REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING∗ JEAN-FRANÇOIS LASLIERa , MATÍAS NÚÑEZb , AND CARLOS PIMIENTAc A BSTRACT. We design a two-player bargaining game where each player simultaneously approves of a set of lotteries over finitely many alternatives. If the two sets have lotteries in common one of these common lotteries is randomly selected. If, on the other hand, the two approved sets do intersect then one of the approved lotteries is randomly selected. The chosen lottery selects the pure alternative to be implemented. We show that this game always has an equilibrium such that players truthfully reveal their preferences. We also prove that every equilibrium is individually rational and consensual. Furthermore, if players are partially honest (Dutta and Sen, 2012) then every equilibrium is Pareto efficient and sincere. We use this result to fully characterize the set of equilibria of the game under partial honesty. K EY WORDS. Approval voting, bargaining, partial honesty, consensual equilibrium. JEL C LASSIFICATION. C70, C72.

1. I NTRODUCTION An elementary version of the bargaining problem involves two parties with complete information who have to decide on the terms of a possible cooperation. The outcome is either an agreement about such terms, or else a conflict, in the case that no agreement is reached. While dynamic bargaining has been extensively explored and often leads to desirable outcomes (in models à la Rubinstein, 1982), the literature on simultaneous bargaining is scant. It has been argued (see, e.g., Osborne and Rubinstein, 1990) that, not to leave room for renegotiation, the bargaining outcome should be Pareto optimal. Furthermore, if both agents are to participate in the bargaining mechanism then the outcome should not be worse than disagreement. ∗

We thank Arnaud Dellis, Marc Fleurbaey, Thibault Gajdos, Don Keenan, Marcus Pivato, Alain

Trannoy, Dimitrios Xefteris and seminar participants for useful comments. Matías acknowledges financial support from the LABEX MME-DII (ANR-11-LBX-0023-01). Carlos acknowledges financial support from the Australian Research Council’s Discovery Projects funding scheme DP140102426. The usual disclaimer applies. a

CNRS & PARIS S CHOOL OF E CONOMICS, F RANCE .

b

CNRS & THEMA, U NIVERSITY OF C ERGY-P ONTOISE , F RANCE .

c

S CHOOL OF E CONOMICS, AUSTRALIAN S CHOOL OF B USINESS, T HE U NIVERSITY OF N EW

S OUTH WALES, S YDNEY, AUSTRALIA . Email addresses: [email protected], [email protected], [email protected]. Corresponding Author: Carlos Pimienta. Date: July 8, 2015. 1

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

We design a simultaneous model of bargaining such that, in equilibrium, parties always reach an agreement. Furthermore, as long as agents are partially honest (Dutta and Sen, 2012) every equilibrium outcome is Pareto efficient. Partial honesty has been recently analyzed by the mechanism design literature and it captures a mild form of preference for honesty. A partially honest agent prefers being sincere over lying whenever sincerity does not lead to a worse outcome.1 In our mechanism, each player simultaneously approves a set of lotteries over the pure alternatives. If some lotteries are approved by both players then the two approved sets intersect. We define the winning set to be such an intersection and we say that the winning set is consensual. Otherwise, if no lottery is approved by both players then we defined the winning set to be the set of lotteries that are approved by at least one agent. In this case, we say that the winning set is nonconsensual. Finally, the mechanism selects a lottery from the winning set using the uniform probability measure over such a winning set. The alternative to be implemented is decided by this selected lottery. Thus, in the same vein as Núñez and Laslier (2015), our model is a reinterpretation of Approval Voting (Brams and Fishburn, 1983; Laslier and Sanver, 2010) as a bargaining mechanism when there are just two voters. Hence, in the sequel, we refer to our bargaining mechanism as Approval Bargaining. Borrowing from the literature on Approval Voting, we define a strategy as sincere if whenever it contains a lottery it also contains every other lottery that she prefers to it.2 Therefore, in our context, a partially honest agent prefers playing a sincere strategy whenever she cannot obtain a better outcome by playing a strategy that is not sincere. In some sense, our Approval Bargaining game is similar to Nash’s (1953) demand game. In the demand game, two players make simultaneous demands and each one receives the payoff she requests if both payoffs are jointly feasible and nothing otherwise. Our model is more complex since strategies are not unidimensional and the threat point is decided endogenously. For example, consider Figure 1. It captures a bargaining situation with three alternatives, each one represented by a degenerate lottery at the corresponding vertex of the simplex. The figure to the left depicts the strategy profile ( s 1 , s 2 ) while the figure to the right shows a situation where players play the strategy profile ( s′1 , s′2 ). Under the strategy profile ( s 1 , s 2 ), Player 1 approves every lottery in the closed subset labeled s 1 and Player 2 approves every lottery in the closed subset labeled s 2 . These two strategies do not intersect, thus they induce the non-consensual winning set s 1 ∪ s 2 . Correspondingly, we say that ( s 1 , s 2 ) is a non-consensual strategy profile. The outcome induced by ( s 1 , s 2 ) is the 1

We discuss how our paper relates to the mechanism design literature at the end of the Introduc-

tion. We do not attempt to give a review on the bargaining literature and simply refer the reader to Serrano (2008). 2 See Merill and Nagel (1987), Brams (2008), and Núñez (2014) for works dealing with sincerity under approval voting.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

b

s1

3

b( s′1 ∩ s′2 )

b

b( s 1 ∪ s 2 ) b

s′1 s2

b( s′1 ∪ s′2 )

s′2

F IGURE 1. A non-consensual (left) and a consensual (right) strategy profile.

uniform probability measure over s 1 ∪ s 2 , and the expectation of such a measure is the barycenter b( s 1 ∪ s 2 ) of the surface formed by the union of these two strategies. This figure suggests that, under a non-consensual strategy profile, players have two joint incentives: (1) approving a large set so that the induced expected outcome is as close as possible to their approved sets and, consequently, (2) playing some sincere strategy that approves every lottery in the upper-contour set of some indifference curve. Note this is the most effective way to obtain a more preferred outcome given the strategy of the opponent. These two incentives work together so that both players approve bigger and bigger sets. The consequence is that a non-consensual strategy profile cannot be an equilibrium. In Section 4, we prove that every equilibrium strategy profile must have a non-empty intersection in the same way as ( s′1 , s′2 ) in the right hand side of Figure 1. The profile ( s′1 , s′2 ) induces the consensual winning set s′1 ∩ s′2 which, in this case, is just the singleton b( s′1 ∩ s′2 ). Note that players can deviate to a nonconsensual strategy and induce an outcome arbitrarily close to b( s′1 ∪ s′2 ). Hence,

b( s′1 ∪ s′2 ) is the endogenous threat point that sustains the equilibrium outcome b( s′1 ∩ s′2 ). Players have a somewhat natural way of playing this game. We show that a player cannot do better than choosing an approved set of the form { p ∈ ∆ | U ( p) ≥ v} where U is the player’s true expected utility function and v corresponds to some utility level. When playing this strategy, the player fully reveals her preferences by announcing one indifference curve and approving every lottery to the side of the indifference curve where her utility increases. Building on these observations, we prove that this approval bargaining game has the following properties. (1) Existence of equilibrium: Every game has an equilibrium in sincere strategies.

