Rationalizable Strategic Behavior

Games

We define a concept of epistemic robustness in the context of an epistemic model of a finite normal game where a player type corresponds to a belief over the profiles of opponent strategies and types. A Cartesian product X of purestrategy subsets is epistemically robust if there is a Cartesian product Y of player type subsets with X as the associated set of best reply profiles such that the set Y i contains all player types that believe with sufficient probability that the others are of types in Y −i and play best replies. This robustness concept provides epistemic foundations for set-valued generalizations of strict Nash equilibrium, applicable also to games without strict Nash equilibria. We relate our concept to closedness under rational behavior and thus to strategic stability and to the best reply property and thus to rationalizability.

Economics and Philosophy

The game theoretic notion of best-response reasoning is sometimes criticized when its application produces multiple solutions of games, some of which seem less compelling than others. The recent development of the theory of team reasoning addresses this by suggesting that interacting players in games may sometimes reason as members of a team-a group of individuals who act together in the attainment of some common goal. A number of properties have been suggested for team-reasoning decision-makers' goals to satisfy, but a few formal representations have been discussed. In this paper we suggest a possible representation of these goals based on the notion of mutual advantage. We propose a method for measuring extents of individual and mutual advantage to the interacting decision-makers, and define team interests as the attainment of outcomes associated with maximum mutual advantage in the games they play.

Games, 2015

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Journal of Economic Surveys, 2012

Establishing existence and characterizing equilibria are both important achievements in the study of auctions. However, we recognize that equilibria existence results form the basis for well accepted characterizations. In this survey, we review the landmark results and highlight open questions regarding equilibria existence and characterizations in auctions. In addition, we review the standard assumptions underlying these results, and discuss the suitability of the Nash equilibrium solution concept. We focus our review on single-object auctions, but also review results in multi-unit, divisible, combinatorial and double auctions.

Journal of Theoretical Politics, 2002

This article analyzes the distinction between two definitions of rationality, standard expected utility maximization and an alternative that says rational individuals maximize their expected utility given their acts. The latter version can explain important political phenomena that challenge the standard approach. Documenting the empirical advantages of the second definition for political science, the article also argues that conditional expected utility maximizers are normatively rational. Moreover, the article demonstrates that some important and recognized work in political science has unwittingly relied on this version’s underlying assumptions. Finally, it formally shows how this decision theory can ground equilibria in two-person games.

The Economic Journal

We are indebted for helpful comments from participants in seminars at Chicago, Columbia,

International Journal of Systems Science, 2013

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Economic Theory, 1994

Many have objected to the use of the Nash equilibrium (or more generally, Bayesian Nash equilibrium) concept in game theory, and similarly to the use of the rational expectations concept in the theory of competitive markets, on the grounds that the theory assumes too much sophistication and coordination of beliefs on the part of decision-markers. The papers in this volume are among those which study the implications of relaxing those assumptions. * The first author thanks the C. V. Starr Center at New York University for their generosity. 2 Indeed, associate with each play of an agent, i, a type, %. Let #~ denote the ex ante belief of player i so that #i (-[%) denotes the beliefs of player-type z~. The Kalai-Lehrer assumption requires that the true play, induced by collection {#~ (. I%)}~, be absolutely continuous with respect to the beliefs of each agent, #~ (. 1%) for each i. It is straightforward to see that this in turn implies that the set of possible types is either finite or countable, which is violated in the coin-tossing example. See Nyarko (1992) for details.

Economics Letters, 1987

A supergame is modeled in which two agents play a game repeatedly and while the finiteness of the repetitions is common knowledge, the number of repetitions is not. Such games can yield non-standard results. This is illustrated with the Prisoner's Dilemma where a cooperative outcome is shown to be possible. * I have benefited from a seminar at the Indian Statistical Institute, Delhi, and from some comments of Debraj Ray.

Mathematical Social Sciences, 1988

Most standard solution concepts of extensive games are history-insensitive in the sense that no matter at which information set one is, it is assumed that each player believes that everybody else is rational. In reality, each history of moves reveals certain traits of the players to one another. It is argued in this paper that solution concepts ought to make use of this fact. An example of such a concept is developed and called a 'reasonable solution' set. It is shown that this can explain cooperation in certain finitely-repeated games like the Prisoner's Dilemma.

