Maximin play in completely mixed strategic games

Since the seminal paper of Nash (Proc Natl Acad Sci USA 36:48–49, 1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax) strategies. In a recent paper of Pruzhansky (Int J Game Theory 40:351–365, 2011) it was shown that under fairy general conditions maximin strategies in completely mixed games can guarantee the same expected payoff as completely mixed Nash equilibrium strategies. Based on this finding, the current paper argues that maximin strategies have important properties. For instance, maximin strategies may refine Nash equilibria in subjective mixed strategies. Further, Bayesian rationality of the players may favor maximin strategies more often than Nash equilibrium strategies. The paper concludes with several suggestions for further experimental research that may shed more light on whether maximin behavior can explain reality better than Nash equilibrium.     

Endogenous correlated equilibria in noncooperative games

Most of the results of modern game theory presuppose that the choices rational agents make in noncooperative games are probabilistically independent. In this paper I argue that there is noa priori reason for rational agents to assume probabilistic independence. I introduce a solution concept for noncooperative games called anendogenous correlated equilibrium, which generalizes the Nash equilibrium concept by dropping probabilistic independence. I contrast the endogenous correlated equilibrium with the correlated equilibrium defined by Aumann (1974, 1987). I conclude that in general the endogenous correlated equilibrium concept is a more appropriate solution concept for noncooperative game theory than the less general Nash equilibrium concept. I close by discussing the relationship between endogenous correlated equilibrium and the game solution concept calledrationalizability introduced by Bernheim (1984) and Pearce (1984).     

Equilibrium and potential in coalitional congestion games

The model of congestion games is widely used to analyze games related to traffic and communication. A central property of these games is that they are potential games and hence posses a pure Nash equilibrium. In reality, it is often the case that some players cooperatively decide on their joint action in order to maximize the coalition’s total utility. This is modeled by Coalitional Congestion Games. Typical settings include truck drivers who work for the same shipping company, or routers that belong to the same ISP. The formation of coalitions will typically imply that the resulting coalitional congestion game will no longer posses a pure Nash equilibrium. In this paper, we provide conditions under which such games are potential games and posses a pure Nash equilibrium.     

A note on the two person exchange game

The two person exchange game is analysed using core theory, the Nash solution and the competitive equilibrium. It is shown for specific utility functions that relative to the other trader, the less wealthy trader does better under the Nash solution than he would under the competitive solution. Conversely, the trader with the lower initial utility level does better under the competitive solution than he would under the Nash solution. If traders have disparate initial endowments, it is argued that outcomes should be restricted to a reduced core containing the Nash and competitive solutions as endpoints.     

An axiomatic analysis of the Nash equilibrium concept

The purpose of this paper is to analyze axiomatically the Nash equilibrium concept. The class of games under study is a (relatively large) subclass of n-person normal form games. Solutions are correspondences which associate to each game a non empty set of strategy vectors of this game. It is shown that if a solution satisfies the axioms Independence of irrelevant alternatives (IIA) and Individual rationality (IR), then all the strategy vectors in this solution are Nash equilibria. This result holds good also if IR is replaced by Strong individual monotonicity (SIM) or Weak principle of fair compromise (WPFC).     

Partial cooperation in strategic multi-sided decision situations

We consider a normal-form game in which there is a single exogenously given coalition of cooperating players that can write a binding agreement on pre-selected actions. The actions representing other dimensions of the strategy space remain under the sovereign, individual control of the players. We consider a standard extension of the Nash equilibrium concept denoted as a partial cooperative equilibrium as well as an equilibrium concept in which the coalition of cooperators has a leadership position. Existence results are stated and we identify conditions under which the various equilibrium concepts are equivalent. We apply this framework to existing models of multi-market oligopolies and international pollution abatement. In a multi-market oligopoly, typically, a merger paradox emerges in the partial cooperative equilibrium. The paradox vanishes if the cartel attains a leadership position. For international pollution abatement treaties, cooperation by a sufficiently large group of countries results in a Pareto improvement over the standard tragedy of the commons outcome described by the Nash equilibrium.     

LEXICOGRAPHIC ADDITIVITY FOR MULTI-ATTRIBUTE PREFERENCES ON FINITE SETS

Payoff dominance, a criterion for choosing between equilibrium points in games, is intuitively compelling, especially in matching games and other games of common interests, but it has not been justified from standard game-theoretic rationality assumptions. A psychological explanation of it is offered in terms of a form of reasoning that we call the Stackelberg heuristic in which players assume that their strategic thinking will be anticipated by their co-player(s). Two-person games are called Stackelberg-soluble if the players' strategies that maximize against their co-players' best replies intersect in a Nash equilibrium. Proofs are given that every game of common interests is Stackelberg-soluble, that a Stackelberg solution is always a payoff-dominant outcome, and that in every game with multiple Nash equilibria a Stackelberg solution is a payoff-dominant equilibrium point. It is argued that the Stackelberg heuristic may be justified by evidentialist reasoning.     