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

(2) Sincere best responses: A player always has a sincere best response against any strategy of her opponent. Hence, if she is partially honest she always plays sincerely. (3) Individual rationality: In every equilibrium, a player obtains at least the same utility as from the uniform lottery over the entire set of alternatives. (4) Consensual equilibria: Every equilibrium is consensual, that is, players always agree on some subset of lotteries. (5) Pareto efficiency and partial honesty: Every sincere equilibrium is Pareto efficient. Thus, if players are partially honest every equilibrium is Pareto efficient. (6) Balanced equilibrium outcomes under partial honesty: If p is an equilibrium outcome then a player’s upper-contour set of p cannot be “too small” unless both players agree on what the best alternative is. More precisely, the last property takes the relative size of the upper-contour sets as a measure of how much the equilibrium outcome favours one player over the other. The smaller a player’s upper-contour set is, the closer the corresponding outcome is to the player’s most preferred alternative. Every equilibrium outcome of the game is balanced in the following sense: If p is an equilibrium outcome and P i is the set of lotteries that Player i prefers to p then both players prefer the equilibrium outcome p to selecting a lottery uniformly from the set P1 ∪ P2 . Furthermore, Properties (5) and (6) characterize the set of equilibrium payoffs (and, therefore, the set of equilibria): If p is Pareto efficient and balanced, then U1 ( p) and

U2 ( p) are equilibrium payoffs. The rest of the paper is structured as follows. After providing a brief account of relevant known results in Mechanism Design, Section 2 presents the model and Section 3 describes the players’ best responses. The game is analyzed in Sections 4 and 5. The latter section focuses on efficiency and partial honesty. The proof of existence of equilibria is contained in the Appendix.

Relationship with the Mechanism Design Literature Maskin (1999) proves that a two-player, Pareto optimal rule defined on the domain of all strong orderings is Nash implementable if and only if it is dictatorial. In view of this result, Moore and Repullo (1990) and Dutta and Sen (1991) characterize Nash implementability with two agents and use such a characterization to find domain restrictions that yield positive results.3 In particular, Dutta and Sen (1991) show that if the set of outcomes is the probability simplex over finitely many alternatives and players have von Neuman-Morgenstern utility functions satisfying

3 See Busetto and Colognato (2009) for a more recent contribution.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

5

suitable conditions then the correspondence that selects the set of Pareto efficient and individually rational lotteries can be implemented.4 As argued by Bagnoli and Lipman (1989), most of the games introduced by the implementation literature are built to be applicable in very general settings rather than for their plausibility. For this reason, these mechanisms are often quite complex. For example, the generalized mechanism used to prove the sufficiency results of the characterization mentioned above uses an “integer game”. The unappealing features of integer games (Jackson, 1992), among other reasons, has stimulated researchers to investigate the implementation problem using different approaches. A recent one explores the scope for implementation when players are partially honest (for a very incomplete list, see Matsushima, 2008a,b; Dutta and Sen, 2012; Kartik and Tercieux, 2012; Kartik et al., 2014; Ortner, 2015). Under partial honesty, a player prefers a truthful message when it does not lead to a strictly worse outcome than what she would obtain otherwise. Dutta and Sen (2012) find necessary and sufficient conditions for implementation under partial honesty when there are two players. However, the existing results do not apply to our setting for different reasons. Some need more than two players (Matsushima, 2008b), some use monetary transfers (Matsushima, 2008a; Kartik et al., 2014) and some propose mechanisms that do not seem suitable to be understood as bargaining protocols (for example Dutta and Sen, 2012 and Kartik and Tercieux, 2012 also use integer games). To consider our setting from a mechanism design viewpoint, define the set of outcomes A to be the probability simplex over the finite set of alternatives. Let the set of states of the world be the set of all pairs D = (D 1 , D 2 ) of linear utility function over A . Let the Social Choice Correspondence (SCC) to be implemented coincide with the equilibrium outcome correspondence f : D → A of the game described in the Introduction. Note that f is the subcorrespondence of the Pareto efficient correspondence that only selects balanced outcomes (see Property (6) above). If both players are partially honest, we can Nash implement f with a direct mechanism where Player i has the simple message space Σ i := D i × R. That is, each agent has to report just her utility function and the utility level she wants to obtain. The outcome of this mechanism maps to each strategy profile either the barycenter of the intersection of the corresponding pair of upper-contour sets (if the intersection is nonempty) or the barycenter of the union of such upper-contour sets (if the intersection is empty). Under partial honesty, every equilibrium of this mechanism is such that each player truthfully reports her utility function and obtains the utility level included in her message.

4 Dutta and Sen assume that no agent is indifferent between two pure alternatives, and that there

is no affine transformation u 1 , u 2 of their Bernoulli utility functions that satisfies either u 1 = u 2 or

u 1 = − u 2 . We do not impose any such restriction.

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

2. T HE G AME Consider two players indexed by i = 1, 2 and a set of alternatives X := { x1 , . . . , xK } with at least two elements. Each Player i is endowed with a Bernoulli utility function u i ∈ R X . Unless stated otherwise, we assume that a Player’s best and a worst K P alternative are associated to different utility levels. Let ∆ := { p ∈ R+ | p i = 1} de-

note the probability simplex over X . Furthermore, we identify an alternative x ∈ X

with the degenerate lottery that assigns probability one to x. Let U i : ∆ → R be Player i ’s corresponding expected utility function.

As mentioned in the Introduction, a strategy for Player i is a subset of lotteries in ∆ that the player approves. If the strategies played by the two players have a nonempty intersection then the outcome of the game is decided by the uniform probability measure over the intersection. If the strategies do not otherwise intersect then the outcome is decided by the uniform probability measure over the union. Therefore, we cannot allow players to play “exotic” subsets of ∆ where the uniform probability measure cannot be defined.5 We let S be the collection of all sets that can be written as the finite union of (not necessarily disjoint) convex and closed (thus compact and Lebesgue measurable) subsets of ∆. The collection of sets S is closed under finite union and finite intersection.6 Lemma 1 below shows that if S is the strategy space of both players then the game is well-defined.7 We give two examples of strategies s i ∈ S . Example 1 (Approving alternatives). Player i can choose a strategy s i ∈ S that approves a subset of X , that is, s i ⊆ X . Any such set s i can be expressed as a finite union of singletons. Note that these strategies coincide with those allowed under standard Approval voting. Example 2 (Approving a half space). Player i can choose a strategy s i ∈ S that contains every lottery p that, for some expected utility function Uˆ i and some v ∈ R

satisfies Uˆ i ( p) ≥ v. If Uˆ i coincides with Player i ’s true expected utility function,

then she approves every lottery in the corresponding upper-contour set associated with the utility level v. A particular case of the strategies given in Example 2 is the collection of sincere strategies. Following the literature on approval voting (see Brams and Fishburn, 5 Not every compact metric space admits a uniform probability measure (see Dembski, 1990). 6 Let A , B ∈ S . If A ∩ B , ; then this intersection can also be written as the finite union of closed

and convex subsets of ∆ because the intersection of two closed and convex sets is also closed and convex. If A ∩ B = ; the same is also true because the empty set is already closed and convex. 7 For the model to be well-defined we need that the pairwise union and intersection of any two strategies admit a uniform distribution. Our restriction of the strategy space is sufficient, but it is not the largest collection of subsets of ∆ satisfying this property.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

7

1983), we say that a strategy of Player i is sincere if it approves every lottery that gives her at least some level of utility. Definition 1 (Sincerity). A strategy s i ∈ S is sincere for Player i if

p ∈ s i and U i ( q) ≥ U i ( p) implies q ∈ s i . Note that a sincere strategy fully reveals the player’s preferences. Given a convex subset A ⊂ ∆, its affine hull aff( A ) is the smallest affine set containing A . The dimension of a nonempty convex subset A , denoted by dim( A ), is the S dimension of its affine hull. The dimension of a finite union of convex sets z∈ Z A z is equal to max z∈ Z dim( A z ) (see Rockafellar, 1997). Let λn be the Lebesgue measure

in Rn . For any n-dimensional set A ∈ S , the uniform measure with support A is given by µ(· | A ) = λn (·)/λn ( A ). Hence, the barycenter b( A ) of A is Z pd µ( p | A ). b( A ) := A

By convention, we let b(∅) = b(∆). Since we work in the probability simplex over X , we will often refer to λK −1 . For simplicity, we simply write λ instead of λK −1 . Given a strategy profile s = ( s 1 , s 2 ) ∈ S , the winning set, to be denoted s 1 ⊗ s 2 , is equal to:  s ∩ s 1 2 s 1 ⊗ s 2 := s ∪ s 1

2

if

s 1 ∩ s 2 , ∅,

otherwise.