Theory and Decision

It is well known that Nash equilibria may not be Pareto-optimal; worse, a unique Nash equilibrium may be Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove a previously conjectured result: every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic equilibrium (NME) in pure strategies, which we define and illustrate. The outcome it gives, which depends on where play starts, may or may not coincide with that given by a Nash equilibrium. We use some simple examples to illustrate properties of NMEs—for instance, that NME outcomes are usually, though not always, maximin—and seem likely to foster cooperation in many games. Other approaches for analyzing farsighted strategic behavior in games are compared with the NME analysis.

Entropy

We study a binary hypothesis testing problem in which a defender must decide whether a test sequence has been drawn from a given memoryless source P 0 , while an attacker strives to impede the correct detection. With respect to previous works, the adversarial setup addressed in this paper considers an attacker who is active under both hypotheses, namely, a fully active attacker, as opposed to a partially active attacker who is active under one hypothesis only. In the fully active setup, the attacker distorts sequences drawn both from P 0 and from an alternative memoryless source P 1 , up to a certain distortion level, which is possibly different under the two hypotheses, to maximize the confusion in distinguishing between the two sources, i.e., to induce both false positive and false negative errors at the detector, also referred to as the defender. We model the defender–attacker interaction as a game and study two versions of this game, the Neyman–Pearson game and the Bayesian game...

Information Fusion in Distributed Sensor Networks with Byzantines

Distributed sensor networks consist of a set of spatially distributed sensors that operate as data collectors or decision makers to monitor a shared phenomenon. This is a common case in many real world situations like air-traffic control, economic and finance, medical diagnosis, electric power networks, wireless sensor networks, cognitive radio networks, online reputation systems, and many others. Usually, in centralized networks, if there are no power, channel, communication or privacy constraints, the sensors can send the full raw information they collect to a FC. However, real life situations are different and several constraints must be considered e.g. when sensors are spatially distributed over a large territorial area, when the channel bandwidth is limited, or even when the sensors are supplied with short life power sources. To address these limitations, sensors must perform some local processing before sending a compressed version of the collected information to the FC. The abstraction level of the information summary can vary a lot. For instance, it can be a quantized set of the raw information, a soft summary statistic like an average or a likelihood value, or even a single information bit. By means of a fusion rule, the FC integrates the information received from the sensors to make a global decision regarding the system or phenomena under observation. The definition of the fusion rule depends on the a-priori information available about the sensors, the transmission channel and the phenomena as well as the type of information provided by the sensors. Fusion rules can be as simple as voting rules or sophisticated statistical rules. Since the behavior of the sensors in the networks could be different due to noise or intentional acts, the interplay between the sensors or between them and the FC could be modeled using Game Theory, which is a mathematical field that studies the situations of competition and/or cooperation, between decision makers or agents. This chapter is divided into two parts. In the first part, we briefly introduce some basic notions of detection theory and outline some detection techniques used locally

American Journal of Sociology, 2001

Social norms that induce us to reward or punish people not for what they did to us but for what they did to other members of one's group have long been thought as sine qua non sociological and thus impossible to explain in terms of rational choice. This article shows how social norms can be deductively derived from principles of (boundedly) rational choice as mechanisms that are necessary to stabilize behaviors in a large class of evolutionary games.

International Game Theory Review

Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower, chooses his strategy after being informed of the irrevocable commitment of the leader (but not of its realization in case it is mixed). Based on a result by Von Stengel and Zamir [2010], the notions of commitment value and commitment optimal strategies for each player are discussed as a possible solution concept. It is shown that in nondegenerate bimatrix games (a) pure commitment optimal strategies together with the follower’s best response constitute Nash equilibria, and (b) strategies that participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies, provided they are not matrix game optimal. For various classes of bimatrix games that generalize zero-sum games, the relationship between the maximin value of the leader’s payoff matrix, the ...

