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.

     

Backward Induction Is Not Robust: The Parity Problem and the Uncertainty Problem

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed.     

Belief formation in a signaling game without common prior: an experiment

Using belief elicitation, the paper investigates the process of belief formation and evolution in a signaling game in which a common prior is not induced. Both prior and posterior beliefs of Receivers about Senders’ types are elicited, as well as beliefs of Senders about Receivers’ strategies. In the experiment, subjects often start with diffuse uniform beliefs and update them in view of observations. However, the speed of updating is influenced by the strength of initial beliefs. An interesting result is that beliefs about the prior distribution of types are updated slower than posterior beliefs, which incorporate Senders’ strategies. In the medium run, for some specifications of game parameters, this leads to outcomes being significantly different from the outcomes of the game in which a common prior is induced. It is also shown that elicitation of beliefs does not considerably change the pattern of play in this game.     

The Backward Induction Argument

The backward induction argument purports to show that rational and suitably informed players will defect throughout a finite sequence of prisoner's dilemmas. It is supposed to be a useful argument for predicting how rational players will behave in a variety of interesting decision situations. Here, I lay out a set of assumptions defining a class of finite sequences of prisoner's dilemmas. Given these assumptions, I suggest how it might appear that backward induction succeeds and why it is actually fallacious. Then, I go on to consider the consequences of adopting a stronger set of assumptions. Focusing my attention on stronger sets that, like the original, obey the informedness condition, I show that any supplementation of the original set that preserves informedness does so at the expense of forcing rational participants in prisoner's dilemma situations to have unexpected beliefs, ones that threaten the usefulness of backward induction.     

Instrumental rationality and cognitive rationality

Beside instrumental rationality, concerned with consistency between available means and pursued objectives, cognitive rationality, concerned with consistency between current beliefs and detained information, is nowadays bursting in economic theory. For an individual decision, the two notions are expressed structurally in similar forms, but they heavily interact in the deliberation process, especially when the agent is searching relevant information on his environment and his own determiners. In a game framework, the players' representations are formalized in a more and more sophisticated way, and are closely interwoven with their preferences in contemporary equilibrium notions as well as in selection between corresponding equilibrium points.     

Self enforceable paths in extensive form games

This paper explores the idea of forward induction for extensive games. It interprets this idea as a general behavioral principle the technical details of which have to be worked out in each specific case. Because of its cooperative ingredient, this approach should be contrasted with the usual approaches of non-cooperative game theory which are rooted in individual rationality.     

Use of data on planned contributions and stated beliefs in the measurement of social preferences

In a series of one-shot linear public goods game, we ask subjects to report their contributions, their contribution plans for the next period, and their first-order beliefs about their present and future partner. We estimate subjects’ preferences from plan data by a finite mixture approach and compare the results with those obtained from contribution data. Controlling for beliefs, which incorporate the information about the others’ decisions, we are able to show that plans convey accurate information about subjects’ preferences and, consequently, are good predictors of their future behavior.     

The gradual decline of cooperation: Endgame effects in evolutionary game theory

The article provides an evolutionary analysis of a finitely iterated Prisoner's Dilemma. The backward induction reasoning for a breakdown of cooperation in this game is transformed to an evolutionary degradation effect. After the introduction of random variations in the strategies' population size, however, cyclical variations of cooperativeness may appear. A breakdown of cooperation is no longer inevitable. An analysis for all possible payoff relations in Prisoner's Dilemma matrices shows that only four qualitatively different dynamical flows can emerge.     

FORWARD INDUCTION IN GAMES WITH AN OUTSIDE OPTION

We provide eductive foundations for the concept of forward induction, in the class of games with an outside option. The formulation presented tries to capture in a static notion the rest point of an introspective process, achievable from some restricted preliminary beliefs. The former requisite is met by requiring the rest point to be a Nash equilibrium that yields a higher payoff than the outside option. With respect to the beliefs, we propose the Incentive Dominance Criterion. Players should consider one action more likely than another whenever the former is better than getting the outside option for more conjectures over his rival's actions. We apply this model to the case where the subgame is a coordination game with a conflict between payoff dominance and risk dominance. Our results provide support for dominance solvability, but not for Van Damme's notion of forward induction. We show how the forward induction logic helps to select the Pareto dominant equilibrium. This is the case whenever player 1's act of giving up the outside option reverses the incentive dominance relations among 1's pure actions in the subgame.     

On Stalnaker's Notion of Strong Rationalizability and Nash Equilibrium in Perfect Information Games

Counterexamples to two results by Stalnaker (Theory and Decision, 1994) are given and a corrected version of one of the two results is proved. Stalnaker's proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition strong rationalizability), then the true (or actual) strategy profile is path equivalent to a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C , the set of profiles that survive the iterative deletion of inferior profiles.     

The chain store paradox

The chain store game is a simple game in extensive form which produces an inconsistency between game theoretical reasoning and plausible human behavior. Well-informed players must be expected to disobey game theoretical recommendations.The chain store paradox throws new light on the well-known difficulties arising in connection with finite repetitions of the prisoners dilemma game. Whereas these difficulties can be resolved by the assumption of secondary utilities arising in the course of playing the game, a similar approach to the chain store paradox is less satisfactory.It is argued that the explanation of the paradox requires a limited rationality view of human decision behavior. For this purpose a three-level theory of decision making is developed, where decisions can be made on different levels of rationality. This theory explains why insight into the rational solution of a decision problem does not necessarily mean that the corresponding course of action will be taken.     