If s 1 ∩ s 2 , ∅ then the strategy profile s is consensual. If s 1 ∩ s 2 = ∅ then the strategy profile s is non-consensual. Ties are broken randomly so that, given the strategy profile s = ( s 1 , s 2 ), the expected outcome is b( s 1 ⊗ s 2 ). The rules described above define the simultaneous Approval Bargaining game

Φ = (S, S, u 1 , u 2 ). With abuse of notation, for any A ∈ S , we write U i ( A ) instead of U i ( b( A )). The following lemma implies that the game Φ is well-defined. Lemma 1. For any ( s 1 , s 2 ) ∈ S , the point b( s 1 ⊗ s 2 ) always exists and belongs to ∆. Proof. We already argued that S is closed under finite union and finite intersection. Furthermore, any nonempty A ∈ S has a well-defined dimension so that, for any strategy profile ( s 1 , s 2 ), the measure µ(· | s 1 ⊗ s 2 ) is well-defined. Since the convex hull of the support of µ(· | s 1 ⊗ s 2 ) is always a subset of ∆ we have b( s 1 ⊗ s 2 ) ∈ ∆. Finally, if both players play the empty set we already decided that b(∅) = b(∆) ∈

∆.



Definition 2 (Equilibrium). A strategy profile s = ( s 1 , s 2 ) is an equilibrium if, for every Player i and every s′i ∈ S , we have U i ( s i ⊗ s j ) ≥ U i ( s′i ⊗ s j ).8 8 Hereinafter, once we introduce Player i we let Player j be the other player so that i , j .

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

3. B EST R ESPONSE A NALYSIS Player i ’s set of best responses against strategy s j ∈ S is BR i ( s j ) := arg max U i ( s i ⊗ s j ). s i ∈S

Given the rules of the game, a best-response s i ∈ BR i ( s j ) can either be consensual (if s i ∩ s j , ;) or non-consensual (if s i ∩ s j = ;). We begin by analyzing consensual best responses to s j , that is, those s i ∈ BR i ( s j ) that satisfy s i ∩ s j , ∅. These strategies can be thought of as “accepting” a subset of lotteries offered in s j . Hence, in a consensual best response, Player i should “accept” only her most preferred lotteries in s j . This implies that every accepted lottery must lead to the same utility level and that, therefore, the set of accepted lotteries has zero λ-measure.9 For any strategy s j ∈ S , we let T i ( s j ) := arg max p∈s j U i ( p) denote the set of most preferred lotteries by Player i in s j . Lemma 2. Let s i ∈ S be a consensual best-response to strategy s j ∈ S . Then µ( s i ∩ T i ( s j ) | s i ∩ s j ) = 1 .

Proof. Assume to the contrary that there is some consensual best-response s i to s j with µ( s i ∩ T i ( s j ) | s i ∩ s j ) < 1. Note that any p ∈ s i ∩ T i ( s j ) satisfies U i ( p) = V¯ i

whereas U i ( p) < V¯ i for any p ∈ s i \ T i ( s j ). Then, Z Ui (s i ∩ s j ) = U i ( p ) d µ( p | s i ∩ s j ) s i ∩s j

=

Z

s i ∩T i ( s j )

U i ( p ) d µ( p | s i ∩ s j ) +

= V¯ i µ( s i ∩ T i ( s j ) | s i ∩ s j ) +

Z

Z

s i ∩( s j \T i ( s j ))

s i ∩( s j \T i ( s j ))

U i ( p ) d µ( p | s i ∩ s j )

U i ( p) d µ( p | s i ∩ s j ).

Since µ( s i ∩ T i ( s j ) | s i ∩ s j ) < 1 and U i ( p) < V¯ i for any p ∈ s i \ T i ( s j ), it follows that U i ( s i ∩ s j ) < V¯ i = U i (T i ( s j ) ∩ s j ). Therefore, s i is not a consensual best response to s j which provides the desired contradiction.



Even if there is no best response that is consensual, there always is a best consensual response. Indeed, no other consensual response to s j does better than the consensual response T i ( s j ). The same property does not hold for non-consensual responses. The next example shows a situation where not only does Player 1 not have a best non-consensual response but also she does not have a best response overall. Example 3. Let players 1 and 2 have strict preferences and let x1 be Players 1’s most preferred alternative. Take Player 2’s strategy to be s 2 = { x2 } so that λ( s 2 ) = 0. Any consensual best response to s 2 by Player 1 includes x2 and, hence, generates utility level u 1 ( x2 ). As far as non-consensual responses are concerned, for any ε > 0 9 Recall that we assumed that each player has a worst and a best alternatives so that indifference

curves are lower-dimensional hyperplanes.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

9

small enough, the sincere strategy sε1 = { p ∈ ∆ |U1 ( p) ≥ u 1 ( x1 ) − ε} generates expected utility

U1 ( sε1 ⊗ s 2 ) = U1 ( sε1 ∪ s 2 ) = U1 ( sε1 ), where the last equality follows from λ( s 2 ) = 0. Hence, U1 ( sε1 ) gets arbitrarily close

to u 1 ( x1 ) as ε decreases. When ε = 0, the strategy sε1 collapses to { x1 } so that U1 ( s01 ⊗

s 2 ) = 12 u 1 ( x1 ) + 12 u 1 ( x2 ) < U1 ( sε1 ⊗ s 2 ) for any ε > 0 small enough. Therefore, Player 1

has no best response to s 2 . In the example, it is critical that Player 2 is playing a lower-dimensional strategy. We will later prove that if s j is a full-dimensional strategy (i.e. of dimension K − 1) then Player i has a well-defined best response to s j . In the meantime, we simply show that if s j is full-dimensional and s i happens to be a non-consensual best response against s j then s i approves every lottery that Player i prefers to b( s i , s j ). For any pair of strategies s i ∈ S and s j ∈ S , let R i ( s i , s j ) := { p ∈ ∆ | U i ( p) ≥ U i ( s i ∪ s j )} be the set of lotteries Player i prefers to b( s i ∪ s j ). Lemma 3. Let s j ∈ S be a full-dimensional strategy and let s i ∈ S be a non-consensual best-response to s j . Then

