Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.     

Bargaining over managerial contracts in delegation games: managerial power,contract disclosure and cartel behavior

Modern corporate governance codes include clauses requiring the disclosure of managerial compensation. Such codes have been installed to protect shareholders' interests. In this paper, we explore the impact of such disclosure on consumer welfare. We consider two‐stage delegation games in which owner‐shareholders negotiate about compensation with their managers in the game's first stage. At the end of the first stage, the managerial compensation contract outcomes of the bargaining process are publicly announced. In the second stage, Cournot competition evolves. We prove that sales delegation generates equilibria radically different from relative performance delegation. Using classical Cournot as the benchmark, contractual bargaining over sales compensation gives tougher product market competition—and hence higher consumer surplus. The opposite holds true for relative performance delegation. Then, cartel behavior is promoted, reducing consumer surplus. Copyright © 2007 John Wiley & Sons, Ltd.     

Repeated games in continuous time as extensive form games

Extensive form games modeling continuous-time decisions are plagued with problems of nonexistence and nonuniqueness of outcomes. As a “second-best”, the literature has imposed extraneous restrictions on the strategy sets, raising questions on the nature and interpretation of the resulting formal object, and on which restrictions are appropriate. We provide a “first-best” framework, formalizing continuous-time repeated games as extensive form games incorporating natural conditions from the onset. Every strategy profile induces a unique outcome, without any restrictions on the strategy sets. Further, the unrestricted strategy sets are equivalent to a specific class of strategies in a more naïvely specified continuous-time game.     

Ambiguity in election games

We construct a model in which the ambiguity of candidates allows them to increase the number of voters to whom they appeal. We focus our analysis on two points that are central to obtain ambiguity in equilibrium: restrictions on the beliefs that candidates can induce in voters, and intensity of voters' preferences. The first is necessary for a pure strategy equilibrium to exist, while the second is necessary for ambiguity in equilibrium when there exists a Condorcet winner in the set of pure alternatives (e.g. the spatial model of electoral competition), and when candidates' only objective is to win the election. In this last case, an ambiguous candidate may offer voters with different preferences the hope that their most preferred alternative will be implemented. We also show that if there are sufficiently many candidates or parties, ambiguity will not be possible in equilibrium, but a larger set of possible policies increases the chance that at least one candidate will choose to be ambiguous in equilibrium. We would like to thank Alberto Alesina, Antonio Cabrales, Steve Coate, Olivier Compte, Tim Feddersen, Itzhak Gilboa, Joe Harrington, Michel Le Breton, Alessandro Lizzeri, George Mailath, Steve Matthews, Steve Morris, Ignacio Ortuno, Tom Palfrey, Larry Samuelson, Murat Sertel, Fernando Vega, Eyal Winter and an anonymous referee for helpful comments. The first author acknowledges financial support from DGICYT-PB 95-0983. This work was done while the first author was visiting the Center in Political Economy at Washington University, and visiting the Center for Basic Research in the Social Sciences at Harvard University. Their hospitality is gratefully acknowledged. The support of the second author's research by the National Science Foundation is also gratefully acknowledged.     

Rationalizability in continuous games

Define a continuous game to be one in which every player's strategy set is a Polish space, and the payoff function of each player is bounded and continuous. We prove that in this class of games the process of sequentially eliminating “never-best-reply” strategies terminates before or at the first uncountable ordinal, and this bound is tight. Also, we examine the connection between this process and common belief of rationality in the universal type space of Mertens and Zamir (1985).     

Survival in Cournot games

In a model that encompasses a general equilibrium framework, we consider a monopolist (a producer) with subjective beliefs that endogenously hedges against fluctuations in input prices in a complete market. We allow for entries and Cournot competition in this economy, and we study how erroneous beliefs affect long-run survival for those firms. We introduce a notion of entropy of beliefs, and we use it to characterize the class of beliefs for which the monopolist eventually disappears almost surely. When disappearance occurs, the whole market power switches to the entrant making the most accurate predictions in our sense. The class of beliefs for which survival occurs is much broader than that of perfectly competitive settings.     

