Showing papers in "International Journal of Game Theory in 2005"

College admissions with affirmative action

Abstract: This paper first shows that when colleges' preferences are substitutable there does not exist any stable matching mechanism that makes truthful revelation of preferences a dominant strategy for every student. The paper introduces student types and captures colleges' preferences for affirmative action via type-specific quotas: A college always prefers a set of students that respects its type-specific quotas to another set that violates them. Then it shows that the student-applying deferred acceptance mechanism makes truthful revelation of preferences a dominant strategy for every student if each college's preferences satisfy responsiveness over acceptable sets of students that respect its type-specific quotas. These results have direct policy implications in several entry-level labor markets (Roth 1991). Furthermore, a fairness notion and the related incentive theory developed here is applied to controlled choice in the context of public school choice by Abdulkadiroglu and Sonmez (2003).

A characterization of the position value

Abstract: We characterize the position value for arbitrary communication situations. The two properties involved in the characterization are component efficiency, which is standard, and balanced link contributions, which is in the same spirit as balanced contributions. Since the Myerson value can be characterized by component efficiency and balanced contributions a comparison between the two allocation rules based on characterizing properties can be made.

A short and constructive proof of Tarski’s fixed-point theorem

Abstract: I give short and constructive proofs of Tarski’s fixed-point theorem, and of Zhou’s extension of Tarski’s fixed-point theorem to set-valued maps.

Ranking participants in generalized tournaments

Abstract: Consider the problem of ranking social alternatives based on the number of voters that prefer one alternative to the other Or consider the problem of ranking chess players by their past performance A wide variety of ranking methods have been proposed to deal with these problems Using six independent axioms, we characterize the fair-bets ranking method proposed by Daniels [4] and Moon and Pullman [14]

An experimental test of alternative models of bidding in ascending auctions

Abstract: The theory and behavior of the clock version of the ascending auction has been well understood for at least 20 years. The more widely used oral outcry version of the ascending auction that allows bidders to submit their own bids has been the subject of some recent controversy mostly in regard to whether or not jump bidding, i.e. bidders submitting bids higher than required by the auctioneer, should be allowed. Isaac, Salmon & Zillante (2005) shows that the standard equilibrium for the clock auction does not apply to the non-clock format and constructs an equilibrium bid function intended to match with field data on ascending auctions. In this study, we will use economic experiments to provide a direct empirical test of that model while simultaneously providing empirical evidence to resolve the policy disputes centered around the place of jump bidding in ascending auctions.

On the survival of some unstable two-sided matching mechanisms

Abstract: In the 1960s, three types of matching mechanisms were adopted in regional entry-level British medical labor markets to prevent unraveling of contract dates. One of these categories of matching mechanisms failed to prevent unraveling. Roth (1991) showed the instability of that failing category. One of the surviving categories was unstable as well, and Roth concluded that features of the environments of these mechanisms are responsible for their survival. However, Unver (2001) demonstrated that the successful yet unstable mechanisms performed better in preventing unraveling than the unsuccessful and unstable category in an artificial-adaptive-agent-based economy. In this paper, we conduct a human subject experiment in addition to short- and long-run artificial agent simulations to understand this puzzle. We find that both the unsuccessful and unstable mechanism and the successful and unstable mechanism perform poorly in preventing unraveling in the experiment and in short-run simulations, while long-run simulations support the previous Unver finding.

Strongly essential coalitions and the nucleolus of peer group games

Abstract: Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespecified collection of size polynomial in the number of players. We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore. As an application, we consider peer group games, and show that they admit at most 2n−1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2n−1. We propose an algorithm that computes the nucleolus of an n-player peer group game in time directly from the data of the underlying peer group situation.

Farsighted Stability in an n-Person Prisoner’s Dilemma

Abstract: Farsighted stability is examined in an n-person prisoner’s dilemma. It is shown that any individually rational and Pareto efficient outcome is a farsighted stable set and no other farsighted stable set exists; it is further shown that the largest consistent set consists of all individually rational outcomes.