R i ( s i , s j ) ⊆ s i and µ(R i ( s i , s j ) | s i ) = 1. Proof. We first prove that if s i is a non-consensual best response to s j then R i ( s i , s j ) is a subset of s i . The set R i ( s i , s j ) coincides with the closure of its interior and s i is a closed set, so it is enough to prove that every point p ∈ int(R i ( s i , s j )) belongs to s i . Assume to the contrary that p ∉ s i . In that case, there is a closed ball B centred at

p such that B ⊂ int(R i ( s i , s j )) and B ∩ s i = ∅. Note that U i (B) > U i ( s i ∪ s j ) and that, consequently, B ∩ s j = ∅ because otherwise B would be a better response to s j than

si. Now consider the expected utility of s i ∪ B against strategy s j which is equal to: ¸ ·Z Z Z 1 U i ( s i ∪ B, s j ) = U i ( p) d λ + U i ( p) d λ + U i ( p) d λ λ( s i ∪ B ∪ s j ) s i sj B Z λ( s i ∪ s j ) 1 = Ui (s i ∪ s j ) + U i ( p) d λ λ( s i ∪ B ∪ s j ) λ( s i ∪ B ∪ s j ) B >

λ( s i ∪ s j )

λ( s i ∪ B ∪ s j )

Ui (s i ∪ s j ) +

λ(B)

λ( s i ∪ B ∪ s j )

Ui (s i ∪ s j )

= U i ( s i ∪ s j ),

where the strict inequality follows from U i (B) > U i ( s i ∪ s j ). Therefore, s i is not a best response to s j , providing the desired contradiction. We now prove that µ(R i ( s i , s j ) | s i ) = 1. Suppose on the contrary that the set

A := s i \ R i ( s i , s j ) has positive measure. Note that the definition of R i ( s i , s j ) implies

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

that U i ( A ) < U i ( s i ∪ s j ). Let s′i := R i ( s i , s j ). Then,

U i ( s i ∪ s j ) = U i ( s′i ∪ A, s j ) =

λ( s′i ∪ s j ) λ( s′i ∪ A ∪ s j )

U i ( s′i ∪ s j ) +

λ( s′i ∪ s j )

U i ( s′i ∪ s j ) + λ( s′i ∪ A ∪ s j ) = U i ( s′i ∪ s j ).

<

1 ′ λ( s i ∪ A ∪ s j )

Z

A

U i ( p) d λ

λ( A ) U i ( s′i ∪ s j ) ′ λ( s i ∪ A ∪ s j )

Thus, s i is not a best response against s j , which provides the desired contradiction and concludes the proof.



A consequence of the description of the best responses given in Lemmas 2 and 3 is that players have a weak incentive to use sincere strategies, that is, to approve the set of lotteries that give her at least some utility level (see Definition 1). Corollary 1. If the set of best responses is non-empty then it always includes a sincere strategy. Proof. If Player i has a consensual best response to s j then by Lemma 2 the strategy © ª p ∈ ∆ | U i ( p) ≥ U i ( q) for any q ∈ T i ( s j ) is a sincere best response to s j . On the other hand, if Player i has a non-consensual best response to s j then by Lemma 3

the strategy s i that satisfies s i = R i ( s i , s j ) is also a sincere best response to s j .



A second consequence of the description of best responses is the following. Corollary 2. The set of best responses cannot include both consensual and nonconsensual strategies. Proof. Assume that Player i ’s set of best responses to s j contains both consensual and non-consensual strategies. Due to the same argument as in the proof of the © ª previous corollary, the strategy p ∈ ∆ | U i ( p) ≥ U i ( q) for any q ∈ T i ( s j ) and the © ª strategy s i that satisfies s i = p ∈ ∆ | U i ( p) ≥ U i ( s i ∪ s j ) are also best responses to

s j . However, both of them must lead to the same utility level so that U i ( s i ∪ s j ) =

U i ( q) for any q ∈ T i ( s j ). In other words, they are both the same strategy. Such a

strategy either intersects with s j or it does not. In the first case, the set of best responses against s j contains only consensual responses to s j and, in the second case, it contains only non-consensual responses.



We conclude this section by proving that players always have a best response against a full-dimensional strategy. To facilitate the analysis, for every strategy profile ( s i , s j ) and each Player i we define the function

Vi ( s i , s j ) =

λ( s i ) λ( s i ) + λ( s j )

Ui (s i ) +

λ( s j ) λ( s i ) + λ( s j )

U i ( s j ).

(3.1)

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

11

In particular, Vi ( s i , s j ) = U i ( s i ⊗ s j ) = U i ( s i ∪ s j ) whenever s i ∩ s j = ∅. A similar argument to the one used in the proof of Lemma 3 shows that, for every fulldimensional strategy s j , the unique sincere strategy that maximizes Vi (·, s j ) is the strategy s i that satisfies: © ª s i = p ∈ ∆ | U i ( p) ≥ Vi ( s i ∪ s j ) .

The next lemma describes the conditions under which the best responses to a full-dimensional strategy are either consensual or non-consensual. Lemma 4. Let s j ∈ S be a full-dimensional strategy. Let s i ∈ S be the unique sincere strategy that maximizes Vi (·, s j ). (1) If s i ∩ s j , ∅ then tPlayer i ’s best response to s j is consensual. (2) If s i ∩ s j = ∅ then Player i ’s best response to s j is non-consensual and, moreover, s i is a best response to s j . Proof. (1) For every non-consensual response s′i to s j we have Vi ( s i , s j ) ≥ Vi ( s′i , s j ) =

U i ( s′i ∪ s j ). By definition, s i approves every lottery that gives Player i a utility larger than Vi ( s i , s j ). Since s i ∩ s j , ∅, the strategy s i includes T i ( s j ). But then,

U i (T i ( s j ) ∩ s j ) = U i (T i ( s j )) ≥ Vi ( s i , s j ). Thus, for every non-consensual strategy s′i we find that the consensual strategy T i ( s j ) satisfies U i (T i ( s j ) ∩ s j ) ≥ U i ( s′i ∪ s j ). We conclude that the best response to s j is consensual. (2) Since s i ∩ s j = ∅ we have U i ( s i ∪ s j ) = Vi ( s i , s j ). Furthermore, the fact that s i maximizes Vi ( s i , s j ) implies that for every non-consensual reply s′i to s j we obtain

U i ( s i ∪ s j ) ≥ U i ( s′i ∪ s j ). Note that s i approves every lottery that Player i prefers to b( s i ∪ s j ). Therefore, U i ( p) ≤ U i ( s i ∪ s j ) for every p ∈ T i ( s j ). This implies that for every consensual response s′′i to s j we have U i ( s i ∪ s j ) ≥ U i ( s′′i ∩ s j ). We conclude that s i is a best response to s j .



As a corollary of Lemma 4, we obtain the following result. Theorem 1. If s j ∈ S is a full-dimensional strategy then BR i ( s j ) is nonempty. Note, however, that players may not play a full-dimensional strategy. Nonetheless, as long as players do not agree on what the best alternative is, there is some incentive to do so. If dim( s i ) < dim( s j ) and the strategy profile s = ( s i , s j ) is nonconsensual then s i is irrelevant when computing the outcome induced by the strategy profile so that b( s i ⊗ s j ) = b( s j ). On the other hand, if both players have a common best alternative then at least one of the players must play a lower-dimensional strategy in any undominated equilibrium of the game.10 This follows because at least one player’s strategy must contain the subset of her most preferred lotteries. It follows that players obtain 10 In this case, letting s := arg max p∈∆ U i ( p), the profile ( s i , s j ) is an undominated Nash i

equilibrium.