Canadian Journal of Economics/Revue canadienne d'économique, 2021

Firms make decisions under uncertainty and differ in their ability to collect and process information. As a result, in changing environments, firms have heterogeneous beliefs on the behaviour of other firms. This heterogeneity in beliefs can have important implications on market outcomes, efficiency and welfare. This paper studies the identification of firms' beliefs using their observed actions-a revealed preference and beliefs approach. I consider a general structural model of market competition where firms have incomplete information and their beliefs and profits are nonparametric functions of decisions and state variables. Beliefs may be out of equilibrium. The framework applies both to continuous and discrete choice games and includes as particular cases models of competition in prices or quantities, auction models, entry games and dynamic games of investment decisions. I focus on identification results that exploit an exclusion restriction that naturally appears in models of competition: an observable variable that affects a firm's cost (or revenue) but does not have a direct effect on other firms' profits. I present identification results under three scenarios-common in empirical industrial organization-on the data available to the researcher. Résumé. Identification des convictions des entreprises dans des modèles structurels de concurrence commerciale. En matière de décision, les entreprises agissent dans l'incertitude et diffèrent dans leurs capacités à recueillir et traiter l'information. Par conséquent, dans des environnements en constante évolution, les convictions des entreprises sur le comportement de leurs concurrents sont hétérogènes. Cette hétérogénéité peut avoir des conséquences importantes sur les effets du marché, le rendement et le bienêtre. Grâce à une approche fondée sur les préférences et les croyances révélées, cet article s'intéresse à l'identification des convictions des entreprises en fonction de leurs actions observées. Nous considérons donc un modèle structurel général de concurrence commerciale au sein duquel les entreprises disposent de données incomplètes et où leurs convictions, de même que leurs profits, sont des fonctions non paramétriques des décisions et des variables d'état. Les convictions peuvent se situer hors équilibre. Ce cadre s'applique à la fois aux jeux à choix continus et aux jeux à choix discrets et comprend, à titre de cas particuliers, des modèles de concurrence de prix ou de quantité, des modèles d'enchères, des jeux d'entrée ainsi que des jeux dynamiques de décisions d'investissement. L'attention est portée sur les résultats d'identification qui exploitent une restriction d'exclusion apparaissant naturellement dans les modèles de concurrence : une variable observable ayant une incidence sur les coûts (ou les recettes) des entreprises mais n'entraînant aucun effet direct sur les profits de leurs concurrents. En matière de

Operations Research

In “Asynchronous Schemes for Stochastic and Misspecified Potential Games and Nonconvex Optimization,” Lei and Shanbhag consider a class of convex stochastic Nash games, possibly corrupted by parametric misspecification and characterized by a possibly nonconvex potential function. The authors present an asynchronous inexact proximal best-response (BR) scheme in which, at any step, a randomly selected player computes an inexact BR step (via stochastic approximation) and other players keep their strategies invariant. Misspecification is addressed by a simultaneous learning process reliant on an increasing batch size of sampled gradients. Almost-sure convergence guarantees are provided to the set of Nash equilibria, and such claims can be extended to delay-afflicted regimes, generalized potential games (with coupled strategy sets), and weighted potential games. In fact, equilibria of this potential game are stationary points of the potential function and asynchronous inexact BR schemes ...

Analyse & Kritik, 2005

It is widely believed that experimental results of behavioral game theory undermine standard economic and game theory. This paper suggests that experimental results present serious theoretical modeling challenges, but do not undermine two pillars of contemporary economic theory: the rational actor model, which holds that individual choice can be modeled as maximization of an objective function subject to informational and material constraints, and the incentive compatibility requirement, which holds that macroeconomic quantities must be derived from the interaction and aggregation of individual choices. However, we must abandon the notion that rationality implies self-regarding behavior and the assumption that contracts are costlessly enforced by third parties.

Lecture Notes in Computer Science, 2016

In this paper we summarize concepts from earlier work and demonstrate how infinite sequential games can be used to model strategic conversations. Such a model allows one to reason about the structure and complexity of various kinds of winning goals that conversationalists might have. We show how to use tools from topology, set-theory and logic to express such goals. We then show how to tie down the notion of a winning condition to specific discourse moves using techniques from Mean Payoff games and discounting. We argue, however, that this still requires another addition from epistemic game theory to define appropriate solution and rationality underlying a conversation.

Lecture Notes in Computer Science, 2016

In this paper we summarize concepts from earlier work and demonstrate how infinite sequential games can be used to model strategic conversations. Such a model allows one to reason about the structure and complexity of various kinds of winning goals that conversationalists might have. We show how to use tools from topology, set-theory and logic to express such goals. Our contribution in this paper is to offer a detailed examination of an example in which a player 'defeats himself' by going inconsistent, and to introduce a simple yet revealing way of talking about unawareness. We then demonstrate how we can use ideas from epistemic game theory to define various solution concepts and justify rationality assumptions underlying a conversation.

