Indefinite terminating points and the iterated Prisoner's Dilemma

Conclusion In conclusion, I shall indicate one consequence of (3.4). The major resultof work on infinitely iterated Prisoner's Dilemma games is that there existcooperative equilibria in such games. I have suggested above that myaccount of finitely, but indefinitely, iterated Prisoner's Dilemma gamesreflects the nature of genuine iterated Prisoner's Dilemmas more accu-rately than accounts involving infinite iterations. If my suggestion iscorrect, then one consequence of (3.4) - of there being only uncooperativeequilibria in finitely, but indefinitely, iterated games - is to call intoquestion the significance of the existence of cooperative equilibria ininfinitely iterated Prisoner's Dilemma games.Thanks to Jules Coleman, Mark Isaac, John Pollock, and Gregory Karva for helpful comments on earlier versions of this paper. Some of these results were presented as part of similar papers given at the 1986 meeting of the society for Exact Philosophy and at Bowling Green University.     

The incoherence of agreeing to disagree

The agreeing-to-disagree theorem of Aumann and the no-expected-gain-from-trade theorem of Milgrom and Stokey are reformulated under an operational definition of Bayesian rationality. Common knowledge of beliefs and preferences is achieved through transactions in a contingent claims market, and mutual expectations of Bayesian rationality are defined by the condition of joint coherence,i.e., the collective avoidance of arbitrage opportunities. The existence of a common prior distribution and the impossibility of agreeing to disagree follow from the joint coherence requirement, but the prior must be interpreted as a risk-neutral distribution: a product of probabilities and marginal utilities for money. The failure of heterogenous information to create disagreements or incentives to trade is shown to be an artifact of overlooking the potential role of trade in constructing the initial state of common knowledge.     

Axiomatic characterizations of solutions for Bayesian games

Bayesian equilibria are characterized by means of consistency and one-person rationality in combination with non-emptiness or converse consistency. Moreover, strong and coalition-proof Bayesian equilibria of extended Bayesian games are introduced and it is seen that these notions can be characterized by means of consistency, one-person rationality, a version of Pareto optimality and a modification of converse consistency. It is shown that, in case of the strong Bayesian equilibrium correspondence, converse consistency can be replaced by non-emptiness. As examples we treat Bayesian potential games and Bayesian congestion games.     

The epistemic structure of a theory of a game

This paper is a contribution to the systematic study of alternative axiom-sets for theories of (normal-form, complete-information) games. It provides an introduction to epistemic logic, describes a formulation in epistemic logic of the structure of a theory of a game (the broad theory of that game), and applies methods of epistemic logic to define strategies for dealing with two disturbing features of game theory, its hyperrationality assumptions and its indeterminacy. The analysis of these problems is conducted in terms of two principles which impregnate much game theory, Cleverness and Cloisteredness (the principles that players know respectively all, and only, the logical consequences of their assumed knowledge). Broad theories allow us to formulate and revise these principles despite their metatheoretical character. It is shown how Cleverness may be weakened by using logics which restrict the Rule of Epistemization, and Cloisteredness by using default logic or autoepistemic logic; the latter is used to characterize Nash equilibrium beliefs as parts of certain autoepistemic extensions of players' knowledge bases, but these particular extensions are rejected as ungrounded.I am grateful for most helpful comments to Robin Cubitt, Joe Halpern, Ernest Geffner, Philippe Mongin, David Squires, Elias Thijsse and Tim Williamson.     

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.     

A theory of forward induction in finitely repeated games

A forward induction solution for finitely repeated games with complete information is developed. This notion is motivated in terms of its implications on the way deviations affect the opponents' expectations about the future behavior of the deviating player. We argue that the inability of the notion of perfect equilibrium to take account of forward induction is a key factor responsible for a number of difficulties encountered in the use of perfect equilibria in repeated games. It is then shown that the solution proposed in this paper remedies some of these problems in the study of three important classes of repeated games: (i) finitely repeated coordination games; (ii) repeated games where one long-term player plays a sequence of short-term players; (iii) repeated battle of the sexes games.     

