;"

And when we integrate 1

e

1 (t"

t)

V

;"

(x; t)

R t" 0

(x; t)

R t" 0

t

ch 1 e

1h

1

e

1 e t ch 1 e

1 (t"

1h

t)

dh

V e

1h

dh t) 1 e

1h

#

dh

(11)

;" (x + 1) t) 1 (t" c (t"

(12)

t)

dh by parts: 1 (t"

t)

V

;"

(x + 1)

1

e

1 (t"

t)

c 1

(13) 18

Since t < t" by our supposition, we can cancel out the term 1 V

;"

(x; t)

V

;"

c

(x + 1)

e

1 (t"

t)

: (14)

:

1

And since this is true for all t < t" , it must be that V

(x)

;"

V

;"

c

(x + 1)

(15)

:

1

Inequality (15) holds for all x such that there exists t < t" (x; t). To prove the proposition, it is su¢ cient to establish the existence of one x that is smaller than c1 and that does not satisfy inequality (15). To build a contradiction, suppose that no such x exists. Then inequality (15) is satis…ed for x 2 f0; 1; :::; mg, where m is the largest integer smaller than c1 . When we sum these inequalities up, we get (m + 1)

c

V

;"

(m + 1)

V

;"

(16)

(0) :

1

Recall that V ( ) is the supremum of the expected equilibrium payo¤ for E, therefore it is bounded by 0 and 1, the minimum and the maximum possible equilibrium payo¤s. This implies that the right hand side of the above inequality is smaller than 1 and therefore 1 . But this is a contradiction to m being the largest integer smaller than c1 . (m + 1) c

6.3

Proof of Proposition 5

We already know that V ;" (x; t) = (x), for all t. Therefore, the claim of the proposition trivially holds for V ;" (x; ). A backward induction argument will be su¢ cient for the proof. Suppose V ;" (x + 1; t) is weakly decreasing in t. We want to show V ;" (x; t) is weakly decreasing in t as well. Take t0 t, and let t = t" (x; t). That is, if E acquired x pieces of evidence until time t, her equilibrium strategy " instructs her to search for evidence for t t more units of time before ending the process, in case that no additional piece of evidence is acquired by then. This behavior induces the following expected continuation payo¤ at node (x; t): "Z Z t t

V

;"

(x; t) =

[V

;"

(x + 1; t + h)

ch] f(x;t) (h) dh +

1

[ (x)

c (t

t)] f(x;t) (h) dh

t t

0

For " to be an equilibrium strategy, t should be the stopping time that maximizes the right hand side of the above equation. Note that if t is strictly larger than t, this maximization implies V ;" (x + 1; t) > (x). Now consider the behavior of E when she is at the decision node (x; t0 ). Suppose that instead of following strategy " , E mimics her behavior at node (x; t). That is, she searches t t more units of time in case that she does not acquire an additional piece of evidence. If she obtains an additional piece of evidence by time t (t t0 ), her

19

#

continuation strategy follows " . Otherwise she ends the process at time t (t t0 ). Such a strategy would give the following expected continuation payo¤ for E at (x; t0 ): Z t t Z 1 0 [V ;" (x + 1; t + h) ch] f(x;t0 ) (h) dh + [ (x) c [t t]] f(x;t0 ) (h) dh t t

0

For the above strategy not to be a pro…table deviation to " at (x; t0 ), V ;" (x; t0 ) should be weakly larger than the last expression above. And since V ;" (x + 1; ) is declining in its second argument and t0 + h < t + h for all h, we can write the following inequality. Z t t Z 1 0 V ;" (x; t ) [V ;" (x + 1; t + h) ch] f(x;t0 ) (h) dh+ [ (x) c [t t]] f(x;t0 ) (h) dh t t

0

Now recall that V ;" (x + 1; t) > (x). By using this and the fact that f(x;t0 ) …rst order stochastically dominates f(x;t) , we conclude that the above inequality holds when we replace f(x;t0 ) with f(x;t) : "Z ^ Z t t

V

;"

0

(x; t )

[V

;"

(x + 1; t + h)

ch] f(x;t) (h) dh +

1

(x)

c t^

t

f(x;t) (h) dh

;"

(17) (x; t).

t^ t

0

The proof follows from the fact that the right-hand side of inequality (17) is V

References [1] Dewatripont, M. and J. Tirole (1999), “Advocates” Journal of Political Economy, 107, 1-39. [2] Farrell, J. (1986) “Voluntary Disclosures: Robustness of the Unraveling Result, and Comments on Its Importance” in “Antitrust and Regulation” (ed: R. Grieson), Lexington Books, Lexington. [3] Froeb, L. M. and B. H. Kobayashi (1996) “Naive, Biased, yet Bayesian: Can Juries Interpret Selectively Produced Evidence” Journal of Law, Economics and Organization, 12, 257-276. [4] Grossman, S. (1981) “The Informational Role of Warranties and Private Disclosure about Product Quality”Journal of Law and Economics, 24, 461-483. [5] Leland, H. E. (1981) “Comments on Grossman”Journal of Law and Economics, 24, 485-489. [6] Mas-Colell, A., M. D. Whinston and J. Green (1995) “Microeconomic Theory”, Oxford University Press, New York. [7] Milgrom, P. (1981) “Good News and Bad News: Representation Theorems and Applications”Bell Journal of Economics, 12, 350-391. 20

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[8] Milgrom, P. and J. Roberts (1986), “Relying on the Information of Interested Parties”Rand Journal of Economics, 17, 18-32. [9] Shin, H. S. (1994a) “The Burden of Proof in a Game of Persuasion” Journal of Economic Theory, 64, 253-264. [10] Shin, H. S. (1994b) “News Management and the Value of Firms” Rand Journal of Economics, 25, 58-71. [11] Shin, H. S. (1996) “Adversarial and Inquisitorial Procedures in Arbitration”unpublished paper. [12] Yilankaya, O. (1998) “A Model of Evidence Production and Optimal Standard of Proof and Penalty in Criminal Trials”unpublished paper.

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