Games without Rules

We introduce the notion of an outcome space, in which strategic interactions are embedded. This allows us to investigate the idea that one strategic interaction might be an expanded version of another interaction. We then characterize the Nash equilibria arising in such extensions and demonstrate a folk-type theorem stating that any individually rational element of the outcome space is a Nash equilibrium.     

Who gains from information asymmetry?

This article considers an asymmetric contest with incomplete information. There are two types of players: informed and uninformed. Each player has a different ability to translate effort into performance in terms of the contest success function. While one player’s type is known to both players, the other is private information and known only to the player himself. We compare the Bayesian Nash equilibrium outcome of a one-sided private information contest to the Nash equilibrium with no private information, in which both players know the type of the other player. We show conditions under which uncertainty increases the investment of the uninformed player and the rent dissipation of the contest, while decreasing the expected net payoff of the informed player. In addition, we consider conditions under which the informed player—before knowing his own type—prefers that the uninformed player knows his type. Moreover, we show conditions for the existence/non-existence of equilibrium in a two-stage contest in which the informed player declares his type (or does not declare) in the first stage and in the second stage the two players play according to the information available to them.     

Pure-strategy Equilibria with Non-expected Utility Players

A pure-strategy equilibrium existence theorem is extended to include games with non-expected utility players. It is shown that to guarantee the existence of a Nash equilibrium in pure strategies, the linearity of preferences in the probabilities can be replaced by the weaker requirement of quasiconvexity in the probabilities.     

Are game theoretic concepts suitable negotiation support tools? From Nash equilibrium refinements toward a cognitive concept of rationality

If game theory is to be used as a negotiation support tool, it should be able to provide unambiguous recommendations for a target to aim at and for actions to reach this target. This need cannot be satisfied with the Nash equilibrium concept, based on the standard instrumental concept of rationality. These equilibria, as is well known, are generally multiple in a game. The concept of substantive or instrumental rationality has proved to be so pregnant, however, that researchers, instead of re-evaluating its use in game theory, have simply tried to design concepts related to the Nash equilibrium, but with the property of being unique in a game — i.e., they have devised ways ofselecting among Nash equilibria. These concepts have been labeledrefined Nash equilibria. The purpose of this paper is to show the following.

1. The different types of refined Nash equilibria, based on the principle of backward induction, can lead to severe contradictions within the framework itself. This makes these concepts utterly unsatisfactory and calls for a new appraisal of the reasoning process of the players.
2. The degree of confidence in the principle of backward induction depends upon the evaluation of potential deviations with respect to the extended Nash equilibrium concept used and upon the possible interpretations of such deviations by the different players. Our goal is to show that the nature of these possible interpretations reinforces the argument that a serious conceptual reappraisal is necessary.
3. Some form of forward induction should then become the real yardstick of rationality, extending Simonianprocedural rationality towards the concept ofcognitive rationality. This could open the way to a renewed game theoretic approach to negotiation support systems. Such a research program, which would be a revision of the basic game theoretic concepts, is dealt with in the end of the paper.

     

Weakly implementable social choice rules

Previous investigations have shown that a social choice function which is partially implementable must be characterized by pervasive veto power. This paper investigates how much additional latitude in the design of social choice functions, and how much relief from this vetoers result, can be achieved by examining multi-valued social choice rules and relaxing the requirement of partial implementability to a requirement that we call weak partial implementability. We find that the power structures which characterize partially implementable social choice functions, including the veto properties, also characterize weakly partially implementable social choice rules. The conclusion is that invoking multi-valuedness and implementation of appealing social choice rules in strong Nash equilibria. Our results apparently exhaust the possibilities for implementation in strong Nash equilibrium. If any implementation possibility results are to be achieved, they can apparently come only by weakening the equilibrium requirement.     

Ellsberg games

In the standard formulation of game theory, agents use mixed strategies in the form of objective and probabilistically precise devices to conceal their actions. We introduce the larger set of probabilistically imprecise devices and study the consequences for the basic results on normal form games. While Nash equilibria remain equilibria in the extended game, there arise new Ellsberg equilibria with distinct outcomes, as we illustrate by negotiation games with three players. We characterize Ellsberg equilibria in two-person conflict and coordination games. These equilibria turn out to be related to experimental deviations from Nash equilibrium play.     