Arbitrage,rationality, and equilibrium

No-arbitrage is the fundamental principle of economic rationality which unifies normative decision theory, game theory, and market theory. In economic environments where money is available as a medium of measurement and exchange, no-arbitrage is synonymous with subjective expected utility maximization in personal decisions, competitive equilibria in capital markets and exchange economies, and correlated equilibria in noncooperative games. The arbitrage principle directly characterizes rationality at the market level; the appearance of deliberate optimization by individual agents is a consequence of their adaptation to the market. Concepts of equilibrium behavior in games and markets can thus be reconciled with the ideas that individual rationality is bounded, that agents use evolutionarily-shaped decision rules rather than numerical optimization algorithms, and that personal probabilities and utilities are inseparable and to some extent indeterminate. Risk-neutral probability distributions, interpretable as products of probabilities and marginal utilities, play a central role as observable quantities in economic systems.     

Impossibility theorems for normal form games

Two recent papers (Cubitt and Sugden, 1994; Samuelson, 1992) have established impossibility results which cast doubt on the coherence of the assumption of common knowledge of rationality'. It is shown that the Cubitt–Sugden result is the more powerful of the two impossibilities. Second, it is proved that the existence of a quasi-strict equilibrium is sufficient to construct sets which satisfy the Cubitt–Sugden axioms. This fact is used to establish that their impossibility result cannot arise in 2-player games. Finally, it is shown that if a weak symmetry postulate is added, a new impossibility result arises for this class of games.     

Dutch books and agent rationality

According to the Dutch Book Argument (DBA), if an agent's subjective probabilities fail to satisfy the axioms of the probability calculus and so make the agent vulnerable to a Dutch Book, the agent's subjective probabilities are incoherent and the agent is therefore irrational. Critics of DBA have argued, however, that probabilistic incoherence is compatible with various kinds of rationality – logico-semantic, epistemic, instrumental and prudential. In this paper, I provide an interpretation of DBA on which it is true that probabilistic incoherence entails agent irrationality. Articulating this interpretation requires the specification of some of the connections among the varieties of rationality. Once this is done, it becomes possible to vindicate a modest version of DBA.     

On the logic of common belief and common knowledge

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge — whether individual or common — is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought — onlymonotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the infinitary nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they aredecidable in the logician's sense.     

When learning meets salience

Behavior in one-shot coordination games with common knowledge labels can be described by theories of salience and focal points. Behavior in repeated games, including coordination games, can be explained by theories of learning. This paper considers games in which both theories apply, repeated coordination games with common knowledge labels. The research question asks how players combine the two sources of information—salience and the history of play—when making their choices. We specifically ask whether salience, normally considered as a one-shot strategy, continues to influence players’ actions beyond the first round, even while the player might learn from the history of play. We explore two possible mechanisms for such a continuing effect of salience: via an influence on prior beliefs, and/or via a bias, given beliefs. Regression analysis of individual-level choices shows that salience, normally considered only in the context of one-shot games, does exert a lasting effect, with the precise mechanism depending on the details of the game.     

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).     

A Critique of an Epistemic Intellectual Culture: Cartesianism,Normativism and Modern Crises

The so‐called epistemological turn of the Descartes‐Locke‐Kant tradition (Rorty) is a hallmark of modern philosophy. The broad family of normativism constitutes one major response to the Cartesian heritage building upon some version of the idea that human knowledge, action and sociality build fundamentally upon some form of social agreement and standards. Representationalism and the Cartesian picture more generally have been challenged by normativists but this paper argues that, even where these challenges by normativism have been taken to heart, our intellectual culture remains fundamentally epistemic in certain problematic senses. Two problems are highlighted: first, normativism remains functionally Cartesian, for human action and sociality appear as processes driven by the shared understandings by competent contributors (regardless of how these are constructed naturalistically), and second, normativism is unable to account for forms of human action and sociality other than those occurring in the relatively small worlds of normatively regulated conceptual spaces of mutual access and listening. These points are illustrated by an applied discussion of the blind spots of normativist accounts of the emerging environmental and the on‐going economic crises.     