Sociotropes, Systematic Bias, and Political Failure: Reflections on the Survey of Americans and Economists on the Economy

Objectives . Economic models of politics typically make two assumptions about voters: first, their motives are egocentric, not sociotropic; second, their beliefs are rational, not subject to systematic bias. Political scientists have presented strong evidence against the first assumption (Mansbridge, 1990), but have become increasingly willing to accept the second (Page and Shapiro, 1992; Marcus and Hanson, 1993). This article tests these two assumptions, then explores the tests' broader implications. Methods . I use the Survey of Americans and Economists on the Economy to test for egocentricity of motivation and rationality of belief. Results . Both standard assumptions fail for the case where the economic approach would seemingly be most relevant: economic beliefs. Conclusions . This is not necessarily cause for greater optimism about the efficiency of democracy: sociotropic voters with biased economic beliefs are more likely to produce severe political failures than are selfish voters with rational expectations.     

宗教世俗化及其未来

随着人类自我意识的发展,宗教世俗化问题被提上日程,尤其是近代以来,片面强调科学理性,忽视价值理性。使现代人价值失落,信仰危机,新兴宗教、膜拜团体出现。变革传统宗教,加快世俗化步伐,宗教才能不断适应现代社会人们的精神需求。     

理性与非理性的博弈:现代投资决策理论的演进

理性与非理性的纷争构成了现代投资决策理论演化的主要脉络。本文从古典经济学对理性的最初描述开始,系统讨论了博弈论框架下的理性特征、理性预期对现代金融理论发展的影响、有限理性修正、凯恩斯批判、噪音交易的存在性以及行为金融学关于投资者非理性行为的解释,最后分析了理性与非理性的经济学本质,对现代金融决策理论发展的脉络进行了全面理论梳理。     

Optimism in the Face of Despair: Black‐White Differences in Beliefs About School as a Means for Upward Social Mobility*

Objective. This study aims to provide a better understanding of how beliefs about the system of social mobility affect students' schooling outcomes. Previous studies reach conflicting conclusions because they conflate two forms of beliefs about social mobility (i.e., perceived value of school and perceived barriers despite schooling). Methods. The Maryland Adolescence Development In Context Study (MADICS) is used to examine black‐white differences in beliefs about the value of school and barriers to upward mobility despite schooling and how these beliefs predict academic achievement and educational attainment. Results. The analyses show that relative to whites, blacks hold stronger beliefs in both the value of school and barriers to social mobility, and have greater affective attitudes toward schooling. However, belief in barriers to social mobility is not consequential for academic outcomes. Conclusions. Beliefs about upward mobility are mechanisms by which the opportunity structure influences individuals' schooling behaviors and making clear distinctions between various beliefs about the system of social mobility can refine the understanding of this link. This study suggests that individuals make nuanced distinctions about the role of schooling for upward mobility, each with separate effects on academic outcomes.     

A theory of unanimous jury voting with an ambiguous likelihood

Theory and Decision - We examine collective decision-making in a jury voting game under the unanimity rule when voters have ambiguous beliefs. Unlike in existing studies (Ellis in Theoretical...     

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.     

Five legitimate definitions of correlated equilibrium in games with incomplete information

Aumann's (1987) theorem shows that correlated equilibrium is an expression of Bayesian rationality. We extend this result to games with incomplete information.First, we rely on Harsanyi's (1967) model and represent the underlying multiperson decision problem as a fixed game with imperfect information. We survey four definitions of correlated equilibrium which have appeared in the literature. We show that these definitions are not equivalent to each other. We prove that one of them fits Aumann's framework; the agents normal form correlated equilibrium is an expression of Bayesian rationality in games with incomplete information.We also follow a universal Bayesian approach based on Mertens and Zamir's (1985) construction of the universal beliefs space. Hierarchies of beliefs over independent variables (states of nature) and dependent variables (actions) are then constructed simultaneously. We establish that the universal set of Bayesian solutions satisfies another extension of Aumann's theorem.We get the following corollary: once the types of the players are not fixed by the model, the various definitions of correlated equilibrium previously considered are equivalent.     

The Developmental Progression of Understanding of Mind during a Hiding Game

In this longitudinal study, 52 typically developing preschoolers engaged in a hiding game with their mothers when children were 42‐, 54‐, and 66‐months old. Children's understanding of mind, positive affect, and engagement with the task were rated, and mothers' utterances were coded for role and content. Analyses confirmed that some facets of children's understanding of mind developed sequentially; specifically, they expressed an understanding of knowledge access before an understanding of deception and false beliefs, and expressed an understanding of deception before an understanding of false beliefs. Children's understanding of mind increased across visits and positively correlated with false belief task performance. Results suggest that mothers may tailor the content of their utterances to the child's growing expertise, but the role of mothers' utterances did not change. Observing preschoolers engaged in a playful hiding game revealed that children's understanding of mind not only increased with age but also developed sequentially.     

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.