Epistemic characterizations of iterated deletion of inferior strategy profiles in preference-based type spaces

Bonanno (Logics and the foundations of game and decision theory, Amsterdam University Press, Amsterdam, 2008) provides an epistemic characterization for the solution concept of iterated deletion of inferior strategy profiles (IDIP) by embedding strategic-form games with ordinal payoffs in non-probabilistic epistemic models which are built on Kripke frames. In this paper, we will follow the event-based approach to epistemic game theory and supplement strategic games with type space models, where each type is associated with a preference relation on the state space. In such a framework, IDIP can be characterized by the conditions that at least one player has correct beliefs about the state of the world and that there is common belief that every player is rational, has correct beliefs about the state of the world and has strictly monotone preferences. Moreover, we shall compare the epistemic motivations for IDIP and its mixed strategy variant known as strong rationalizability (SR). Presuppose the above conditions. Whenever there is also common belief that players’ preferences are representable by some expected utility function IDIP still applies. But if there is common belief that players’ preferences are representable by some expected payoff function, then SR results.     

An epistemic analysis of the Harsanyi transformation

Harsanyi (1967–68) proposed a method for transforming uncertainty over the strategy sets of players into uncertainty over their payoffs. The transformation appears to rely on an assumption that the players are rational, or, indeed, that they are rational and that there is common belief of rationality. Such an assumption would be awkward from the perspective of the epistemic program, which is often interested in the implications of irrationality or a lack of common belief of rationality. This paper shows that without common belief of rationality, such implications are not necessarily maintained under a Harsanyi transformation. The paper then shows how, with the belief-system model of Aumann and Brandenburger (1995), such implications can be maintained in the absence of common belief of rationality. Received: December 2000/Revised: February 2002     

Core and Ancillary Epistemic Virtues

We argue, primarily by appeal to phenomenological considerations related to the experiential aspects of agency, that belief fixation is broadly agentive; although it is rarely (if ever) voluntary, nonetheless, it is phenomenologically agentive because of its significant phenomenological similarities to voluntary-agency experience. An important consequence is that epistemic rationality, as a central feature of belief fixation, is an agentive notion. This enables us to introduce and develop a distinction between core and ancillary epistemic virtues. Core epistemic virtues involve several inter-related kinds of epistemic rationality in belief fixation. Other “habits of mind” pertinent to belief fixation constitute ancillary epistemic virtues. Finally, we discuss the relationship between both kinds of virtues, offering a unified account of epistemic virtuousness.     

Proper belief revision and rationalizability in dynamic games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.     

Belief revision conditionals: basic iterated systems

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints.Recent work has shown that in spite of these negative results, the question ‘how to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved.In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing.     

A framework for managing uncertain inputs: An axiomization of rewarding

The success postulate in belief revision ensures that new evidence (input) is always trusted. However, admitting uncertain input has been questioned by many researchers. Darwiche and Pearl argued that strengths of evidence should be introduced to determine the outcome of belief change, and provided a preliminary definition towards this thought. In this paper, we start with Darwiche and Pearl’s idea aiming to develop a framework that can capture the influence of the strengths of inputs with some rational assumptions. To achieve this, we first define epistemic states to represent beliefs attached with strength, and then present a set of postulates to describe the change process on epistemic states that is determined by the strengths of input and establish representation theorems to characterize these postulates. As a result, we obtain a unique rewarding operator which is proved to be a merging operator that is in line with many other works. We also investigate existing postulates on belief merging and compare them with our postulates. In addition, we show that from an epistemic state, a corresponding ordinal conditional function by Spohn can be derived and the result of combining two epistemic states is thus reduced to the result of combining two corresponding ordinal conditional functions proposed by Laverny and Lang. Furthermore, when reduced to the belief revision situation, we prove that our results induce all the Darwiche and Pearl’s postulates as well as the Recalcitrance postulate and the Independence postulate.     

On the concepts of rationalizability in games

Rationalizability arises when the decision situations and rational behaviors of the players are common knowledge among them. We extend the notion of rationalizability, introduced by Bernheim [5] and Pearce [18] for Bayesian behavior, to some another kinds of player's behavior. We also present a representation of common knowledge consisting in introducing an additional player who sends messages to the players. This revised version was published online in June 2006 with corrections to the Cover Date.     

