A Strategic Implementation of the Shapley Value for the Nested Cost‐Sharing Problem

When agents have different needs for a public facility but serving a given agent allows serving all agents with smaller needs than his without any extra cost, how should the agents divide the cost of the facility among themselves? We provide a strategic implementation of the Shapley value for this class of cost‐sharing problems. We introduce a three‐stage extensive form game that respects individual rationality and show that there is one and only one subgame‐perfect equilibrium outcome of the game. Moreover, it is the allocation assigned by the Shapley value.     

Bilateral commitment

We consider non-cooperative environments in which two players have the power to gradually and unilaterally rule out some of their actions. Formally, we embed a strategic-form game into a multi-stage game, in which players can restrict their action spaces in all but the final stage, and select among the remaining actions in the last stage. We say that an action profile is implementable by commitment if this action profile is played in the last stage of a subgame-perfect equilibrium path. We provide a complete characterization of all implementable action profiles and a simple method to find them. It turns out that the set of implementable profiles does not depend on the length of the commitment process. We show, furthermore, that commitments can have social value in the sense that in some games there are implementable action profiles that dominate all Nash equilibria of the original game.     

Feasible elimination procedures in social choice: An axiomatic characterization

Feasible elimination procedures (Peleg, 1978) play a central role in constructing social choice functions which have the following property: in the associated game form, for any preference profile there exists a strong Nash equilibrium resulting in the sincere outcome. In this paper we provide an axiomatic characterization of the social choice correspondences resulting from applying feasible elimination procedures. The axioms are anonymity, Maskin monotonicity, and independent blocking. We also show that these axioms are logically independent.     

The museum pass game and its value

We discuss a game in which service providers (e.g., museums) team up in offering a limited time access pass allowing unlimited usage of their services. In this game, a natural way to allocate the joint income from the sale of passes among the service providers is by using the Shapley value. We show that, for the particular game considered, the Shapley value takes a very intuitive and computationally simple form.     

A Theory of Endogenous Coalition Structures

Consider an environment with widespread externalities, and suppose that binding agreements can be written. We study coalition formation in such a setting. Our analysis proceeds by defining on a partition function an extensive-form bargaining game. We establish the existence of a stationary subgame perfect equilibrium for such a game. Our main results are concerned with the characterization of equilibriumcoalition structures. We develop an algorithm that generates (under certain conditions) an equilibrium coalition structure. Our characterization results are especially sharp forsymmetricpartition functions. In particular, we provide a uniqueness theorem and apply our results to a Cournot oligopoly.Journal of Economic LiteratureClassification Numbers: C71, C72, C78, D62.     

Cyclic Games: An Introduction and Some Examples

We introduce a model of a cyclic game. Designed to take advantage of the recurring nature of certain economic and social situations, a cyclic game differs from an extensive form game in that a cyclic game does not necessarily have an end. The same situations, although with different players, may be repeated infinitely often. We provide an example showing that, even though a cyclic game has, in a sense, perfect information, it may not have an equilibrium in pure strategies. We demonstrate existence of equilibrium and illustrate the application of our model to an oligopolistic industry. Journal of Economic Literature Classification Numbers: C72, C73, D43.     

Repetition, Communication, and Coordination Failure

We examine repetition as an institution that affects coordination failure in a game with and without pre-play communication. We use probit regression with random effects to test hypotheses regarding the frequency and form of coordination failure in the presence of repeated play versus one-shot games. Our results indicate that repetition without pre-play communication results in a lower frequency of coordination failure relative to one-shot game outcomes. This result is reversed when pre-play communication is allowed. Our evidence also suggests that repeated play coordination failures tend to be suboptimal Nash equilibria, whereas one-shot game coordination failures are disequilibria regardless of the presence of pre-play communication.     

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.     

Secrecy, two-sided bias and the value of evidence

We provide necessary and sufficient conditions on players' preferences for a verifiable message game to possess a separating equilibrium and/or a pooling equilibrium, as well as sufficient conditions for every equilibrium of such a game to be pooling. We also characterize all nonseparating equilibria, and use this property to show that less information might be conveyed in any equilibrium of a verifiable message game than in some equilibrium of the cheap talk game with the same parameters.     

A three way equivalence

In view of the well known core equivalence results in atomless economies, coincidence of market game equilibrium allocations with competitive allocations is tantamount to a three way equivalence between market game mechanisms, competitive equilibria and the core. Based on this idea we study equilibrium refinements of market games, which allow us to use the core equivalence machinery in order to provide an exact market game characterization of competitive equilibria.     