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

their maximum utility in every such undominated equilibrium. Henceforth, unless stated otherwise, we make the following assumption that ensures that there is no common best alternative: Assumption A. There is no alternative that both players consider at least as good as any other alternative. 4. E QUILIBRIUM P ROPERTIES Every game has an equilibrium in sincere strategies. This result does not follow from standard existence results because of the complexity of the strategy space—it is not finitely dimensional. Additionally, utility functions are not continuous.11 Indeed, the outcome of the game (i.e. b( s 1 ⊗ s 2 )) “jumps" discontinuously whenever the limit of a sequence of non-consensual strategy profiles is a consensual strategy profile. The proof of existence consists of approximating the game Φ using a sequence of finite two-player approval games whose set of alternatives contains the set of pure alternatives X and larger and larger (finite) subsets of ∆. Thus, each game in this sequence is a standard Approval voting game with two players and a richer strategy space. Each such game admits an equilibrium in pure and sincere strategies as proved by Núñez and Laslier (2015). The limit of such a sequence of sincere equilibrium strategies, appropriately extended, is an equilibrium of our game Φ. The details of the proof can be found in the Appendix. Theorem 2. Every Approval Bargaining game Φ has an equilibrium in sincere strategies. We turn to describing the equilibrium properties of the game. Theorem 3. Players play full-dimensional strategies in equilibrium. Proof. Let s = ( s i , s j ) be an equilibrium and let v¯ i := max p∈∆ U i ( p) with i = 1, 2. Proceeding by contradiction, assume first that m := max{dim( s 1 ), dim( s 2 )} < K − 1. Given Assumption A, there is a Player i such that U i ( s i ⊗ s j ) < v¯ i . Let sεi denote the sincere strategy sεi := { p ∈ ∆ | U i ( p) ≥ v¯ i − ε}. Note that sεi is a full-dimensional strategy. Moreover, when ε is small enough, sεi ∩ s j = ; because U i ( s i ⊗ s j ) < v¯ i . Therefore, as ε decreases U i ( sεi ⊗ s j ) becomes arbitrarily close to v¯ i . This implies that Player i has a profitable deviation, proving that ( s i , s j ) is not an equilibrium. Therefore m = K − 1. Analogously, assume now l := min{dim( s 1 ), dim( s 2 )} < K − 1. Let dim( s j ) < K − 1. If U i ( s i ⊗ s j ) < v¯ i then, using the same definition for sεi as before, Player i can make

U i ( sεi ⊗ s j ) be arbitrarily close to v¯ i , proving that she does not have a best response and contradicting that ( s i , s j ) is an equilibrium. If U i ( s i ⊗ s j ) = v¯ i then, in turn, 11

We have not specified a topology on the strategy space. However, the informal argument that

follows should be sufficiently clear.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

13

Player j is not playing a best response to s i . Indeed, playing a non-consensual strategy which contains all lotteries p with U j ( p) > U j ( s i ⊗ s j ) strictly increases her utility. Therefore l = K − 1 as we wanted.



In an intuitive sense, this property is related to the next equilibrium property which specifies that every equilibrium of the game is consensual. Each player plays a full-dimensional strategy in equilibrium so that her opponent does not find it profitable to deviate to a non-consensual strategy. Put differently, the outcome of any potential deviation by Player j to a non-consensual strategy is less harmful to Player i the “larger” the strategy that she plays is. Thus, for any equilibrium strategy ( s 1 , s 2 ), the equilibrium outcome is b( s 1 ∩ s 2 ) while the threat point sustaining such an equilibrium is b( s 1 ∪ s 2 ). Theorem 4. Every equilibrium is consensual. Proof. Suppose to the contrary that there is a non-consensual equilibrium ( s 1 , s 2 ). By Theorem 3, players play full-dimensional strategies. Thus, we can use Lemma 3 to obtain both b( s 1 ∪ s 2 ) ∈ s 1 and b( s 1 ∪ s 2 ) ∈ s 2 . But this implies s 1 ∩ s 2 , ;. Hence, any equilibrium must be consensual.



The next property deals with the minimal utility level that a player can obtain from an ex ante viewpoint. Theorem 5. Each Player i gets at least U i (∆) in equilibrium. Proof. The sincere strategy s∗i := { p ∈ ∆ : U i ( p) ≥ U i (∆)} guarantees a payoff of at least U i (∆) to Player i regardless of the strategy s j played by Player j . This is clear if s∗i ∩ s j , ∅. In turn, if Player j plays a non-consensual response to s∗i then she plays a closed subset of ∆ \ s∗i , that is, a (strict) subset of the set of lotteries that are less preferred than b(∆) by Player i . Hence, U i (∆) = U i ( s∗i ∪ (∆ \ s∗i )) > U i ( s∗i ∪ s j ) for any strategy s j that satisfies s j ⊂ ∆ \ s∗i .



If the Bernoulli utility functions of the players satisfy u 1 = − u 2 up to some affine transformation of utilities then we say that the Players have opposing preferences. In this case, the game has a unique equilibrium outcome. Corollary 3. If players have opposing preferences then the unique equilibrium outcome is the barycenter of the simplex b(∆). Corollary 3 implies that the lower-bound on equilibrium payoffs given in Theorem 5 is the highest payoff that the game can guarantee players for any utility profile. In fact, when we consider a utility profile where players have opposing preferences we can see that the same statement is true for any mechanism whose set of possible outcomes is ∆. Nonetheless, typically, the game has a continuum of Nash equilibrium outcomes such that both players obtain strictly more than their utility values under b(∆). At

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

the end of Section 5, after we characterize the set of equilibria in sincere strategies, we present an example illustrating this fact. 5. E FFICIENCY AND PARTIAL HONESTY We turn to the efficiency properties of equilibria of the game Φ. Definition 3 (Efficiency). A lottery p ∈ ∆ is (ex-ante) efficient if there is no q ∈ ∆ such that U i ( q) ≥ U i ( p) for i = 1, 2 with U i ( q) > U i ( p) for at least some i . If a lottery is Pareto efficient then it only gives positive probability to Pareto efficient alternatives. If, say, alternative x1 is Pareto dominated by alternative x2 a lottery p with p 1 > 0 is Pareto dominated by the lottery q that satisfies:

q′1 = 0, q′2 = p 1 + p 2 , and q′k = p k for k = 3, . . . , K. This shows that any lottery that assigns positive probability to inefficient alternatives is inefficient. If an efficient lottery is the equilibrium outcome of the game then, ex-post, players would never have a common incentive to renegotiate once the equilibrium outcome has been realized into some alternative in X . Theorem 2 guarantees that the game has at least one sincere equilibrium. We now show that such an equilibrium is necessarily efficient. Proposition 1. Every sincere equilibrium outcome is efficient. Proof. Let ( s 1 , s 2 ) be a sincere equilibrium strategy. Every equilibrium is consensual (Theorem 4) so s 1 ∩ s 2 , ∅. Lemma 2 implies that Players i ’s utility level associated with the sincere strategy s i is v i := max p∈s j U i ( p) and that, moreover, for every

p ∈ s 1 ∩ s 2 we have U i ( p) = v i . Suppose there is a q ∈ ∆ such that U i ( q) ≥ v i for i = 1, 2, with strict inequality for at least one player. Then q is both in s 1 and s 2 because they are sincere strategies. But this contradicts our definition of v i for at least one i = 1, 2. Thus, every lottery in the winning set of a sincere equilibrium is Pareto efficient.



