Belief Hierarchies Research Articles

Generalized Nash equilibrium with stable belief hierarchies in static games with unawareness

In game theory, recently models and solution concepts of games with unawareness have been developed. Focusing on static games with unawareness, this paper discusses generalized Nash equilibrium, an existing equilibrium concept. Some generalized Nash equilibria can be unstable in the sense that, once an equilibrium is played, some agent’s belief is falsified at some level of someone’s perception hierarchy. Based on the observation, we characterize a particular class of generalized Nash equilibrium that expresses stable belief hierarchies so that it can avoid such a problem. This class of equilibrium can be motivated as a stable convention of the game. We also study how unawareness can affect the agents’ behaviors in a stationary state.

Common belief foundations of global games

We study coordination games under general type spaces. We characterize rationalizable actions in terms of the properties of the belief hierarchies and show that there is a unique rationalizable action played whenever there is approximate common certainty of rank beliefs, defined as the probability the players assign to their payoff parameters being higher than their opponents'. We argue that this is the driving force behind selection results for the specific type spaces in the global games literature.

Bayesian Markov Games with Explicit Finite-Level Types

In impromptu or ad hoc settings, participating players are precluded from precoordination. Subsequently, each player's own model is private and includes some uncertainty about the others' types or behaviors. Harsanyi's formulation of a Bayesian game lays emphasis on this uncertainty while the players each play exactly one turn. We propose a new game-theoretic framework where Bayesian players engage in a Markov game and each has private but imperfect information regarding other players' types. Consequently, we construct player types whose structure is explicit and includes a finite level belief hierarchy instead of utilizing Harsanyi's abstract types and a common prior distribution. We formalize this new framework and demonstrate its effectiveness on two standard ad hoc teamwork domains involving two or more ad hoc players.

Admissibility and assumption

Brandenburger et al. (2008) defined the epistemic condition rationality and common assumption of rationality (RCAR) in lexicographic type structures. We define RCAR in the space of coherent lexicographic belief hierarchies and show that it is an epistemic condition for iterated admissibility. We also show that some coherent lexicographic belief hierarchies cannot be represented as types in lexicographic type structures. An advantage of this approach is that RCAR is possible in a canonical hierarchy space.

A Homeomorphism Theorem for the Universal Type Space with the Uniform Weak Topology

Kolmogorov’s extension theorem provides a natural mapping from the space of coherent hierarchies of an agent’s first-order, second-order, etc. beliefs to the space of probability measures over the exogenous parameters and the other agents' belief hierarchies. Mertens and Zamir (1985) showed that, if the spaces of belief hierarchies are endowed with the product topology, then this mapping is a homeomorphism. This paper shows that this mapping is also a homeomorphism if the spaces of belief hierarchies are endowed with the uniform weak topology of Chen et al. (2010) or the universal strategic topology of Dekel et al. (2006), both of which ensure that strategic behaviour exhibits desirable continuity properties.

COGNITIVE DISCOURSE RELEASE OF INTERCULTURAL COMMUNICATING: SOCIAL POINT

The paper discusses the cognitive aspects of discourse process focusing on the concept of the socially shared knowledge structure underlying culturally determined discourse behavior. It is argued that culture specific differences in the cognitive schemata of the speech events determine the difference in the choice of the interactive strategies and in the repertoires of conventional linguistic means employed for performing these speech events in various languages. Discursive behavior, pertaining to a certain language and sociocultural intercourse, is based on individual, culturally determined socio relevant sense of verbal and non-verbal natural communication. Presented discursive analysis in the process of intercultural communication provides an opportunity not only to distinguish culturally determined mental aspects of different language behavior, but to obtain certain data connected to general patterns of language communication. More than that the process of communication can not be considered adequate without cognitive process of comprehension which takes place in the minds of communication participants while them produce and accept speech. The necessity of linguistic and relevant external communicative parameters research and mental representations becomes evident. The speakers, usually unaware of such a difference in their knowledge structures, subconsciously base their strategic choices on the native hierarchy of beliefs and assumptions in code switching which might result in a communication conflict and even culture clash.

