Uniqueness of stationary equilibria in a one-dimensional model of bargaining

We prove uniqueness of stationary equilibria in a one-dimensional model of bargaining with quadratic utilities, for an arbitrary common discount factor. For general concave utilities, we prove existence and uniqueness of a “minimal” stationary equilibrium and of a “maximal” stationary equilibrium. We provide an example of multiple stationary equilibria with concave (nonquadratic) utilities.     

A three-player dynamic majoritarian bargaining game

We analyze an infinitely repeated divide-the-dollar bargaining game with an endogenous reversion point. In each period a new dollar is divided among three legislators according to the proposal of a randomly recognized member—if a majority prefer so—or according to previous period's allocation otherwise. Although current existence theorems for Markovian equilibria do not apply for this dynamic game, we fully characterize a Markov equilibrium. The equilibrium is such that irrespective of the discount factor or the initial division of the dollar, the proposer eventually extracts the whole dollar in all periods. We also show that proposal strategies are weakly continuous in the status quo that equilibrium expected utility is not quasi-concave, and the correspondence of voters’ acceptance set (the set of allocations weakly preferred over the status quo) fails lower hemicontinuity.     

Credulity, lies, and costly talk

This paper studies a model of strategic communication by an informed and upwardly biased sender to one or more receivers. Applications include situations in which (i) it is costly for the sender to misrepresent information, due to legal, technological, or moral constraints, or (ii) receivers may be credulous and blindly believe the sender's recommendation. In contrast to the predictions obtained in the benchmark cheap talk model, our model admits a fully separating equilibrium, provided that the state space is unbounded above. The language used in equilibrium is inflated and naive receivers are deceived.     

Coordination and bargaining power in contracting with externalities

Building on Genicot and Ray [G. Genicot, D. Ray, Contracts and externalities: How things fall apart, J. Econ. Theory 131 (2006) 71-100] we develop a model of non-cooperative bargaining that combines the two main approaches in the literature of contracting with externalities: the offer game (in which the principal makes simultaneous offers to the agents) and the bidding game (in which the agents make simultaneous offers to the principal). Allowing for agent coordination, we show that the outcome of our bargaining procedure may differ remarkably from those of the offer and the bidding games. In particular, we find that bargaining can break agents' coordination and that the principal's payoff can be decreasing in his own bargaining power.     

Optimal mechanism design with resale via bargaining

In this paper, we examine the optimal mechanism design of selling an indivisible object to one regular buyer and one publicly known buyer, where inter-buyer resale cannot be prohibited. The resale market is modeled as a stochastic ultimatum bargaining game between the two buyers. We fully characterize an optimal mechanism under general conditions. Surprisingly, in this optimal mechanism, the seller never allocates the object to the regular buyer regardless of his bargaining power in the resale market. The seller sells only to the publicly known buyer, and reveals no additional information to the resale market. The possibility of resale causes the seller to sometimes hold back the object, which under our setup is never optimal if resale is prohibited. We find that the seller?s revenue is increasing in the publicly known buyer?s bargaining power in the resale market. When the publicly known buyer has full bargaining power, Myerson?s optimal revenue is achieved; when the publicly known buyer has no bargaining power, a conditionally efficient mechanism prevails.     

Trees and extensive forms

This paper addresses the question of what it takes to obtain a well-defined extensive form game. Without relying on simplifying finiteness or discreteness assumptions, we characterize the class of game trees for which all pure strategy combinations induce unique outcomes. The generality of the set-up covers “exotic” cases, like stochastic games or decision problems in continuous time (differential games). We find that the latter class, though a well-defined problem, fails this test.     

Dynamic group formation in the repeated prisoner?s dilemma

We consider dynamic group formation in repeated n-person prisoner?s dilemma. Agreements in coalitional bargaining are self-binding in that they are supported as subgame perfect equilibria of repeated games. Individuals are allowed to renegotiate the cooperating group agreement through a process of voluntary participation. We prove that a cooperating group forms as an absorbing state of a Markov perfect equilibrium after a finite number of renegotiations if and only if the group is Pareto efficient, provided that individuals are patient. The cooperating group can only expand.     

Optimal voting schemes with costly information acquisition

A group of individuals with identical preferences must make a decision under uncertainty about which decision is best. Before the decision is made, each agent can privately acquire a costly and imperfect signal. We discuss how to design a mechanism for eliciting and aggregating the collected information so as to maximize ex-ante social welfare.We first show that, of all mechanisms, a sequential one is optimal and works as follows. At random, one agent at a time is selected to acquire information and report the resulting signal. Agents are informed of neither their position in the sequence nor of other reports. Acquiring information when called upon and reporting truthfully is an equilibrium.We next characterize the ex-ante optimal scheme among all ex-post efficient mechanisms. In this mechanism, a decision is made when the precision of the posterior exceeds a cut-off that decreases with each additional report. The restriction to ex-post efficiency is shown to be without loss when the available signals are sufficiently imprecise. On the other hand, ex-post efficient mechanisms are shown to be suboptimal when the cost of information acquisition is sufficiently small.     

Search theory,competitive equilibrium,and the Nash bargaining solution

We investigate a canonical search-theoretic model without entry. Two agents are randomly matched with a long side being rationed. The matched agents face a pair of randomly drawn non-transferable payoffs, and then choose whether or not to form a partnership subject to a small probability of exogenous break down. As this probability and friction vanish, the Nash bargaining solution emerges as the unique undominated strategy equilibrium outcome if the mass of each party is the same. If the size of one party is larger than the other, the short side extracts the entire surplus, a sharp contrast to Rubinstein and Wolinsky (1985) [16].     

