Interactive beliefs,epistemic independence and strong rationalizability

We use a universal, extensive form interactive beliefs system to provide an epistemic characterization of a weak and a strong notion of rationalizability with independent beliefs. The weak solution concept is equivalent to backward induction in generic perfect information games where no player moves more than once in any play. The strong solution concept is related to explicability (Reny,22) and is outcome-equivalent to backward induction in generic games of perfect information.     

Common belief of rationality in games of perfect information

Aumann (1995) showed that for games with perfect information common knowledge of substantive rationality implies backward induction. Substantive rationality is defined in epistemic terms, that is, in terms of knowledge. We show that when substantive rationality is defined in doxastic terms, that is, in terms of belief, then common belief of substantive rationality implies backward induction. Aumann (1998) showed that material rationality implies backward induction in the centipede game. This result does not hold when rationality is defined doxastically. However, if beliefs are interpersonally consistent then common belief of material rationality in the centipede game implies common belief of backward induction.     

Epistemic equivalence of extended belief hierarchies

In this paper, we introduce a notion of epistemic equivalence between hierarchies of conditional beliefs and hierarchies of lexicographic beliefs, thus extending the standard equivalence results of Halpern (2010) and Brandenburger et al. (2007) to an interactive setting, and we show that there is a Borel surjective function, mapping each conditional belief hierarchy to its epistemically equivalent lexicographic belief hierarchy. Then, using our equivalence result we construct a terminal type space model for lexicographic belief hierarchies. Finally, we show that whenever we restrict attention to full-support beliefs, epistemic equivalence between a lexicographic belief hierarchy and a conditional belief hierarchy implies that an arbitrary Borel event is commonly assumed under the lexicographic belief hierarchy if and only if it is commonly strongly believed under the conditional belief hierarchy. This is the first result in the literature directly linking common assumption in rationality (Brandenburger et al., 2008) with common strong belief in rationality (Battigalli and Siniscalchi, 2002).     

Belief and higher-order belief in the centipede games: An experimental investigation

This paper experimentally explores the epistemic conditions behind people's non-equilibrium behaviour in the centipede games. We propose a novel design of laboratory experiment to elicit people's first- and second-order beliefs regarding their opponents' choices and beliefs. The measured beliefs, together with the choice data, help us to estimate people's level of rationality, belief of rationality and second-order belief of rationality. To examine how these epistemic variables are affected by the social-efficiency property of the classic increasing-sum centipede game, we revisit the constant-sum centipede and compare the measured epistemic conditions from the constant-sum with those from the classic centipede. We find that people's non-backward induction behaviour may be attributed to the diffusion of beliefs and higher-order beliefs in the increasing-sum centipede. We consider a behavioural model in which people's preferences for social efficiency are incorporated into the extended utility maximization problem. Our analytical and estimation results indicate that the presence of efficiency-oriented players and people's belief towards the uncertain portion of such type of players may play a part in the non-backward-induction outcomes in experimental centipede games.     

Belief in the opponentsʼ future rationality

For dynamic games we consider the idea that a player, at every stage of the game, will always believe that his opponents will choose rationally in the future. This is the basis for the concept of common belief in future rationality, which we formalize within an epistemic model. We present an iterative procedure, backward dominance, that proceeds by eliminating strategies from the game, based on strict dominance arguments. We show that the backward dominance procedure selects precisely those strategies that can rationally be chosen under common belief in future rationality if we would not impose (common belief in) Bayesian updating.     

Pairwise epistemic conditions for Nash equilibrium

We introduce a framework for modeling pairwise interactive beliefs and provide an epistemic foundation for Nash equilibrium in terms of pairwise epistemic conditions locally imposed on only some pairs of players. Our main result considerably weakens not only the standard sufficient conditions by Aumann and Brandenburger (1995), but also the subsequent generalization by Barelli (2009). Surprisingly, our conditions do not require nor imply mutual belief in rationality.     

Conservative belief and rationality

Playersʼ beliefs may be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require playersʼ beliefs to be conservative, in the sense that all players must ascribe the actual world positive probability. We show that common conservative belief of rationality (CCBR) characterizes strategies in the support of a subjective correlated equilibrium where all playersʼ beliefs have common support. We also define a notion of strong rationalizability, and show that it is characterized by CCBR.     

Extensive and strategic forms: Games and models for games

This paper aims to make precise, in the context of epistemic models for games, some relations between the normal or strategic form representation of a game and the extensive or dynamic form representation. It is argued, first, that epistemic models defined for strategic form representations provide all the materials necessary for defining models for corresponding extensive form representations of the game, models that provide information about the way the game is played that is sufficient to evaluate the rationality of the choices that the players make, and are disposed to make, in the course of playing the dynamic game. Second, two definitions of rationality are compared — one for strategy choices in the normal form representation, and one for the individual choices that the player is disposed to make in the course of playing the dynamic game. It is shown that they are essentially equivalent in games with perfect recall. The main focus is on the intuitive foundational assumptions about rationality and dynamic choice that are needed to motivate the definitions. It is argued that to evaluate the rationality of a player's choices in a dynamic context, it is essential to distinguish passive knowledge (knowledge about nature and about the prior beliefs and strategy choices of other players that is based on observation and inference) fromactive knowledge (knowledge about one's own choices and future choices that is grounded in one's decisions).     