The context of the game

We study games of incomplete information and argue that it is important to correctly specify the “context” within which hierarchies of beliefs lie. We consider a situation where the players understand more than the analyst: It is transparent to the players—but not to the analyst—that certain hierarchies of beliefs are precluded. In particular, the players’ type structure can be viewed as a strict subset of the analyst’s type structure. How does this affect a Bayesian equilibrium analysis? One natural conjecture is that this doesn’t change the analysis—i.e., every equilibrium of the players’ type structure can be associated with an equilibrium of the analyst’s type structure. We show that this conjecture is wrong. Bayesian equilibrium may fail an Extension Property. This can occur even in the case where the game is finite and the analyst uses the so-called universal structure (to analyze the game)—and, even, if the associated Bayesian game has an equilibrium. We go on to explore specific situations in which the Extension Property is satisfied.

Hierarchies of beliefs and the belief-invariant Bayesian solution

The belief-invariant Bayesian solution is a notion of correlated equilibrium in games with incomplete information proposed by Forges (1993), and hierarchy of beliefs over conditional beliefs is introduced by Ely and Peski (2006) in their study of interim independent rationalizability. We study the connection between the two concepts. We partially characterize the correlations embedded among type spaces with the same set of hierarchies of beliefs over conditional beliefs with partially correlating devices, which send correlated signals to players in a way that preserves each player’s belief about others’ types. Since the belief-invariant Bayesian solution is also implemented by such correlating devices, we then establish that it is invariant on equivalent type space.

Interim partially correlated rationalizability

We formalize a solution concept called interim partially correlated rationalizability (IPCR), which was implicitly discussed in both Ely and Peski (2006) and Dekel et al. (2007). IPCR allows for interim correlations, i.e., correlations that depend on opponents' types but not on the state of nature. As a direct extension of Ely and Peski's main result, we show that hierarchies of beliefs over conditional beliefs are necessary and sufficient for the identification of IPCR. We use new proof techniques that better illustrate the connection between higher order beliefs and interim rationalizability.

Graph of a Reflexive Game and Bélles-léttres

Abstract The authors consider reflexive games that describe the interaction of subjects (agents) making decisions based on an awareness structure, i.e., a hierarchy of beliefs about essential parameters, beliefs about beliefs, and so on. It was shown that the language of graphs of reflexive games represents a convenient uniform description method for reflexion effects in bélles-léttres.

Status Consciousness

This research examined individual differences in how people think about social status via a scale with eight proposed factors. Items designed to measure these factors were administered to an online sample (n = 1,009). A factor analysis revealed eight meaningful factors: rejection of status, high-perceived status, respect for hierarchy, low-perceived status, status display, egalitarianism, belief in hierarchy, and enjoyment of status. The 40 items forming these eight factors were then administered to a new sample of online participants (n = 303) alongside measures of self-esteem, social dominance orientation, competitiveness, assertiveness, social comparison orientation, narcissism, and hypersensitive narcissism. Confirmatory factor analyses from this subsequent study supported the model derived in the first study. A preliminary analysis of the construct validity of this new “Status Consciousness Scale” scale was undertaken by examining the correlations between the factors and other personality variables that were predicted to relate to each factor.

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).

UNIVERSALLY RATIONAL BELIEF HIERARCHIES

In a recent paper, Tsakas [2013 Rational belief hierarchies, Journal of Mathematical Economics, Maastricht University] introduced the notion of rational beliefs. These are Borel probability measures that assign a rational probability to every Borel event. Then, he constructed the corresponding Harsanyi type space model that represents the rational belief hierarchies. As he showed, there are rational types that are associated with a non-rational probability measure over the product of the underlying space of uncertainty and the opponent's types. In this paper, we define the universally rational belief hierarchies, as those that do not exhibit this property. Then, we characterize them in terms of a natural restriction imposed directly on the belief hierarchies.

FROM CLASSICAL TO EPISTEMIC GAME THEORY

In this paper, we give a historical overview of the transition from classical game theory to epistemic game theory. To that purpose we will discuss how important notions such as reasoning about the opponents, belief hierarchies, common belief, and the concept of common belief in rationality arose, and gradually entered the game theoretic picture, thereby giving birth to the field of epistemic game theory. We will also address the question why it took game theory so long before it finally incorporated the natural aspect of "reasoning" into its analysis. To answer the latter question we will have a close look at the earliest results in game theory, and see how they shaped our approach to game theory for many years to come.

Universal type structures with unawareness

Infinite hierarchies of awareness and beliefs arise in games with unawareness, similarly to belief hierarchies in standard games. A natural question is whether each hierarchy describes the playerʼs awareness of the hierarchies of other players and beliefs over these, or whether the reasoning can continue indefinitely. This paper constructs the universal type structure with unawareness where each type has an awareness level and a belief over types. Countable hierarchies are therefore sufficient to describe all uncertainty in games with unawareness.