Educational Dimension

This paper is about people's strategic behavior as observed through experiments. The research question posed in this paper is how players (primarily children) learn in complex strategic situations that they have never encountered before. We examine data from various games played during popular game theory lectures and present findings about players' strategic learning progress while competing with other players. Four ``pick a number'' games with similar-looking rules but very different properties were investigated. These games were presented to various groups of listeners. The collected data is available for replication and analysis in an open repository. In this paper, we analyze data and propose an agent-based model of a beauty pageant game to explain observed behavior. Finally, we discuss the findings, hypotheses to test, and open questions for future research.

Economics Letters, 2013

h i g h l i g h t s • This paper implements multidimensional eductive learning with non-atomistic agents. • It shows the link between the results by Basu (1991) and Guesnerie (1992). • It explores Cournot economies in the limit of atomistic players.

Journal of Economic Methodology, 2017

Mindreading and endogenous beliefs in games Abstract (100 words): we argue that a Bayesian explanation of strategic choices in games requires introducing a psychological theory of belief formation. We highlight that beliefs in epistemic game theory are derived from the actual choice of the players, and cannot therefore explain why Bayesian rational players should play the strategy they actually chose. We introduce the players' capacity of mindreading in a game theoretical framework with the simulation theory, and characterise the beliefs that Bayes rational players could endogenously form in games. We show in particular that those beliefs need not be ratifiable, and therefore that rational players can form action-dependent beliefs.

Economic Theory, 2012

We study how asymmetric information affects the set of rationalizable solutions in a linear setup where the outcome is determined by forecasts about this same outcome. The unique rational expectations equilibrium is also the unique rationalizable solution when the sensitivity of the outcome to agents' forecasts is less than one, provided that this sensitivity is common knowledge. Relaxing this common knowledge assumption, multiple rationalizable solutions arise when the proportion of agents who know the sensitivity is large, and the uninformed agents believe it is possible that the sensitivity is greater than one. Instability is equivalent to existence of some kind of sunspot equilibria.

Advances in Complex Systems, 2018

We introduce a new solution concept, called periodicity, for selecting optimal strategies in strategic form games. This periodicity solution concept yields new insight into nontrivial games. In mixed strategy strategic form games, periodic solutions yield values for the utility function of each player that are equal to the Nash equilibrium ones. In contrast to the Nash strategies, here the payoffs of each player are robust against what the opponent plays. Sometimes, periodicity strategies yield higher utilities, and sometimes the Nash strategies do, but often the utilities of these two strategies coincide. We formally define and study periodic strategies in two player perfect information strategic form games with pure strategies and we prove that every nontrivial finite game has at least one periodic strategy, with nontrivial meaning nondegenerate payoffs. In some classes of games where mixed strategies are used, we identify quantitative features. Particularly interesting are the im...

Games and Economic Behavior

In this paper we reconsider the classic stable matching problem. Using the concept of a matching problem's normal form, the preference lists with information irrelevant to the set of stable matchings discarded, we show that a matching problem possesses a unique stable matching if and only if preferences on the normal form possess the well-known acyclic condition. This is the first necessary and sufficient condition for a unique stable matching.

Operations Research Letters, 1996

We analyze the long run dynamics of queues in which customers undergo self selec tion. We describe the structure and local stability of equilibria for the various capacity adjustment procedures and solve the problem of global stability for the limiting cases. The intermediate cases can be quite complicated. We show that one such case leads to chaotic dynamics.

Journal of Physics: Conference Series, 2013

We define and study periodic strategies in two player finite strategic form games. This concept can arise from some epistemic analysis of the rationalizability concept of Bernheim and Pearce. We analyze in detail the pure strategies and mixed strategies cases. In the pure strategies case, we prove that every two player finite action game has at least one periodic strategy, making the periodic strategies an inherent characteristic of these games. Applying the algorithm of periodic strategies in the case where mixed strategies are used, we find some very interesting outcomes with useful quantitative features for some classes of games. Particularly interesting are the implications of the algorithm to collective action games, for which we were able to establish the result that the collective action strategy can be incorporated in a purely non-cooperative context. Moreover, we address the periodicity issue for the case the players have a continuum set of strategies available. We also discuss whether periodic strategies can imply any sort of cooperativity. In addition, we put the periodic strategies in an epistemic framework.

