Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

Belief consistency and trade consistency

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agentsʼ beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinsteinʼs email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.     

Bargaining with random implementation: An experimental study

We use a laboratory experiment to study bargaining with random implementation. We modify the standard Nash demand game so that incompatible demands do not necessarily lead to the disagreement outcome. Rather, with exogenous probability q, one bargainer receives his/her demand, with the other getting the remainder. We use an asymmetric bargaining set (favouring one bargainer) and disagreement payoffs of zero, and we vary q over several values.Our results mostly support game theoryʼs directional predictions. As with conventional arbitration, we observe a strong chilling effect on bargaining for q near one: extreme demands and low agreement rates. Increasing q reinforces the gameʼs built-in asymmetry – giving the favoured player an increasingly large share of payoffs – but also raising efficiency. These effects are non-uniform: over sizable ranges, increases in q have minimal effect, but for some q, small additional increases lead to sharp changes in results.     

Null or nullifying players: The difference between the Shapley value and equal division solutions

A famous solution for cooperative transferable utility games is the Shapley value. Most axiomatic characterizations of this value use some axiom related to null players, i.e. players who contribute zero to any coalition. We show that replacing null players with nullifying players characterizes the equal division solution distributing the worth of the ‘grand coalition’ equally among all players. A player is nullifying if every coalition containing this player earns zero worth. Using invariance we provide similar characterizations of the equal surplus division solution assigning to every player its own worth, and distributing the remaining surplus equally among all players.     

How Much Common Belief Is Necessary for a Convention?

We study Bayesian coordination games in which players choose actions conditional on the realization of their respective signals. Due to differential information, the players do not have common knowledge that a particular game is being played. However, they do have common beliefs with specified probabilities concerning their environment. In our framework, any equilibrium set of rules must be simple enough so that the actions of all players are common belief with probability 1 at every state. Common belief with probability close to 1 will not do.Journal of Economic LiteratureClassification Numbers: C72, D82.     

The effect of environmental uncertainty on the tragedy of the commons

We model a common pool resource game under environmental uncertainty, where individuals in a symmetric group face the dilemma of sharing a common resource. Each player chooses a consumption level and obtains a corresponding share of that resource, but if total consumption exceeds a sustainable level then the resource deteriorates and all players are worse-off. We consider the effect of uncertainty about the sustainable resource size on the outcome of this game. Assuming a general dynamic for resource deterioration, we study the effect of increased ambiguity (i.e., uncertain probabilities pertaining to the common resourceʼs sustainable size). We show that whereas increased risk may lead to more selfish behavior (i.e., to more consumption), increased ambiguity may have the opposite effect.     

Universal type structures with unawareness

Infinite hierarchies of awareness and beliefs arise in games with unawareness, similarly to belief hierarchies in standard games. A natural question is whether each hierarchy describes the playerʼs awareness of the hierarchies of other players and beliefs over these, or whether the reasoning can continue indefinitely. This paper constructs the universal type structure with unawareness where each type has an awareness level and a belief over types. Countable hierarchies are therefore sufficient to describe all uncertainty in games with unawareness.     

Strategic information exchange

We analyze a toy class of two-player repeated games with two-sided incomplete information. In our model, two players are facing independent decision problems and each of them holds information that is potentially valuable to the other player. We study to what extent, and how, information can be exchanged at equilibrium. We show that, provided oneʼs initial information is valuable to the other player, equilibria exist at which an arbitrary amount of information is exchanged at an arbitrary high rate. The construction relies on an indefinite, reciprocated, exchange.     

Uniqueness of Stationary Equilibrium Payoffs in the Baron-Ferejohn Model

We consider a multilateral sequential bargaining model in which the players may differ in their probability of being selected as the proposer and the rate at which they discount future payoffs. For games in which agreement requires less than unanimous consent, we characterize the set of stationary subgame perfect equilibrium payoffs. With this characterization, we establish the uniqueness of the equilibrium payoffs. For the case where the players have the same discount factor, we show that the payoff to a player is nondecreasing in his probability of being selected as the proposer. For the case where the players have the same probability of being selected as the proposer, we show that the payoff to a player is nondecreasing in his discount factor. Journal of Economic Literature Classification numbers: C72, C78, D70.     

Alternating-offer games with final-offer arbitration

I analyze an alternating-offer model that integrates the common practice of having an arbitrator determine the outcomes if both playersʼ offers are rejected. I assume that the arbitrator uses final-offer arbitration (as in professional baseball). I find that if the arbitrator does not excessively favor one player, then the unique subgame-perfect equilibrium always coincides with the subgame-perfect equilibrium outcome in Rubinsteinʼs infinite-horizon alternating-offer game. However, if the arbitrator sufficiently favors the player making the initial offer, then delay occurs in equilibrium.     

Nonparametric learning rules from bandit experiments: The eyes have it!