A framework for robustness to ambiguity of higher-order beliefs

Standard type spaces induce belief structures defined by precise beliefs. This paper proposes and analyzes simple procedures for constructing perturbations of such belief structures in which beliefs have a degree of ambiguity. Specifically, we construct ambiguous type spaces whose induced (ambiguous) belief hierarchies approximate the standard, precise, belief hierarchies corresponding to the initial type space. Based on a metric that captures the resulting approximation, two alternative procedures to construct such perturbations are introduced, and are shown to yield a simple and intuitive characterization of convergence to the initial unperturbed environment. As a special case, one of these procedures is shown to characterize the set of all finite perturbations. The introduced perturbations and their convergence properties provide conceptual foundations for the analysis of robustness to ambiguity of various solutions concepts, and for various decision rules under ambiguity.

Rational belief hierarchies

We consider agents who attach a rational probability to every Borel event. We call these Borel probability measures rational and introduce the notion of a rational belief hierarchy, where the first order beliefs are described by a rational measure over the fundamental space of uncertainty, the second order beliefs are described by a rational measure over the product of the fundamental space of uncertainty and the opponent’s first order rational beliefs, and so on. Then, we derive the corresponding rational type space model, thus providing a Bayesian representation of rational belief hierarchies. Our main result shows that this type-based representation has the counterintuitive property that some rational types are associated with non-rational beliefs over the product of the fundamental space of uncertainty and the opponent’s types, thus implying that the agent may attach an irrational probability to some Borel event even if she has a rational belief hierarchy.

Incomplete-Information Models of Guilt Aversion in the Trust Game

In the theory of psychological games it is assumed that players’ preferences on material consequences depend on endogenous beliefs. Most of the applications of this theoretical framework assume that the psychological utility functions representing such preferences are common knowledge. However, this is often unrealistic. In particular, it cannot be true in experimental games where players are subjects drawn at random from a population. Therefore, an incomplete-information methodology is needed. We take a first step in this direction, focusing on guilt aversion in the Trust Game. In our models, agents have heterogeneous belief hierarchies. We characterize equilibria where trust occurs with positive probability. Our analysis illustrates the incomplete-information approach to psychological games and can help to organize experimental results in the Trust Game. This paper was accepted by Uri Gneezy, behavioral economics.

On the structure of rationalizability for arbitrary spaces of uncertainty

Weinstein and Yildiz (2007) show that only very weak predictions are robust to misspecifications of higher order beliefs. Whenever a type has multiple rationalizable actions, any of these actions is uniquely rationalizable for some arbitrarily close type. Hence, refinements of rationalizability are not robust. This negative result is obtained under a richness condition, which essentially means that all common knowledge assumptions on payoffs are relaxed. In many settings, this condition entails an unnecessarily demanding robustness test. It is, therefore, natural to explore the structure of rationalizability when arbitrary common knowledge assumptions are relaxed (i.e., without assuming richness). For arbitrary spaces of uncertainty and for every player i, I construct a set A ∞ of actions that are uniquely rationalizable for some hierarchy of beliefs. The main result shows that for any type ti and any action ai rationalizable for ti ,i fai belongs to A ∞ and is justified by conjectures concentrated on A ∞ , then there exists a sequence of types converging to ti for which ai is uniquely rationalizable. This result significantly generalizes Weinstein and Yildiz’s. Some of its implications are discussed in the context of auctions and equilibrium refinements, and in connection with the literature on global games.

The economic approach to ‘theory of mind’

Theory of mind (ToM) is a great evolutionary achievement. It is a special intelligence that can assess not only one's own desires and beliefs, but also those of others. Whether it is uniquely human or not is controversial, but it is clear that humans are, at least, significantly better at ToM than any other animal. Economists and game theorists have developed sophisticated and powerful models of ToM and we provide a detailed summary of this here. This economic ToM entails a hierarchy of beliefs. I know my preferences, and I have beliefs (a probabilistic distribution) about your preferences, beliefs about your beliefs about my preferences, and so on. We then contrast this economic ToM with the theoretical approaches of neuroscience and with empirical data in general. Although this economic view provides a benchmark and makes useful suggestions about empirical tendencies, it does not always generate a close fit with the data. This provides an opportunity for a synergistic interdisciplinary production of a falsifiable theory of bounded rationality. In particular, a ToM that is founded on evolutionary biology might well be sufficiently structured to have predictive power, while remaining quite general. We sketch two papers that represent preliminary steps in this direction.