Between-game rule learning in dissimilar symmetric normal-form games

Rule learning posits that decision makers, rather than choosing over actions, choose over behavioral rules with different levels of sophistication. Rules are reinforced over time based on their historically observed payoffs in a given game. Past works on rule learning have shown that when playing a single game over a number of rounds, players can learn to form sophisticated beliefs about others. Here we are interested in learning that occurs between games where the set of actions is not directly comparable from one game to the next. We study a sequence of ten thrice-played dissimilar games. Using experimental data, we find that our rule learning model captures the ability of players to learn to reason across games. However, this learning appears different from within-game rule learning as previously documented. The main adjustment in sophistication occurs by switching from non-belief-based strategies to belief-based strategies. The sophistication of the beliefs themselves increases only slightly over time.     

Information structures with unawareness

I construct a state space model with unawareness following [R.J. Aumann, Agreeing to disagree, Ann. Stat. 76 (1976) 1236-1239]. Dekel et al. [E. Dekel, B.L. Lipman, A. Rustichini, Standard state-space models preclude unawareness, Econometrica 66 (1998) 159-173] show that standard state space models are incapable of representing unawareness. The model circumvents the impossibility result by endowing the agent with a subjective state space that differs from the full state space when he has the unawareness problem. Information is modeled as a pair, consisting of both factual information and awareness information. The model preserves the central properties of the standard information partition model.     

Would rational voters acquire costly information?

We analyze an election in which voters are uncertain about which of two alternatives is better for them. Voters can acquire some costly information about the alternatives. In agreement with Downs's rational ignorance hypothesis, individual investment in political information declines to zero as the number of voters increases. However, if the marginal cost of information is near zero for nearly irrelevant information, there is a sequence of equilibria such that the election outcome is likely to correspond to the interests of the majority for arbitrarily large numbers of voters. Thus, “rationally ignorant” voters are consistent with a well-informed electorate.     

Contest success functions

Summary Tournaments, conflict, and rent-seeking have been modelled as contests in which participants exert effort to increase their probability of winning a prize. A Contest Success Function (CSF) provides each player's probability of winning as a function of all players' efforts. In this paper the additive CSF employed in most contests is axiomatized, with an independence from irrelevant alternatives property as the key axiom. Two frequently used functional forms are also axiomatized: one in which winning probabilities depend on the ratio of players' efforts and the other in which winning probabilities depend on the difference in efforts.I am grateful to three anonymous referees, Jack Hirshleifer, Guillermo Owen, and especially Duncan Luce for their helpful comments, and to the UC Irvine Academic Senate and the NSF for financial support.     

Bayesian coalitional rationalizability

In this paper we extend Ambrus's [A. Ambrus, Coalitional rationalizability, Quart. J. Econ. 121 (2006) 903-929] concept of “coalitional rationalizability (c-rationalizability)” to situations where, in seeking mutually beneficial interests, players in groups (i) make use of Bayes' rule in expectation calculations and (ii) contemplate various deviations, i.e., the validity of deviation is checked against any arbitrary sets of strategies, and not just only against restricted subsets of strategies. We offer an alternative notion of Bayesian c-rationalizability suitable for such complex social interactions. We show that Bayesian c-rationalizability possesses nice properties similar to those of conventional rationalizability.     

One-dimensional bargaining with Markov recognition probabilities

We study a process of bargaining over alternatives represented by points in the unit interval. The paper focuses on the asymptotic behavior of the subgame perfect equilibrium in stationary strategies as the continuation probability approaches one. We give a complete characterization of the limit of the equilibrium proposals as the generalized fixed point of the decumulative distribution of the players' ideal points as induced by the recognition probabilities. In contrast to the existing literature, we find no general relationship between the limit equilibrium proposals and either the Nash bargaining solution or the median voter outcome.     

On the learnability of majority rule

We establish how large a sample of past decisions is required to predict future decisions of a committee with few members. The committee uses majority rule to choose between pairs of alternatives. Each member's vote is derived from a linear ordering over all the alternatives. We prove that there are cases in which an observer cannot predict precisely any decision of a committee based on its past decisions. Nonetheless, approximate prediction is possible after observing relatively few random past decisions.     

Correlated equilibria and local interactions

Summary This paper shows that Nash equilibria of a local-interaction game are equivalent to correlated equilibria of the underlying game.This work was done while George Mailath and Larry Samuelson were visiting the University of Bonn, whose hospitality is gratefully acknowledged. We thank Ken Binmore for helpful discussions. This is a revision of Section 2 of Mailath, Samuelson, and Shaked [6]. Financial support from the National Science Foundation and the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn, is gratefully acknowledged.     

Political bargaining in a federation: Buchanan meets Coase

In recent decades, the issues of federalism and political integration have gained prominence in public debate as well as in the academic realm. A frequently made point is that allowing free secession may protect the minority's interests, thus providing it with an incentive to enter the federation. This paper explicitly considers the political process in the federation arguing that the option to secede may distort the political choices made by the individual regions to improve their bargaining positions. As a result, the allocation of resources in the federation could well end up being inefficient and unattractive for the minority region. In contrast, limiting the secession possibilities by requiring the consent of a majority of voters through a regional referendum, rather than leaving it to the discretion of the legislature, restores efficiency.     

The strategy of manipulating joint decision-making

We study a model of strategic persuasion based on the theory of cheap talk, in which a better-informed agent manipulates two decision-makers’ joint decision on alternative proposals. With the heterogeneity of two decision-makers’ value of the outside option, only the decision-maker with the better outside option is critical in determining whether communication is truthful, overselling, or ineffective.