Trembling hand perfection for mixed quantal/best response equilibria

The quantal response equilibrium (QRE) is a powerful alternative to full rationality equilibrium concepts. At a QRE, all joint moves have non-zero probability. However in “mixed scenarios”, where some players use quantal response and some use best response, equilibrium strategy profiles can have joint moves with zero probability. This raises the question of applying the trembling hand refinement to such mixed scenarios. To address this I first show how to reformulate the QRE as a “best response” equilibrium where expected utilities are replaced by more general objective functions. I then show that under this reformulation the two popular types of trembling hand perfection can differ when some players use quantal response and some use best response. I end by showing that one of those types of trembling hand perfection cannot be used to remove certain troubling kinds of equilibrium in such mixed scenarios, while the other type can. The conclusion is that only the one type of trembling hand perfection should be applied when we allow some players to be quantal response and some to be best response.     

Characterizing solution concepts in terms of common knowledge of rationality

Characterizations of Nash equilibrium, correlated equilibrium, and rationalizability in terms of common knowledge of rationality are well known. Analogous characterizations of sequential equilibrium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using earlier results of Halpern characterizing these solution concepts using non-Archimedean fields.     

Distributed consensus in noncooperative inventory games

This paper deals with repeated nonsymmetric congestion games in which the players cannot observe their payoffs at each stage. Examples of applications come from sharing facilities by multiple users. We show that these games present a unique Pareto optimal Nash equilibrium that dominates all other Nash equilibria and consequently it is also the social optimum among all equilibria, as it minimizes the sum of all the players’ costs. We assume that the players adopt a best response strategy. At each stage, they construct their belief concerning others probable behavior, and then, simultaneously make a decision by optimizing their payoff based on their beliefs. Within this context, we provide a consensus protocol that allows the convergence of the players’ strategies to the Pareto optimal Nash equilibrium. The protocol allows each player to construct its belief by exchanging only some aggregate but sufficient information with a restricted number of neighbor players. Such a networked information structure has the advantages of being scalable to systems with a large number of players and of reducing each player’s data exposure to the competitors.     

An analysis of the a priori and a posteriori

I present and defend a unified, non-reductive analysis of the a priori and a posteriori. It is a mistake to remove all epistemic conditions from the analysis of the a priori (as, for example, Alvin Goldman has recently suggested doing). We can keep epistemic conditions (like unrevisability) in the analysis as long as we insist that a priori and a posteriori justification admit of degrees. I recommend making the degree to which a belief’s justification is a priori or a posteriori solely dependent on the revisability relations that obtain among the faculties that deliver the belief and all other faculties.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Bounded rationality and correlated equilibria

We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability p such that all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality \(\overline{p}\) that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.     

The power of paradox: some recent developments in interactive epistemology

Paradoxes of game-theoretic reasoning have played an important role in spurring developments in interactive epistemology, the area in game theory that studies the role of the players’ beliefs, knowledge, etc. This paper describes two such paradoxes – one concerning backward induction, the other iterated weak dominance. We start with the basic epistemic condition of "rationality and common belief of rationality" in a game, describe various ‘refinements’ of this condition that have been proposed, and explain how these refinements resolve the two paradoxes. We will see that a unified epistemic picture of game theory emerges. We end with some new foundational questions uncovered by the epistemic program. This survey owes a great deal to joint work and many conversations with Robert Aumann, Amanda Friedenberg, Jerry Keisler, and Harborne Stuart. Scott Ashworth, John Asker, Carliss Baldwin, Heski Bar-Isaac, Pierpaolo Battigalli, Ken Corts, Konrad Grabiszewski, Joe Halpern, Rena Henderson, Martin Meier, Martin Rechenauer, and participants in various seminars provided important input. The associate editor and referees made very helpful suggestions and observations. Financial support from Harvard Business School and the Stern School of Business is gratefully acknowledged.     