The core cover in relation to the nucleolus and the Weber set

Abstract: The class of games for which the core coincides with the core cover (compromise stable games) is characterized. Moreover, an easy explicit formula for the nucleolus of this class of games is developed, using an approach based on bankruptcy problems. Also, the class of convex and compromise stable games is characterized. The relation between the core cover and the Weber set is studied and it is proved that under a weak condition their intersection is nonempty.

On the existence of equilibria in discontinuous games: three counterexamples

Abstract: We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy ɛ-equilibria for all ɛ>0. We show by examples that there are: 1. quasiconcave, payoff secure games without pure strategy ɛ-equilibria for small enough ɛ>0 (and hence, without pure strategy Nash equilibria), 2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ɛ-equilibria for small enough ɛ>0, and 3. payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe [6], also shows that non-quasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.

The Nash rationing problem

Abstract: This paper studies the problem of allocating utility losses among n agents with cardinal non-comparable utility functions. This problem is referred to as the Nash rationing problem, as it can be regarded as the translation of the Nash bargaining problem to a rationing scenario. We show that there is no single-valued solution satisfying the obvious reformulation of Nash’s axioms, nor a multivalued solution satisfying a certain extension of these axioms. However, there is a multivalued solution that is characterised by an appropriate extension of the axioms. We thus call this mapping the Nash rationing solution. It associates with each rationing problem the set of points that maximises a weighted sum of utilities, in which weights are chosen so that all agents’ weighted losses are equal.

Improvement dynamics in games with strategic complementarities

Abstract: In a finite game with strategic complementarities, every strategy profile is connected to a Nash equilibrium with a finite individual improvement path. If, additionally, the strategies are scalar, then every strategy profile is connected to a Nash equilibrium with a finite individual best response improvement path.

An axiomatization of the consistent non-transferable utility value

Abstract: The Maschler–Owen consistent value for non-transferable utility games is axiomatized, by means of a marginality axiom.

The Böhm–Bawerk horse market: a cooperative analysis

Abstract: Single–valued solutions for the case of two-sided market games without product differentiation, also known as Bohm–Bawerk horse market games, are analyzed. The nucleolus is proved to coincide with the τ value, and is thus the midpoint of the core. The Shapley value is in the core only if the game is a square glove market, and in this case also coincides with the two aforementioned solutions.

A reinforcement learning process in extensive form games

Abstract: The CPR (“cumulative proportional reinforcement”) learning rule stipulates that an agent chooses a move with a probability proportional to the cumulative payoff she obtained in the past with that move. Previously considered for strategies in normal form games (Laslier, Topol and Walliser, Games and Econ. Behav., 2001), the CPR rule is here adapted for actions in perfect information extensive form games. The paper shows that the action-based CPR process converges with probability one to the (unique) subgame perfect equilibrium.

Values for strategic games in which players cooperate

Abstract: In this paper we propose a new method to associate a coalitional game with each strategic game. The method is based on the lower value of finite two-player zero-sum games. We axiomatically characterize this new method, as well as the method that was described in Von Neumann and Morgenstern (1944). As an intermediate step, we provide axiomatic characterizations of the value and the lower value of matrix games and finite two-player zero-sum games, respectively.

A characterization of plurality-like rules based on non-manipulability, restricted efficiency, and anonymity

Abstract: A society needs to decide which issues (laws, public projects, public facilities, etc.) in an agenda to accept. The decision can be any subset of the agenda but must reflect the preferences of its members, which are assumed to be “separable weak orderings”. We characterize a family of plurality-like rules based on strategy-proofness, restricted efficiency, anonymity, and two weak axioms pertaining to the variable agenda feature of our model, called dummy independence and division indifference. We also characterize a wide spectrum of rules dropping anonymity or restricted efficiency.

Stable cores of large games

Abstract: We give general conditions, based on the largeness of the core, under which cores of exact TU games are their unique von Neumann-Morgenstern stable sets. We show that this condition is satisfied by convex games and by nonatomic exact market games. In this way, we extend and unify earlier results existing in literature. Under some additional conditions we also prove the equivalence between the core and the Mas-Colell bargaining set.