被忽略的认知运气--论葛梯尔问题的实质与德性知识论

葛梯尔问题的实质是认知运气。葛梯尔问题被提出以后，知识论者们试图寻找第四个条件来解决此问题，从而形成了以JTB+X的方式来定义知识。 X看似解决了葛梯尔问题，实际上它们都在解决认知运气的问题，同时自身也面临着认知运气的威胁。不论X的提出还是受到的质疑与批判等，都以认知运气为焦点。当JTB+X式的知识分析不能解决问题的时候，一些知识论者对知识的分析产生了质疑，从而走上了对知识的规范性解释，产生了德性知识论。不论是以格雷科为代表的强德性知识论者还是以普理查德为代表的温和的德性知识论者，都致力于认知运气的解决，正是认知运气推动了德性知识论的产生和发展。     

Effective choice in the single-shot Prisoner's Dilemma tournament

This paper discusses the results of a single-shot Prisoner's Dilemma computer tournament. In the single-shot Prisoner's Dilemma tournament each pair of players interacts only once. But players can establish and detect reputations because they know how their current opponent has behaved in previous games with other players. The results show that cooperation is worthwhile, even in single-shot games, provided the outcomes of previous games are common knowledge.Reality is a game, but you shouldn't take a game for reality(Pragmaticist Proverb)     

Stochastic Evolution of Rules for Playing Finite Normal Form Games

The evolution of boundedly rational rules for playing normal form games is studied within stationary environments of stochastically changing games. Rules are viewed as algorithms prescribing strategies for the different normal form games that arise. It is shown that many of the “folk results” of evolutionary game theory, typically obtained with a fixed game and fixed strategies, carry over to the present environments. The results are also related to some recent experiments on rules and games.      

Mutual Intention

This paper takes as its starting point the problem of characterizing, in a precise way, situations in which two people collaborate to achieve a common goal. It is suggested that collaboration is normally based on an apparently paradoxical state of mind which I call "mutual intention". Mutual intention is a concept belonging to the same family as Lewis's and Schiffer's "mutual knowledge". These concepts have the paradoxical feature that they require, for their definition, an infinite series of propositions of the form X knows (or intends, or assumes) p, where X is a single agent and p is a proposition. The source of these infinite series is traced, and it is shown that they can be represented in a plausible and enlightening way by means of a recursive notation. Finally, three applications of the concept of "mutual intention" are given: (i) in the semantic analysis of certain sentences with plural subjects; (ii) in the analysis of agreement and related speech acts; and (iii) in the clarification of the phenomenon of "implicit agreement".     

Doxastic Conditions for Backward Induction

The problem of finding sufficient doxastic conditions for backward induction in games of perfect information is analyzed in a syntactic framework with subjunctive conditionals. This allows to describe the structure of the game by a logical formula and consequently to treat beliefs about this structure in the same way as beliefs about rationality. A backward induction and a non-Nash equilibrium result based on higher level belief in rationality and the structure of the game are derived.     

Learning to cooperate with Pavlov an adaptive strategy for the iterated Prisoner's Dilemma with noise

Conflict of interest may be modeled, heuristically, by the iterated Prisoner's Dilemma game. Although several researchers have shown that the Tit-For-Tat strategy can encourage the evolution of cooperation, this strategy can never outscore any opponent and it does poorly against its clone in a noisy environment. Here we examine the family of Pavlovian strategies which adapts its play by positive and negative conditioning, much as many animals do. Mutual cooperation will evolve in a contest with Pavlov against a wide variety of opponents and in particular against its clone. And the strategy is quite stable in a noisy environment. Although this strategy cooperates and retaliates, as does Tit-For-Tat, it is not forgiving; Pavlov will exploit altruistic strategies until he is punished by mutual defection. Moreover, Pavlovian strategies are natural models for many real life conflict-of-interest encounters as well as human and computer simulations.