Order of Play, Forward Induction, and Presentation Effects in Two-Person Games

We investigate the effects of order-of-play (simultaneous, unobserved sequential and fully observed sequential play) and form of presentation (extensive vs. normal) in three simple two person games: battle-of-the-sexes with and without outside option and a three strategy game which differentiates between virtual observability (VO) and iterated elimination of dominated strategies as principles of equilibrium selection. VO predicts that knowledge of the order of play alone will affect the distribution of strategies chosen. We contrast this with the predictions of iterated elimination of dominated strategies. We report results from 1800 one-shot games conducted in 6 sessions with 120 subjects and analysed as panel data. The form of presentation strongly affects the distribution of outcomes and strategies. Information about order of play shifts the distribution of strategies away from the distribution in simultaneous play and towards the distribution in fully observed play, especially in the less complicated games presented in normal order. Order-of-play effects are less evident as complexity of the game increases. Extensive form presentation appears to induce sequential thinking even in simultaneously played games.     

Uniqueness of stationary equilibria in bargaining one-dimensional policies under (super) majority rules

We consider negotiations selecting one-dimensional policies. Individuals have instantaneous preferences represented by continuous, concave and single-peaked utility functions, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. We provide sufficient conditions that guarantee the existence of a unique stationary subgame perfect equilibrium, and we provide its explicit characterization. The uniqueness of the equilibrium permits an analysis of the set of Pareto optimal voting rules. For symmetric distributions of peaks and uniform recognition probabilities unanimity is the unanimously preferred majority rule.     

Investment dynamics with common and private values

We study a dynamic investment game with two-dimensional signals, where each firm observes its continuously distributed idiosyncratic cost of investment and a discrete signal correlated with common investment returns. We demonstrate that the one-step property holds and provide an equilibrium existence/characterization result. “Reversals” are possible, where a large number of firms investing in a given round becomes bad news about investment returns. Welfare is compared to static and rigid-timing benchmarks, and computed for large economies.     

Reciprocity, Trust, and Payoff Privacy in Extensive Form Bargaining

We report decision making in two-person extensive form game trees, using six treatments that vary matching protocol, payoffs, and payoff information. Our objective is to examine game theoretic hypotheses of decision making based on dominance and backward induction in comparison with the culturally or biologically derived hypothesis that reciprocity supports more cooperation than predicted by game theory. We find strong support for cooperation under complete information, even in single-play treatments and in games of trust, unreinforced by the prospect of punishment for defection from reciprocity. Only under private information do we observe strong support for noncooperative game theory.Journal of Economic LiteratureClassification Numbers: C78, C92.     

Equilibrium play and best response to (stated) beliefs in normal form games

We report experimental results on a series of ten one-shot two-person 3×3 normal form games with unique equilibrium in pure strategies played by non-economists. In contrast to previous experiments in which game theory predictions fail dramatically, a majority of actions taken coincided with the equilibrium prediction (70.2%) and were best-responses to subjects' stated beliefs (67.2%). In constant-sum games, 78% of actions taken were predicted by the equilibrium model, outperforming simple K-level reasoning models. We discuss how non-trivial game characteristics related to risk aversion, efficiency concerns and social preferences may affect the predictive value of different models in simple normal form games.     

Dynamics in near-potential games

We consider discrete-time learning dynamics in finite strategic form games, and show that games that are close to a potential game inherit many of the dynamical properties of potential games. We first study the evolution of the sequence of pure strategy profiles under better/best response dynamics. We show that this sequence converges to a (pure) approximate equilibrium set whose size is a function of the “distance” to a given nearby potential game. We then focus on logit response dynamics, and provide a characterization of the limiting outcome in terms of the distance of the game to a given potential game and the corresponding potential function. Finally, we turn attention to fictitious play, and establish that in near-potential games the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria, where the size of the neighborhood increases according to the distance to the set of potential games.     

Multiperson Bargaining over Two Alternatives

We present a continuous-time model of multiperson bargaining with incomplete information, where only two agreements are possible. We show that under mild informational assumptions, players behave as if they were representing the other players who prefer the same alternative, individually playing a war-of-attrition against a conglomerate player of the opposite side. We also provide a characterization of the perfect Bayesian equilibria in undominated strategies of the game, which yields a unique outcome in most cases.Journal of Economic LiteratureClassification Numbers: C72, C78, D72, D74.     

On the generic finiteness of equilibrium outcome distributions in bimatrix game forms

We provide an example of an outcome game form with two players for which there is an open set of utilities for both players such that, in each of the associated games, the set of Nash equilibria induces a continuum of outcome distributions.     

The strategic value of recall

This work studies the value of two-person zero-sum repeated games in which at least one of the players is restricted to (mixtures of) bounded recall strategies. A (pure) k-recall strategy is a strategy that relies only on the last k periods of history. This work improves previous results ( [Lehrer, 1988] and [Neyman and Okada, 2009]) on repeated games with bounded recall. We provide an explicit formula for the asymptotic value of the repeated game as a function of the one-stage game, the duration of the repeated game, and the recall of the agents.