An axiomatization of minimal curb sets

Abstract: Norde et al.[Games Econ.Behav. 12 (1996) 219] proved that none of the equilibrium concepts in the literature on equilibrium selection in finite strategic games satisfying existence is consistent.A transition to set-valued solution concepts overcomes the inconsistency problem: there is a multiplicity of consistent set-valued solution concepts that satisfy nonemptiness and recommend utility maximization in one-player games.The minimal curb sets of Basu and Weibull [Econ.Letters 36 (1991) 141] constitute one such solution concept; this solution concept is axiomatized in this article.

On the non-emptiness of the fuzzy core

Abstract: The seminal contribution of Debreu and Scarf (Int Econ Rev 4:235–246, 1963) connects the two concepts of core and competitive equilibrium in exchange economies. In effect, their core-equilibrium equivalence result states that, when the set of economic agents is replicated, the set of core allocations of the replica economy shrinks to the set of competitive allocations. Florenzano (J Math Anal Appl 153:18–36, 1990) defines the fuzzy core as the set of allocations which cannot be blocked by any coalition with an arbitrary rate of participation and then shows the asymptotic limit of cores of replica economies coincides with the fuzzy core. In this note, we provide an elementary proof of the non-emptiness of the fuzzy core for an exchange economy. Hence, in motivation, our result is similar to the contribution of Vohra (On Scarf’s theorem on the non-emptiness of the core: a direct proof through Kakutani’s fixed point theorem. Brown University Working Paper, 1987) and Shapley and Vohra (Econ Theory 1:108–116, 1991) for the core. Unlike the classical Debreu–Scarf limit theorem (Debreu and Scarf in Int Econ Rev 4:235–246, 1963) and its numerous extensions our result does not require any asymptotic intersection—or limit—of the set of core allocations of replica economies.

Stability of pure strategy sampling equilibria

Abstract: The idea of procedurally rational players was introduced in [4]. Among other procedures, the sampling procedure was proposed. The resulting equilibrium was called a sampling equilibrium. Evolutionary approach to the notion of a sampling equilibrium was developed in [7], where some instability results were proved. Using the concept of first-order stochastic dominance, we introduce the notion of a superior strategy and prove a stability result.

Compensatory Transfers in Two-Player Decision Problems

Abstract: This paper presents an axiomatic characterization of a family of solutions to two-player quasi-linear social choice problems. In these problems the players select a single action from a set available to them. They may also transfer money between themselves. The solutions form a one-parameter family, where the parameter is a non-negative number, t. The solutions can be interpreted as follows: Any efficient action can be selected. Based on this action, compute for each player a “best claim for compensation”. A claim for compensation is the difference between the value of an alternative action and the selected efficient action, minus a penalty proportional to the extent to which the alternative action is inefficient. The coefficient of proportionality of this penalty is t. The best claim for compensation for a player is the maximum of this computed claim over all possible alternative actions. The solution, at the parameter value t, is to implement the chosen efficient action and make a monetary transfer equal to the average of these two best claims. The characterization relies on three main axioms. The paper presents and justifies these axioms and compares them to related conditions used in other bargaining contexts. In Nash Bargaining Theory, the axioms analogous to these three are in conflict with each other. In contrast, in the quasi-linear social choice setting of this paper, all three conditions can be satisfied simultaneously.

Multi-Utilitarianism in Two-Agent Quasilinear Social Choice

Abstract: We introduce a new class of rules for resolving quasilinear social choice problems. These rules extend those of Green. We call such rules multi-utilitarian rules. Each multi-utilitarian rule is associated with a probability measure over the set of weighted utilitarian rules, and is derived as the expectation of this probability. These rules are characterized by the axioms efficiency, translation invariance, monotonicity, continuity, and additivity. By adding recursive invariance, we obtain a class of asymmetric rules generalizing those Green characterizes. A multi-utilitarian rule satisfying strong monotonicity has an associated probability measure with full support.

A dynamic approach to the Shapley value based on associated games

Abstract: We propose a dynamic process leading to the Shapley value of TU games or any solution satisfying Inessential Game (IG) and Continuity (CONT), based on a modified version of Hamiache's notion of an associated game.