Rationality and the definition of consistent pairs

A consistent pair specifies a set of rational strategies for both players such that a strategy is rational if and only if it is a best reply to a Bayesian belief that gives positive probability to every rational strategy of the opponent and probability zero otherwise. Although the idea underlying consistent pairs is quite intuitive, the original definition suffers from non-existence problems. In this article, we propose an alternative formalization of consistent pairs. According to our definition, a strategy is rational if and only if it is a best reply to some lexicographic probability system that satisfies certain consistency conditions. These conditions imply in particular that a player's probability system gives infinitely more weight to rational strategies than to other strategies. We show that modified consistent pairs exist for every game.This article is based on Chapter 3 of my Ph.D. thesis finished at the University of Bonn in fulfillment of the requirements of the European Doctoral Programme. For helpful comments and discussions, I would like to thank Eddie Dekel, Larry Blume, Tilman Börgers, Martin Dufwenberg, Frank Schuhmacher, Ariel Rubinstein, Avner Shaked, and seminar participants at Tel Aviv and Iowa City. Financial assistance by the German Academic Exchange Service is gratefully acknowledged.     

Understanding, Integration, and Epistemic Value

Understanding enjoys a special kind of value, one not held by lesser epistemic states such as knowledge and true belief. I explain the value of understanding via a seemingly unrelated topic, the implausibility of veritism. Veritism holds that true belief is the sole ultimate epistemic good and all other epistemic goods derive their value from the epistemic value of true belief. Veritism entails that if you have a true belief that p, you have all the epistemic good qua p. Veritism is a plausible and widely held view; I argue that it is untenable. I argue that integration among beliefs possesses epistemic value independent from the good of true belief, and so has value veritism cannot account for. I argue further that this integration among beliefs comprises the distinctive epistemic value of understanding.     

Preference-based belief operators

We show how different kinds of belief operators derived from preferences can be defined in terms an accessibility relation of epistemic priority, and characterized by means of a vector of nested accessibility relations. The semantic structure enables us to compare and reconcile certain non-standard notions of belief that have recently been used in epistemic analyses of games.     

The dynamics of Bowley's model with bounded rationality

A nonlinear dynamical system which describe the time evolution of n-competitors in a Cournot game (Bowley's model) with bounded rationality is analyzed. The existence and stability of the equilibria of this system is studied. The stability conditions of the steady states for two and three players are explicitly computed. Complex behavior such as cycles and chaotic behavior are observed by numerical simulation. Delayed Bowley's with bounded rationality in monopoly is studied. We show that firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium.     

Fuzzy play,matching devices and coordination failures

We revisit n-player coordination games with Pareto-ranked Nash equilibria. As a novelty, we introduce fuzzy play and a matching device. By fuzzy play we mean that each player does not choose which pure strategy to play, but instead chooses a nonempty subset of his strategy set that he submits to the matching device. The matching device is a very simple one. It randomly selects a match if possible, and it selects randomly some strategy belonging to the strategy set sent by each player otherwise. That is, it does not impose that the best alternatives are matched. Using the concepts of perfect Nash equilibrium and of trembling-hand perfect rationalizability, we show that players coordinate directly on the Pareto optimal outcome. This implies that they neither use the option of fuzzy play, nor make use of the matching device.We thank an anonymous referee and an Associate Editor for valuable comments. Jean-Jacques Herings would like to thank the Netherlands Organisation for Scientific Research (NWO) for financial support. Vincent Vannetelbosch is Chercheur Qualifié at the Fonds National de la Recherche Scientifique. The research of Ana Mauleon has been made possible by a fellowship of the Fonds Européen du Développement Economique Régional (FEDER). Financial support from the Belgian French Communitys program Action de Recherches Concertée 99/04-235 (IRES, Université catholique de Louvain) is gratefully acknowledged.     

Suboptimality of M-step back strategies in Bayesian games

We consider the Bayes optimal strategy for repeated two player games where moves are made simultaneously. In these games we look at models where one player assumes that the other player is employing a strategy depending only on the previousm-move pairs (as discussed in Wilson, 1986). We show that, under very unrestrictive conditions, such an assumption is not consistent with the assumption of rationality of one's opponent. Indeed, we show that by employing such a model a player is implicitly assuming that his opponent is not playing rationally,with probability one. We argue that, in the context of experimental games, thesem-step back models must be inferior to models which are consistent with the assumption that an opponent can be rational.