Dynamic Games and Applications, 2022

We propose a solution concept for social environments called social rationalizability with mediation that identifies the consequences of common knowledge of rationality and farsightedness. In a social environment several coalitions may and could be willing to move at the same time. Individuals not only hold conjectures about the behaviors of other individuals but also about how a mediator is going to solve conflicts of interest. The set of socially rationalizable outcomes with mediation is shown to be non-empty for all social environments, and it can be computed by an iterative reduction procedure. We show that social rationalizability with mediation does not necessarily satisfy coalitional rationality when the number of coalition members is greater than two.

Synthese, 2012

In this paper we study substantive assumptions in social interaction. By substantive assumptions we mean contingent assumptions about what the players know and believe about each other's choices and information. We explain why such assumptions are important, we show how to identify and compare them, formally, and we also show that there exist contexts where no substantive assumptions are being made. Towards the end of the paper we briefly explain the relation between such structures and a number of other "large" structures studied in epistemic game theory. 1 This terminology comes from philosophical epistemology. See e.g. [Huber and Schmidt-Petri, 2009]. 2 We give a formal definition below. 3 See for instance the discussion in [Samuelson, 2004] and the references in [Moscati, 2009]. 4 Of course the quote opens by saying that these are "not assumptions". We rather view them as assumptions of a different kind, hence our terminology.

Theory and decision library, 1997

Noncooperative game theory is often interpreted as a theory of how games would be played if players were rational. On this view, its central project is to discover which strategies are rationally justifiable and which are not, in any game. In this paper, we consider whether the conventional assumptions of modern game theory equip it to fulfil this project. We present an impossibility result which concerns the extent to which they can do so. The result can be seen as strengthening a claim established by Borgers and Samuelson (1992) and Samuelson (1992). 13.2 AN IMPOSSIBILITY RESULT We will consider finite, normal form, noncooperative games; the set of all such games is denoted by G. We shall say that a strategy is justifiable for a player if and only if it would be an optimal choice for her, given some coherent set of beliefs. A set of beliefs is coherent if and only if it is internally consistent and 293

Revue D Economie Politique, 2021

The paper puts emphasis on the so-called "eductive" approach for a critical assessment of the Rational Expectations hypothesis. Section 2 makes an intuitive unformal presentation, aimed at comparing the approach and the results of "eductive" learning and "real time" learning in two polar models, (a two period partial equilibrium model and a simple Real Business Cycle mode). A segment of theoretical literature, taking an eductive view of stability in the fields of finance, trade, general equilibrium, short term or long term macroeconomics….is reviewed in Section 3. Résumé Le texte met en exergue ce que l'on peut appeler l'approche « divinatoire » pour un examen critique de l'Hypothèse d'Anticipations Rationnelles. Le paragraphe 2 compare de façon informelle et intuitive les approches et résultats de l'apprentissage « divinatoire » et de l'apprentissage « en temps réel » dans deux cas polaires (un modèle à deux périodes d'équilibre partiel et un modèle simple de cycles réels). Le paragraphe 3 passe en revue partie de la littérature théorique analysant la stabilité « divinatoire » dans les champs de la finance, du commerce international, de l'équilibre général, de la macroéconomie de court ou plus long terme

1996

This paper surveys the recent literature on evolutionary game theory from the viewpoint of economic theory. The aim is to underst and to what extent evolutionary arguments can substitute rationality-based arguments as a foundation for Nash equilibrium and other non-cooperative solution concepts. We conclude that there is an intimate connection between the attractors of evolutionary processes and Nash equilibrium even in environments which are much mor e general than the simple setting of the standard biological model. Furthermore, even when the evolutionary process does not converge, it's long-run evolution will, in continuous time, necessarily eliminate dominated and iteratively dominated strategies. However, we also find that the current evolutionary setup needs substantial generalization if it is to incorporate plausible adaptive behavior by intelligent but boundedly rational players. We suggest some directions for such generalizations and show by example that in more general environments the connection between evolutionary outcomes and Nash equilibrium may be weaker.