The modiclus and core stability

Abstract: The modiclus, a relative of the prenucleolus, assigns a singleton to any cooperative TU game. We show that the modiclus selects a member of the core for any exact orthogonal game and for any assignment game that has a stable core. Moreover, by means of an example we show that there is an exact TU game with a stable core that does not contain the modiclus.

Stochastic dominance equilibria in two-person noncooperative games

Abstract: Two-person noncooperative games with finitely many pure strategies are considered, in which the players have linear orderings over sure outcomes but incomplete preferences over probability distributions resulting from mixed strategies. These probability distributions are evaluated according to t-degree stochastic dominance. A t-best reply is a strategy that induces a t-degree stochastically undominated distribution, and a t-equilibrium is a pair of t-best replies. The paper provides a characterization and an existence proof of t-equilibria in terms of representing utility functions, and shows that for large t behavior converges to a form of max–min play. Specifically, increased aversion to bad outcomes makes each player put all weight on a strategy that maximizes the worst outcome for the opponent, within the supports of the strategies in the limiting sequence of t-equilibria.

Implementation of the Walrasian Correspondence: The Boundary Problem

Abstract: Consider exchange economies in which preferences are continuous, convex and strongly monotonic. It is well known that the Walrasian correspondence is not Nash implementable. Maskin monotonicity (Maskin, 1999) is violated for allocations at the boundary of the feasible set. We derive an impossibility result showing that it is in fact not implementable in any solution concept. Next, we construct a sequential mechanism based on price-allocation announcements that fits the very description of Walrasian Equilibrium. Imposing an additional domain restriction, we show that it fully implements the Walrasian correspondence in subgame perfect and strong subgame perfect equilibrium. We thus take care of the boundary problem that was prominent in the Nash implementation literature.

Bargaining sets in exchange economies with indivisibilities and money

Abstract: This paper introduces and investigates the semireactive bargaining set (Sudholter and Potters (2001)) and the reactive bargaining set (Granot (1994)), which are originally solution concepts for TU-games, in economies in which agents exchange indivisible goods and one perfectly divisible good (money). Under the assumptions that the preferences of the agents are quasi-linear and the endowments satisfy the Total Abundance condition, a condition on the amounts of money agents initially have, it is shown that the (semi)reactive bargaining set is nonempty. In addition, we prove that in such an economy the (semi)reactive bargaining set and the (strong) core coincide if and only if the (semi)reactive bargaining set and the core of the underlying TU-game coincide.

Nonsymmetric values under Hart-Mass-Colell consistency

Abstract: Let f be a single valued solution for cooperative TU games that satisfies inessential game property, efficiency, Hart Mas-Colell consistency and for two person games is strictly monotonic and individually unbounded. Then there exists a family of strictly increasing functions associated with players that completely determines f. For two person games, both players have equal differences between their functions at the solution point and at the values of characteristic function of their singletons. This solution for two person games is uniquely extended to n person games due to consistency and efficiency. The extension uses the potential with respect to the family of functions and generalizes potentials introduced by Hart and Mas Colell [6]. The weighted Shapley values, the proportional value described by Ortmann [11], and new values generated by power functions are among these solutions.

Equilibrium with strategy-dependent trembles

Abstract: Trembling-hand arguments, introduced by Selten, where players make errors in implementing a strategy, are a successful and intuitively plausible approach to eliminating some equilibria. Typically, these errors are independent of the strategy the player aims to implement. In many contexts, however, when a player aims at one strategy, some errors are more likely than if a different strategy is the target. We introduce the notion of strategy-dependent trembles where each strategy intended by a player implies a different probability vector of errors over other strategies. This seemingly minor but plausible change makes a major difference. The focus becomes a player's own errors rather than the errors of others. A weakly dominated strategy is not necessarily eliminated if less costly errors arise when aiming at it than at the dominating strategy. Almost every Nash equilibrium in pure strategies is a strategy-dependent trembling-hand perfect equilibrium.