However, as the next example demonstrates, not every equilibrium of the game is efficient. Example 4. In Figure 2 we represent a bargaining game with three alternatives and a sincere equilibrium ( s 1 , s 2 ). The intersection of the equilibrium strategies

s 1 and s 2 consists of only one point p and the strategies are defined by s i := { r ∈

∆ : U i ( r ) ≥ U i ( p)} for i = 1, 2. The lottery p′ := b( s 1 ∪ s 2 ) is the threat point of the equilibrium ( s 1 , s 2 ). Either player can induce an outcome as close as they wish to p′ by deviating to a sincere non-consensual strategy. But both players prefer p to

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

b

15

p

b

q s1

b

p′ b

q′

t1

t2

s2

F IGURE 2. An inefficient equilibrium.

p′ , thus confirming that ( s 1 , s 2 ) is a equilibrium. Such an equilibrium is clearly efficient. We now construct an inefficient equilibrium by first considering indifference curves associated with slightly lower utility levels for both players. These new indifference curves cross at the lottery q in the interior of the simplex. We obtain the strategy profile ( t 1 , t 2 ) inducing the consensual outcome q by bending the indifference curves at q to obtain t 1 as the area to the north-west of the dotted line and t 2 as the area to the south-east of the dashed line. Note that no player can profitably deviate to a different consensual strategy. The new threat point q′ := b( t 1 ∪ t 2 ) is close-by to the old threat point due to a continuity argument and, therefore, no player can profitably deviate to a non-consensual strategy either. Hence ( t 1 , t 2 ) is an equilibrium inducing the inefficient lottery q. In the inefficient equilibrium of the previous example, both players are indifferent between playing their insincere equilibrium strategy and some sincere strategy. The inefficient outcome arises because players coordinate in their insincere strategies. However, if we slightly refine rationality and assume that players always play a sincere strategy whenever they have one available in their set of best responses then this sort of equilibria disappears. This assumption is equivalent to saying that players are partially honest, an assumption recently proposed in the implementation literature. We follow the formal definition of partial honesty given by Dutta and Sen (2012). Henceforth, the set of sincere strategies for Player i is denoted by S i . We denote by º i Player i ’s ordering over the set of strategy profiles S when she is partially honest. Its asymmetric component is denoted by ≻ i . Definition 4. Player i is partially honest if for any two ( s i , s j ), ( s′i , s j ) ∈ S . (1) If U i ( s i ⊗ s j ) ≥ U i ( s′i ⊗ s j ) and s i ∈ S i , s′i ∉ S i , then ( s i , s j ) ≻ i ( s′i , s j ).

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

(2) In all other cases, ( s i , s j ) º i ( s′i , s j ) if and only if U i ( s i ⊗ s j ) ≥ U i ( s′i ⊗ s j ). The first part of the definition represents the individual’s partial preference for honesty. She strictly prefers the strategy profile ( s i , s j ) to ( s′i , s j ) when s i is a sincere strategy and s′i is not, provided that the outcome corresponding to ( s i , s j ) is at least as good as the one corresponding to ( s i , s j ). The second part of the definition implies that in every other case, the player’s preference ordering over the corresponding strategy profiles is not altered. The preference profile (º1 , º2 ) now defines a modified normal form game. We omit formal definitions for the sake of brevity. The next proposition is a trivial and important implication of Corollary 1 and Proposition 1. Proposition 2. In the game with partially honest players, a player’s best response is sincere, and every equilibrium sincere and Pareto efficient. Assuming partial honesty allows us to focus, for each Player i , on her set of sincere strategies S i ⊂ S . Such a subset of strategies has a simple characterization. For each Player i let v¯ i := max x∈ X u i ( x) and v i := min x∈ X u i ( x). To each utility value ª © v i ∈ [v i , v¯ i ] we associate the sincere strategy s i (v i ) := p ∈ ∆ : U i ( p) ≥ v i .

We turn to characterizing the set of equilibria under partial honesty. Given a

sincere strategy of a player, the other player’s best response is either consensual or non-consensual. Since every equilibrium is consensual, to show that a given strategy profile is an equilibrium we need to prove (1) that both players are playing their best consensual response, and that (2) they do not gain by deviating to a nonconsensual response. We now study how the best consensual and non-consensual responses of a player behave as the opponent changes her strategy. For each v j ∈ (v j , v¯ j ), we let CU i (v j ) denote Player i ’s utility value from the best sincere consensual response to s j (v j ). Instead of working with the analogous expression for Player i ’s best sincere non-consensual response (that, as we argued before, might not exist) we let NU i (v j ) denote Player i ’s maximal value of Vi (·, s j (v j )) (see Equation (3.1)). Recall that if the sincere strategy s i (v i ) maximizes Vi (·, s j (v j )) and s i (v i ) ∩ s j (v j ) = ∅ then s i (v i ) is the best response to s j (v j ) and, therefore, also the best non-consensual response to s j (v j ). Note that CU i and NU i are continuous functions on (v j , v¯ j ). Furthermore, CU i is nonincreasing in v j (because s j (v j ) ⊂ s j (v′j ) whenever v′j > v j ). Proposition 3. In the game with partially honest players, for each Player i there exists a unique η i ∈ (v j , v¯ j ) such that: CU i (v j ) ≥ NU i (v j ) if and only if v j ≤ η i . Proof. We already argued (proof of Lemma 4) that if v j ∈ (v j , v¯ j ) and the best response to s j ( s j ) is consensual then CU(v j ) ≥ NU(v j ) and that if v j ∈ (v j , v¯ j ) and the best response to s j (v j ) is non-consensual then NU(v j ) ≥ CU(v j ). If v j is close enough

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

17

to v j then Player i ’s best response to s j (v j ) is consensual because she can obtain a payoff close to v¯ i by playing a consensual best response while she can only get a payoff close to U i (∆) by playing a non-consensual response. In turn, if v j is close enough to v¯ j then Player i ’s best response to s j (v j ) is non-consensual because Player i can obtain a utility close to v¯ i by playing a non-consensual strategy (in a similar vein as in Example 3) whereas she can only get, at most, a utility close to the one corresponding to her second most preferred alternative if she plays a consensual best response (due to Assumption A). The continuity of CU i and NU i as functions of v j implies the existence of some η i ∈ (v j , v¯ j ) for which CU i (η i ) = NU i (η i ). To prove uniqueness, suppose that CU i (v j ) = NU i (v j ) for some v j > η i . Since

s j (v j ) ⊂ s j (η i ) and s j (η i ) \ s j (v j ) is a set with positive measure that only contains lotteries that give Player i utility less than η i we have NU i (v j ) > NU i (η i ). Moreover CU i is nonincreasing on v j so that CU i (η i ) ≥ CU i (v j ). Hence, for any v j > η i , we have NU i (v j ) > CU i (v j ).



We can now complete the full characterization of equilibria in our Approval Bargaining game under partial honesty. Proposition 4. In the game with partially honest players, let (v1 , v2 ) be a utility profile derived from some Pareto efficient lottery. The profile ( s 1 (v1 ), s 2 (v2 )) is an equilibrium payoff if and only if v j ≤ η i for both i = 1, 2. Proof. Let p ∈ ∆ be Pareto efficient and let v i = U i ( p) for i = 1, 2. Consider the strategy profile ( s 1 (v1 ), s 2 (v2 )). We have p ∈ s 1 (v1 ) ∩ s 2 (v2 ) and, because p is Pareto efficient, such an intersection has an empty interior. Thus, no player has an incentive to deviate to a different consensual strategy. Furthermore, since v1 ≤ η2 and

v2 ≤ η1 , the previous proposition implies that no player has an incentive to deviate to a non-consensual strategy. On the other hand, let (v1 , v2 ) be an equilibrium payoff. From Lemma 2 we know that players are playing ( s 1 (v1 ), s 2 (v2 )) which, by Theorem 4, is a consensual strategy profile. Because players do not have an incentive to deviate to a non-consensual strategy we have v1 ≤ η2 and v2 ≤ η1 .