On the evaluation of solution concepts

It is proposed that solution concepts for games should be evaluated in a way that is analogous to the way a logic is evaluated by a model theory for the language. A solution concept defines a set of strategy profiles, as a logic defines a set of theorems. A model theoretic analysis for a game defines a class of models, which are abstract representations of particular plays of the game. Given an appropriate definition of a model, one can show that various solution concepts are characterized by intuitively natural classes of models in the same sense that the set of theorems of a logic is characterized by a class of models of the language. Sketches of characterization results of this kind are given for rationalizability, Nash equilibrium, and for a refinement of rationalizability —strong rationalizability — that has some features of an equilibrium concept. It is shown that strong rationalizability is equivalent to Nash equilibrium in perfect information games. Extensions of the model theoretic framework that represent belief revision and that permit the characterization of other solution concepts are explored informally.     

Either with us or against us: experimental evidence on partial cartels

This paper analyzes the coordination challenge a partial cartel faces when payoff asymmetries between potential cartel insiders and potential cartel outsiders are large. We introduce two experimental treatments: a standard treatment where a complete cartel can be supported in a Nash equilibrium and a modified treatment where a complete cartel and a partial cartel can both be supported in a Nash equilibrium. To assess the role of communication both treatments are additionally run with a “chat option,” yielding four treatments in total. Our results show that subjects frequently reject the formation of partial cartels in the modified treatments. In all treatments with communication subjects are more likely to form complete cartels than partial cartels. The implications of these results are important for antitrust: payoff asymmetries between cartel members and outsiders may jeopardize the formation of partial cartels. Yet complete cartels may be formed instead, if institutional mechanisms with frequent communication are used to form cartels.

     

Risk aversion and the relationship between Nash's solution and subgame perfect equilibrium of sequential bargaining

This article presents some new, intuitive derivations of several results in the bargaining literature. These new derivations clarify the relationships among these results and allow them to be understood in a unified way. These results concern the way in which the risk posture of the bargainers affects the outcome of bargaining as predicted by Nash's (axiomatic) solution of a static bargaining model (Nash, 1950) and by the subgame perfect equilibrium of the infinite horizon sequential bargaining game analyzed by Rubinstein (1982). The analogous, experimentally testable predictions for finite horizon sequential bargaining games are also presented.     

Implementation without rationality assumptions

Hurwicz (Social Choice and public decision making. Essays in honor of Kenneth J. Arrow, Cambridge University Press, Cambridge, 1986) was the first to study an approach to implementation theory based on choice functions instead of preference relations. We argue that the solution concept used by him, the generalized Nash equilibrium, is not really compatible with the idea that individual behavior is describable by a choice function. A new solution concept that better fits the choice function framework is then introduced. Using this, we investigate what behavioral assumptions are needed for the full characterizations of Nash implementable social choice correspondences to still hold. We will show that a condition known as Property α is central.     

A Note on Implementation of Bargaining Solutions

Miyagawa (Games and Economics Behavior 41(2), 292–308 [2002]) provides a simple extensive game form that implements a large class of two-agent bargaining solutions in subgame-perfect equilibrium. This class includes all of the Nash, Kalai–Smorodinsky, and relative utilitarian solutions. This note extends Miyagawa’s result to multi-agent bargaining problems.     

地方政府债务风险规制的博弈分析

以博弈论为分析工具,文章建立了中央政府与地方政府的静态博弈支付矩阵,通过引入地方政府过度举债被中央政府监管发现后所付出的成本、地方政府债务风险被媒体曝光而导致公众的担忧等变量,在一定程度上解释了中央政府和地方政府在债务风险控制上达成一致的均衡条件;由于地方政府间在发展经济过程中存在竞争关系,因此文章考察了地方政府间举债发展的重复博弈模型,通过引入奖励变量,改变了地方政府过度举债的纳什均衡解。结论为中央政府严格监管与适当奖励的搭配使用可以把地方政府的债务风险规制在合理的水平。     

Now or Later?--Endogenous Timing of Threats

Whereas early threats are chosen before bargaining, late threats are determined after bargaining ended in conflict. Instead of exogenously imposing the timing of threats, these are derived endogenously as in indirect evolution or endogenous timing. Based on a duopoly market, we first derive the equilibrium for all possible constellations regarding the timing of threats. Our analysis surprisingly justifies the early timing of threats as suggested by Nash (1953).