How do people learn? We assess, in a model-free manner, subjectsʼ belief dynamics in a two-armed bandit learning experiment. A novel feature of our approach is to supplement the choice and reward data with subjectsʼ eye movements during the experiment to pin down estimates of subjectsʼ beliefs. Estimates show that subjects are more reluctant to “update down” following unsuccessful choices, than “update up” following successful choices. The profits from following the estimated learning and decision rules are smaller (by about 25% of average earnings by subjects in this experiment) than what would be obtained from a fully-rational Bayesian learning model, but comparable to the profits from alternative non-Bayesian learning models, including reinforcement learning and a simple “win-stay” choice heuristic.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Reputation with observed actions

Summary A single long-run player plays a fixed stage game (simultaneous orsequential move) against an infinite sequence of short-run opponents that play only once but can observe all past realized actions. Assuming that the probability distributions over types of long and short-run players have full support, we show that the long-run player can always establish a reputation for theStackelberg strategy and is therefore guaranteed almost his Stackelberg payoff in all Nash equilibria of the repeated game.The financial support of the National Science Foundation, Grant SES 90-7999, and of Consiglio Nazionale delle Ricerche is gratefully acknowledged. I wish to thank David Levine, Wolfgang Pesendorfer and Seminar Participants at UCLA, Universidad Carlos III de Madrid and University of Naples for useful discussions and suggestions.     

Coordination,timing and common knowledge

There is tight link between coordination and common knowledge. The role of higher order beliefs in static incomplete information games has been widely studied. In particular, information frictions break down common knowledge. A large body of literature in economics examine dynamic coordination problems when there are timing frictions, in the sense that players do not all move at once. Timing frictions in dynamic coordination games play a role that is closely analogous to information frictions in static coordination games.This paper makes explicit the role of higher order beliefs about timing in dynamic coordination games with timing frictions. An event is said to be effectively known if a player knew the event when he last had an option to change his behavior. The lack of effective common knowledge of the time drives results of dynamic coordination games.     

Aggregate comparative statics

In aggregative games, each playerʼs payoff depends on her own actions and an aggregate of the actions of all the players. Many common games in industrial organization, political economy, public economics, and macroeconomics can be cast as aggregative games. This paper provides a general and tractable framework for comparative static results in aggregative games. We focus on two classes of games: (1) aggregative games with strategic substitutes and (2) nice aggregative games, where payoff functions are continuous and concave in own strategies. We provide simple sufficient conditions under which positive shocks to individual players increase their own actions and have monotone effects on the aggregate. The results are illustrated with applications to public good provision, contests, Cournot competition and technology choices in oligopoly.     

Smooth multibidding mechanisms

We propose a smooth multibidding mechanism for environments where a group of agents have to choose one out of several projects. Our proposal is related to the multibidding mechanism (Pérez-Castrillo and Wettstein, 2002) but it is “smoother” in the sense that small variations in an agentʼs bids do not lead to dramatic changes in the probability of selecting a project. This mechanism is shown to possess several interesting properties. First, the equilibrium outcome is unique. Second, it ensures an equal sharing of the surplus that it induces. Finally, it enables reaching an outcome as close to efficiency as is desired.     

Team reasoning theory: an experimental analysis of common reason to believe and social distance

The aim of this paper is to study the role of the “common reason to believe” (Sugden in Philos Explor 16:165–181, 2003) and the reduction of social distance within the theory of team reasoning. The analysis draws on data collected through a Traveler’s Dilemma experiment. To study the role of the common reason to believe, players’ beliefs in their counterparts’ choices are elicited, and the correlation between the endorsement of team reasoning and beliefs is considered. The relation between social distance and team reasoning is analyzed by introducing a meeting between the two players after the game. We show that the common reason to believe appropriately explains the internal logic of team reasoning and that a reduction of social distance does not produce any statistically significant effect on the probability that team reasoning will be used.     

Social norms and trust among strangers

We study the development of a social norm of trust and reciprocity among a group of strangers via the “contagious strategy” as defined in Kandori (1992). Over an infinite horizon, the players anonymously and randomly meet each other and play a binary trust game. In order to provide the investors with proper incentives to follow the contagious strategy, there is a sufficient condition that requires that there exists an outside option for the investors. Moreover, the investorsʼ payoff from the outside option must converge to the payoff from trust and reciprocity as the group size goes to infinity. We show that this sufficient condition is also a necessary condition to sustain any sequential equilibrium in which the trustees adopt the contagious strategy. Our results imply that a contagious equilibrium only supports trust if trust contributes almost nothing to the investorsʼ payoffs.     

Implementation with partial provability

We extend implementation theory by allowing the social choice function to depend on more than just the preferences of the agents and allowing agents to support their statements with hard evidence. We show that a simple condition on evidence is necessary for the implementation of a social choice function f when the preferences of the agents are state independent and sufficient for implementation for any preferences (including state dependent) with at least three agents if the social planner can perform small monetary transfers beyond those called for by f. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under a boundedness assumption. For both results, transfers only occur out of equilibrium. The use of evidence enables implementation which is robust in the sense that the planner needs little information about agents? preferences or beliefs and agents need little information about each others? preferences. Our results are robust to evidence forgery at any strictly positive cost.     

Rock Paper Scissors

Rock Paper Scissors is used to resolve conflict when a compromise is not possible. Individuals playing ``games' tend to have biases they prefer to play. I show that Rock Paper Scissors with biased players results in a player with a high valuation for winning being victorious with a probability greater than a player with a low valuation. Thus, it frequently achieves the efficient outcome. Furthermore, as the benefit to victory increases for all players, victory for each player becomes equally likely. Therefore, it is often efficient for minor conflicts while it performs no better than a coin-flip for major conflicts.