Belief consistency and trade consistency

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agentsʼ beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinsteinʼs email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.     

On Forward Induction and Evolutionary and Strategic Stability

We analyze which normal form solution concepts capture the notion of forward induction, as defined by E. van Damme (1989, J. Econ. Theory48, 476-496) in the class of generic two player normal form games preceded by an outside option. We find that none of the known strategic stability concepts (including Mertens stable sets and hyperstable sets) capture this form of forward induction. On the other hand, we show that the evolutionary concept of EES set (J. Swinkels, 1992, J. Econ. Theory57, 306-332) is always consistent with forward induction. Journal of Economic Literature Classification Number: C72.     

Recent results on belief,knowledge and the epistemic foundations of game theory

We provide a self-contained, selective overview of the literature on the role of knowledge and beliefs in game theory. We focus on recent results on the epistemic foundations of solution concepts, including correlated equilibrium, rationalizability in dynamic games, forward and backward induction.     

What do uncertainty-averse decision-makers believe?

Summary. This paper introduces the concept of firm belief, which is proposed as a new epistemic model for a wide class of preferences. In particular, firm beliefs are shown to have the following desirable properties: (i) they are derived from preferences according to a plausible rule of epistemic inference; (ii) they satisfy standard logical properties; and (iii) tractable representations of firm belief are available for all (suitably continuous) biseparable preferences [13, 14], including the Choquet expected utility [30] and maxmin expected utility [16] classes. We also use firm belief to construct a generalization of Nash equilibrium for (two-player) normal form games. Received: December 14, 1999; revised version: February 26, 2001     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

Self-admissible sets

Best-response sets (Pearce, 1984 [28]) characterize the epistemic condition of “rationality and common belief of rationality.” When rationality incorporates a weak-dominance (admissibility) requirement, the self-admissible set (SAS) concept (Brandenburger, Friedenberg, and Keisler, 2008 [17]) characterizes “rationality and common assumption of rationality.” We analyze the behavior of SAS's in some games of interest—Centipede, the Finitely Repeated Prisoner's Dilemma, and Chain Store. We then establish some general properties of SAS's, including a characterization in perfect-information games.     

Mindreading and endogenous beliefs in games

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.     

Authority and communication in the laboratory

We report findings from experiments on two delegation–communication games. An uninformed principal chooses whether to fully delegate her decision-making authority to an informed agent or to retain the authority and communicate with the agent via cheap talk to obtain decision-relevant information. In the game in which the delegation outcome is payoff-dominated by both the truthful and the babbling communication outcomes, we find that principal-subjects almost always retain their authority and agent-subjects communicate truthfully. Significantly more choices of delegation than of communication are observed in another game in which the delegation outcome payoff-dominates the unique babbling communication outcome; yet there is a non-negligible fraction of principal-subjects who holds on to their authority and agent-subjects who transmits some information. A level-k analysis of the game indicates that a principal-subject “under-delegates” due to the belief that her less-than-fully-strategic opponent will provide information; such belief is in turn consistent with the actual play.     

Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections

We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a long-standing claim to the same effect, due to R. McKelvey (Amer. J. Polit. Sci.30 (1986), 283-314), in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games. Journal of Economic Literature Classification Numbers: C72, D72.     

A Logic for Suspicious Players: Epistemic Actions and Belief–Updates in Games

This paper introduces a notion of 'epistemic action' to describe changes in the information states of the players in a game. For this, ideas are developed from earlier contributions. The ideas are enriched to cover not just purely epistemic actions, but also fact–changing actions ('real moves', e.g. choosing a card, exchanging cards etc.) and nondeterministic actions and strategies (conditional actions having knowledge tests as conditions). The author considers natural operations with epistemic actions and uses them to describe significant aspects of the interaction between beliefs and actions in a game. A logic is used that combines in a specific way a multiagent epistemic logic with a dynamic logic of 'epistemic actions'. The author presents a complete and decidable proof system for this logic. As an application, the author analyses a specific example of a dialogue game (a version of the Muddy Children Puzzle, in which some of the children can 'cheat' by engaging in secret communication moves, while others may be punished for their credulity). Also presented is a sketch of a 'rule–based' approach to games with imperfect information (allowing 'sneaky' possibilities, such as cheating, being deceived and suspecting the others to be cheating).     

Variational convergence: Approximation and existence of equilibria in discontinuous games

We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.     

Correlated Nash equilibrium

We modify the epistemic conditions for Nash equilibrium only to accommodate Gilboa and Schmeidler's [I. Gilboa, D. Schmeidler, Maxmin expected utility with nonunique prior, J. Math. Econ. 18 (1989) 141-153] maxmin expected utility preferences, and identify the equilibrium concept in n-player strategic games that characterizes the modified epistemic conditions. The epistemic characterization supports the equilibrium concept as a minimal generalization of Nash equilibrium, in the sense that it deviates from Nash equilibrium only in terms of players' attitude towards ambiguity. Consequently, comparing it with Nash equilibrium constitutes a ceteris paribus study of the effects of ambiguity on how a game is played. For example, with ambiguity, (beliefs about) action choices are in general correlated.