Statistical disclosure in general three-dimensional tables

Statistical offices are concerned with problems of protecting confidential information when publishing data in statistical tables. One method to avoid disclosure is the method of cell suppression, in which the values of the sensitive cells in the table are suppressed from publication. To prevent the values of the sensitive cells from being calculated from the tables totals, additional suppressions are necessary. The problem of minimizing the loss of information caused by the additional suppressions is a difficult optimization problem. We present and compare the performance of some heuristics for cell suppression for general three-dimensional tables.     

Recurrent crises in global games

Global games have unique equilibria in which aggregate behavior changes sharply when an underlying random fundamental crosses some threshold. This property relies on the existence of dominance regions: all players have a highest and lowest action that, for some fundamentals, is strictly dominant. But if the fundamental follows a random walk, it eventually spends nearly all of its time in these regions: crises gradually disappear. We obtain recurring crises by adding a single large player who lacks dominance regions. We also show that in order to obtain recurring crises, one must either relax dominance regions or restrict to fundamentals that continually return to or cross over a fixed region.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Non-manipulability in Walrasian cost games

JEL classification: D41, D51 Received: 24 December 1999 / Accepted: 18 October 2001     

Bargaining power in stationary parallelogram games

A stationary variant of the repeated prisoners dilemma in which the game frontier is a parallelogram is analyzed. By using the probabilistic cheap talk concept of [3], the discount factor becomes fungible, and for a critical value of the discount factor a unique Pareto-optimal and Pareto-dominant solution can be found. The relative bargaining power of the players can be quantified in terms of the shape of the parallelogram. If the parallelogram is asymmetric, the solution results in an asymmetric allocation of payoffs. Players with more bargaining power receive a greater share of the allocation. The solution satisfies some standard bargaining axioms within the class of parallelogram games. A characterization is provided in terms of these axioms and one new axiom, weak-monotonicity, which is in the spirit of, but different from, the Kalai-Smorodinsky restricted-monotonicity axiom.Received: 15 January 2001, Accepted: 4 December 2003, We thank Wayne Shafer for interlocution and helpful comments.     

Inefficient entry order in preemption games

In a preemption game, players decide when to take an irreversible action. Delaying the action exogenously increases payoffs, but there is an early mover advantage. Riordan (1992) shows that in a preemption game with two asymmetric players, players act in decreasing order of efficiency. This provides a microfoundation to the assumption that entry in a market occurs in the order of profitability, commonly used in the empirical analysis of market entry. We provide a counterexample showing that with more than two players this intuitive result can be reversed. We present a preemption game of entry into a new market. The potential entrants are three asymmetric firms: one “efficient” firm with high post-entry profits, and two “inefficient firms”. We show that the set of parameters such that the equilibrium entry order does not reflect the efficiency ranking is nonempty, and analyse which changes in post-entry profits preserve this entry order.     

Inefficiency and complementarity in sharing games

A sharing game is a very simple device for partially reconciling an organization’s goal with the interests of its members. Each member chooses an action, bears its cost, and receives a share of the revenue which the members’ actions generate. A (pure-strategy) equilibrium of the game may be inefficient: surplus (revenue minus the sum of costs) may be less than maximal. In a previous paper, we found that for a wide class of reward functions, no one squanders at an inefficient equilibrium (spends more than at an efficient profile) if the revenue function has a complementarity property. In the present paper, we examine the “opposite” of the complementarity property (Substitutes) and we study a class of finite games where squandering equilibria indeed occur if Substitutes holds strongly enough. Squandering equilibria play a key role when one traces the effect of technological improvement on a sharing game’s surplus shortfall. We then turn to the question of choice among reward functions in a principal/agents setting. We find that if we again assume complementarity then strong conclusions can be reached about the reward functions preferred by “society”, by the players (agents), and by the principal.     