Thus, with partial honesty, a player’s set of equilibrium payoffs (and, correspondingly, the set of equilibrium strategies) is a closed interval. Moreover, the previous proposition implies that a player’s upper-contour set of an equilibrium outcome cannot be “too small” relative to that of the other player. To formalize this idea we introduce the following definition: Definition 5. Let P i be Player i ’s upper-contour set of the lottery p. We say that p is a balanced lottery if U i ( p) ≥ U i (P1 ∪ P2 ). That is, a lottery p is balanced if both players prefer that lottery to the expected outcome of the uniform measure over the set of lotteries that at least one agent

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

prefers to p. Proposition 4 implies that a payoff vector is an equilibrium payoff vector if and only if it is generated by an efficient and balanced outcome. In particular, this implies that, unless players have opposing preferences, both players strictly prefer the equilibrium outcome to b(∆).12 We have developed most of the analysis under Assumption A. If otherwise players have a common best alternative then it is not difficult to prove that partial honesty implies that players include their best alternatives in their strategies. Hence, if Assumption A is not satisfied, the equilibrium outcome is also efficient and balanced. We have also assumed from the outset that a Player’s best and worst alternative are associated to different utility levels. If this is not the case for Player i then partial honesty implies that Player i can only play s i = ∆ or s i = ∅. In either case, it follows that the equilibrium outcome is again efficient and balanced. To conclude, we note that the equilibrium characterization can be used to compute the set of Nash equilibria for any given utility profile. We do so in the next example. Example 5. Consider a bargaining situation with set of alternatives X = { x1 , x2 , x3 }. Players 1 and 2 have Bernoulli utility functions u 1 = (10, u, 0) and u 2 = (0, v, 10). If

u + v = 10 then there is a unique equilibrium outcome b(∆) (Corollary 3). Otherwise, the game does not have a unique equilibrium outcome. We consider the family of games where u = v. For each of these games, Proposition 3 gives the maximum equilibrium utilities for Player i (hence, it also gives the minimum equilibrium utility for Player j .) For instance, if u = v = 1, then there is a continuum of equilibrium outcomes in which Player 1 obtains some utility value uˆ ∈ [8.91334, 9.77993] and Player 2 obtains utility (90 − uˆ )/9. Table 1 shows the interval for the equilibrium payoffs for Player 1 for any given u = v ∈ {1, . . . , 10}. In any such situation, if Player 1 obtains payoff uˆ then Player 2 obtains payoff p( uˆ ) with:

p( uˆ ) =

(

100−(10+ uˆ ) u 10− u ˆ 10 u− uu 10− u

u ≤ 5, u > 5.

T ABLE 1. Player 1’s maximum and minimum equilibrium payoffs of Φ for utility profiles u 1 = (10, u, 0) and u 2 = (0, v, 10) when players are partially honest. Values are rounded up to two decimal places.

u=v

1

2

3

4

5

6

7

8

9

10

u1

8.91 8.26 7.01 5.35 5 5.43 6.00 6.71 7.64 10

u1

9.78 9.25 8.35 6.98 5 6.38 7.43 8.32 9.15 10

12 Following the same logic as in the proof of Theorem 5, if Player i plays the sincere strategy

s i (U i (∆)) and U i ( s i (U i (∆)) ∩ s j ) = U i (∆) then U i ( s i (U i (∆)) ∪ s j ) ≥ U i (∆) which, furthermore, holds with strict inequality whenever s 1 ∪ s 2 , ∆.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

19

6. C ONCLUSION This paper develops a simultaneous bargaining mechanism between two players. Among its appealing properties, it is noteworthy that it triggers an agreement between the players in every equilibrium. In order to ensure that both players reach an agreement, the mechanism gives them the following incentives. If they announce non-disjoint sets of approved lotteries, then one of these consensual lotteries is randomly chosen. Otherwise, if the players fail to agree, one of the proposals advocated by either of the players gets selected at random so that a disagreement lottery occurs over both approved sets. This strong threat forces both players to reach an agreement. Indeed, in the event of this disagreement lottery, each player prefers to add to her approved set all the lotteries she prefers to the barycenter of the approved sets. Since this barycenter is common to both players, this leads to the desired agreement. Nonetheless, the mechanism may lead to Pareto inefficient equilibria. This is line with Maskin’s (1999) impossibility result for scenarios with two players which underlines a tension between implementation in pure strategies and Pareto efficiency. However, as we prove, this inefficiency is quite mild since it disappears if players have a slight preference for sincerity (i.e. partial honesty). Indeed, under this refinement of rationality, the players’ best responses must be sincere which, in turn, implies that any equilibrium outcome is Pareto efficient. Additionally, this refinement of rationality allows us to derive a clear-cut characterization of equilibrium outcomes: an outcome is an equilibrium one if and only if it is Pareto efficient and balanced. A natural research question that arises is whether this mechanism can be extended to many players. The answer to this question seems far from obvious. With two players, they either agree or they do not. However, this duality is lost with three or more players. The main problem seems to be what the rules of the game should specify to determine the outcome when some but not all players agree on some set of lotteries. While one might think of several possible extensions, none of them seems to conveniently extend the properties of the current approval bargaining game. A PPENDIX : E XISTENCE OF EQUILIBRIUM The proof of existence of equilibrium builds a sequence of finite games that suitably approximate our game Φ. Each game in this sequence is an Approval voting game with two players. This class of games is analyzed by Núñez and Laslier (2015). Each player selects a subset of the finite set of alternatives that she approves. If the intersection of these two subsets is non-empty then the outcome is determined by a uniform lottery over the intersection. If the intersection of the two subsets is empty then the outcome is decided by the uniform lottery over the union. We need the following properties proved in Núñez and Laslier (2015).

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JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

(α) Every two-player approval voting game has an equilibrium in sincere strategies. That is, an equilibrium where if a player approves some alternative then she also approves every alternative that she prefers to it. (β) If an equilibrium outcome is non-consensual then each player approves every alternative that she prefers to the equilibrium outcome. (γ) In every sincere equilibrium, each player only approves alternatives that she prefers to the equilibrium outcome. As we construct the sequence of finite two-player approval games we also construct a sequence of measures to approximate outcomes in Φ with sequences of outcomes of the approval games. We embed the (K − 1)-dimensional simplex ∆ in RK −1 and consider the smallest hypercube I ⊂ RK −1 containing ∆. We construct a sequence of probability measures {λ t } on I iteratively. We first set I 0 := I and let c be the barycenter of I 0 and

C 0 := { c}. The probability measure λ0 gives probability 1 to c ∈ I 0 . For each t > 0, let I t be the collection of hypercubes that one obtains by dividing each hypercube in

I t−1 into 2K −1 equally sized hypercubes. Each one of the 2K −1 hypercubes h ∈ I t has a barycenter c( h). Let C t := { c( h) : h ∈ I t }. The probability measure λ t gives probability 1/#C t to each c( h) such that h ∈ I t . Furthermore, the game Γ t is defined as the approval voting game with 2-players and set of alternatives X t := C t ∩ ∆. Player’s utilities over elements in X t are computed by extending linearly their Bernoulli utility function over the original set of alternatives X . The next lemma will be used to approximate outcomes in the game Φ with a sequence of outcomes of the finite approval games constructed above. The proof consists of showing that the sequence of probability measures {λ t } converges weakly to the uniform measure λ(·)/λ( I ) over the hypercube I . There are several equivalent definitions of weak convergence but for our purposes we only need two.13 Given the hypercube I (with its Borel σ-algebra) the bounded sequence of positive finite measures {λ t } on I converges weakly to the finite positive measure λ(·)/λ( I ) if any of the following equivalent conditions is true: • lim λ t (E ) = λ(E )/λ( I ) for every set E whose boundary ∂E satisfies λ(∂E ) = 0. R R • lim I f d λ t = λ(1I ) I f d λ for every bounded and uniformly continuous func-

tion f .

Lemma 5. Let E ⊂ ∆ satisfy λ(E ) > 0 and λ(∂E ) = 0, and define E t := X t ∩ E . Then R P e∈E t e E pd λ = . lim t→∞ #E t λ(E ) Proof. As we announced previously, we actually prove that the sequence of probability measures {λ t } converges weakly to the uniform measure λ(·)/λ( I ) over I . A consequence is that conditional probabilities induced by members of {λ t } on subsets 13 See Theorem 25.8 in Billingsley (1986) for equivalent definitions of weak convergence.

REACHING CONSENSUS THROUGH SIMULTANEOUS BARGAINING

21

E ⊂ I whose boundary has zero Lebesgue measure also converge to the corresponding uniform probability measures over those subsets (and, hence, also their means). Take some hypercube h ∈ I t and note that, if c( h) is its barycenter, λ t ( c( h)) = 1/#C t = λ( h)/λ( I ). That is, the probability of c( h) coincides with the volume of h normalized by the volume of I . For any bounded, uniformly continuous function

f : I → R, Z I

Z 1 X t→∞ 1 f dλ = f ( c( h))λ( h) −→ f d λ, λ( I ) h∈ I t λ( I ) I t

which means that {λ t } converges weakly to the measure λ(·)/λ( I ).



Now we can finally prove: Theorem 2. Every game Φ has an equilibrium in sincere strategies. Proof. Given property (α) we can take a sequence {( s 1t , s 2t )}∞ t=1 of pairs of finite subsets of ∆ such that ( s 1t , s 2t ) is a sincere equilibrium of Γ t for every t. For i = 1, 2 and for every t define v it := min p∈s t U i ( p). The utility to Player i from every lottery in i

s ti is at least v it . The sequence {(v1t , v2t )}∞ t=1 is contained in a compact set, therefore, it has a subsequence that converges to some (v1∗ , v2∗ ). For each i = 1, 2 define the © ª sincere strategy s∗i := p ∈ ∆ : U i ( p) ≥ v∗i . We claim that ( s∗1 , s∗2 ) is an equilibrium of

Φ. We proceed in three steps. Step 1: ( s∗1 , s∗2 ) induces a consensual outcome. We prove this step by contradiction. Suppose that s∗1 ∩ s∗2 = ∅. Since lim(v1t , v2t ) = (v1∗ , v2∗ ) continuity of the utility functions on ∆ implies that, passing to a subsequence if necessary, for every t high enough we also have s 1t ∩ s 2t = ∅. Because ( s 1t , s 2t ) is a non-consensual equilibrium of Γ t , Property (β) above implies that the strategy s ti contains every lottery that Player i prefers to b( s 1t ∪ b 2t ). For i = 1, 2, let q ti := arg min p t ∈s t k p ti , b( s 1t ∪ s 2t )k be the lottery approved by Player i in the strati

i

egy s ti that is closest to the outcome b( s 1t ∪ b 2t ). Clearly, for i = 1, 2, the sequence k q ti , b( s 1t ∪ s 2t )k∞ t=1 converges to zero. The triangular inequality implies that the se∗ ∗ quence k q 1t , q 2t k∞ t=0 also converges to zero. This contradicts s 1 ∩ s 2 = ∅ proving that

( s∗1 , s∗2 ) induces a consensual outcome. Step 2: ( s∗1 , s∗2 ) generates expected payoffs (v1∗ , v2∗ ). To the contrary and without loss of generality, assume that Player 1 gets a payoff strictly higher than v1∗ under the strategy profile ( s∗1 , s∗2 ) so that U1 ( s∗1 ∩ s∗2 ) > v1∗ . There must be a pˆ ∈ s∗2 such that U1 ( pˆ ) > v1∗ . Such an inequality also holds for every point in some closed neighborhood P of pˆ . Thus, for t high enough, we can choose a pˆ t ∈ S t ∩ P such that U1 ( pˆ t ) > v1∗ and pˆ t ∈ int( s∗2 ) (i.e. U2 ( pˆ t ) > v2∗ ). This means that pˆ t ∈ s 2t for sufficiently high t. Therefore, U1 ( s 1t ⊗ s 2t ) ≥ U1 ( pˆ t ) for any sincere equilibrium ( s 1t , s 2t ) of Γ t . But then, also for every sufficiently high t,

v1t ≥ U1 ( s 1t ⊗ s 2t ) ≥ U1 ( pˆ t ) > v1∗ ,

(A.1)

22

JEAN-FRANÇOIS LASLIER, MATÍAS NÚÑEZ, AND CARLOS PIMIENTA

where the first inequality follows from (γ). But this is impossible because v1∗ is the limit point of the sequence {v1t }∞ t=1 . This provides a contradiction so we can conclude that ( s∗1 , s∗2 ) generates expected payoffs (v1∗ , v2∗ ). Step 3: ( s∗1 , s∗2 ) is an equilibrium. Suppose again the the contrary that ( s∗1 , s∗2 ) is not an equilibrium of Φ. Without loss of generality, let there be an sˆ1 such that U1 ( sˆ1 ⊗ s∗2 ) > v1∗ . The fact that ( s∗1 , s∗2 ) induces the consensual outcome b( s∗1 ⊗ s∗2 ) that generates the vector of utility levels (v1∗ , v2∗ ), implies that Player 1’s deviation to sˆ1 induces a non-consensual outcome

b( sˆ1 ∪ s∗2 ). For each t, consider the strategy sˆ1t that approves every lottery available in Γ t that belongs to sˆ1 . By construction, the outcome b( sˆ1t ⊗ s 2t ) is non-consensual and Lemma 5 guarantees that lim b( sˆ1t ∪ s 2t ) = b( sˆ1 ∪ s∗2 ). Hence, for every t high enough and some ε > 0 we obtain

U1 ( sˆ1t ∪ s 2t ) > v1∗ + ε.

(A.2)

Since each member of the sequence {( s 1t , s 2t )}∞ t=0 is an equilibrium of the corresponding game Γ t , property (γ) implies that U1 ( s 1t ∪ s 2t ) ≤ v1t for every t. It follows that lim U1 ( s 1t ∪ s 2t ) ≤ v1∗ and, for every t high enough, U1 ( s 1t ∪ s 2t ) ≤ v1∗ + ε. But this last inequality combined with (A.2) implies that ( s 1t , s 2t ) is not an equilibrium of Γ t . This is a contradiction so ( s∗1 , s∗2 ) must be an equilibrium of Φ.



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