Interim correlated rationalizability in infinite games

In a Bayesian game, assume that the type space is a complete, separable metric space, the action space is a compact metric space, and the payoff functions are continuous. We show that the iterative and fixed-point definitions of interim correlated rationalizability (ICR) coincide, and ICR is non-empty-valued and upper hemicontinuous. This extends the finite-game results of Dekel et al. (2007), who introduced ICR. Our result applies, for instance, to discounted infinite-horizon dynamic games.     

Subgame-perfect equilibria in stochastic timing games

We develop a notion of subgames and the related notion of subgame-perfect equilibrium – possibly in mixed strategies – for stochastic timing games. To capture all situations that can arise in continuous-time models, it is necessary to consider stopping times as the starting dates of subgames. We generalize Fudenberg and Tirole’s (Rev. Econom. Stud. 52, 383–401, 1985) mixed-strategy extensions to make them applicable to stochastic timing games and thereby provide a sound basis for subgame-perfect equilibria of preemption games. Sufficient conditions for equilibrium existence are presented, and examples illustrate their application as well as the fact that intuitive arguments can break down in the presence of stochastic processes with jumps.     

Coalition formation in general NTU games

A general nontransferable utility (NTU) game is interpreted as a collection of pure bargaining games that can be played by individual coalitions. The threatpoints or claims points respectively, in these pure bargaining games reflect the players' opportunities outside a given coalition. We develop a solution concept for general NTU games that is consistent in the sense that the players' outside opportunities are determined by the solution to a suitably defined reduced game. For any general NTU game the solution predicts which coalitions are formed and how the payoffs are distributed among the players.     

Two-sided communication in competing mechanism games

We consider multiple-principal multiple-agent games of incomplete information. We identify a class of two-way communication mechanisms which mirror those considered in the single principal analysis of Myerson (1982). In such mechanisms, every agent truthfully reveals her type to all principals, and obeys the private recommendations she receives from each of them. We show that there is a rationale in restricting attention to this class of mechanisms: if principals use these mechanisms, there is no unilateral incentive to deviate towards more sophisticated ones. We develop three examples to analyze possible extensions and limits of our approach. The first two examples show that the restriction to direct and incentive compatible mechanisms is not sufficient to provide a complete characterization of all pure strategy equilibria. The third one shows that private recommendations play a fundamental role in competing mechanism games, suggesting that one cannot safely restrict to one-sided communication mechanisms.     

Endogenous institutions in bureaucratic compliance games

We consider a set-up where two governments have either conflicting or matching preferences on the provision of differentiated (local) goods supplied by a common monopoly bureau. We develop a two-stage game. At stage-1, the two governments decide whether or not to merge into a single institution. At stage-2, all players simultaneously and independently take their decisions in terms of production and rents, with perfect knowledge of the other players' strategies. We solve the subgame perfect Nash equilibrium of this game, and show that, if the bureau immediately updates its objective function to institutional changes, then the governments always prefer merging. However, if there is an initial bureaucratic inertia in adjusting the bureau's objective function to the institutional change, then ruling politicians may prefer decentralisation to centralisation, depending on the strategic properties of the compliance game and on their own discounting. Received: May 1999 / Accepted April 2000     

Network design in games with spillovers

How should an organization be designed in order to provide its members with minimal incentives to defect? And how does the optimal design depend on the type of strategic interaction between defectors and remaining organizational members? This paper addresses such issues in a game theoretic model of cooperation, in which an organization is formally represented by a connected network, and where gains from cooperation are given by a partition function. We show that critical structural features of the organization depend in a clear-cut way on the sign of spillovers. In particular, positive spillovers favor the adoption of dispersed and centralized forms, while negative spillovers favor cohesive and horizontal ones. Moreover, if the organizational form determines all the communication possibilities of members, a highly centralized organization—the star—emerges under positive spillovers, whereas two horizontal architectures—the circle and the complete—emerge under negative spillovers.