PIERPAOLO BATTIGALLI

Abstract

We study strategic reasoning in a signaling game where players initially have common belief in an outcome distribution and in the event that the sender’s beliefs are independent of her type. We characterize the behavioral implications of these epistemic hypotheses by means of a rationalizability procedure with belief restrictions. Our solution concept is related to, but weaker than Divine Equilibrium (Banks and Sobel, 1987). First, we do not obtain a sequential equilibrium, but just a perfect Bayesian equilibrium with heterogeneous off-path beliefs (Fudenberg and He, 2018). Second, when we model how the receiver may rationalize a particular deviation, we take into account that some types could have preferred a different deviation, and we show this is natural and relevant via an economic example.

DOI: 10.1016/j.jet.2024.105903

Working paper: IGIER Working Paper n. 681

Abstract

We analyze the infinite repetition with imperfect feedback of a simultaneous or sequential game, assuming that players are strategically sophisticated—but impatient— expected-utility maximizers. Sophisticated strategic reasoning in the repeated game is combined with belief updating to provide a foundation for a refinement of self-confirming equilibrium. In particular, we model strategic sophistication as rationality and common strong belief in rationality. Then, we combine belief updating and sophisticated reasoning to provide sufficient conditions for a kind of learning—that is, the ability, in the limit, to exactly forecast the sequence of future observations—thus showing that impatient agents end up playing a sequence of self-confirming equilibria in strongly rationalizable conjectures of the one-period game.

JEL codes: C72; C73; D82; D83.

DOI: 10.1007/s00199-024-01588-3

Abstract

The standard extensive-form partitional representation of information in sequential games fails to distinguish the description of the rules of interaction from the description of players' personal traits. Indeed, this representation does not model how the information given to players as per the rules of the game blends with players' cognitive abilities. We propose a representation of sequential games that explicitly describes the flow of information accruing to players rather than the stock of information retained by players encoded in information partitions. Then, we add a game-independent description of players' mnemonic abilities. If players have perfect memory, our flow representation gives rise to information partitions satisfying the perfect recall property, but different combinations of information flows and players' mnemonic abilities may induce the same information partitions. We show how to use our framework to explicitly model a wide array of cognitive limitations and embed them in the representation of game situations.

Keywords: Sequential games, extensive form, information flows, memory, information partition, possibility correspondence, separation principle.

DOI: 10.1016/j.geb.2024.03.018

Working paper: IGIER Working Paper n. 678

Abstract

This paper investigates how private information affects the joint determination of the payment method and the bid premium in M&As. The focus is on the uncertainty of the stand‐alone valuations of the firms involved in the transaction induced by their opacity. First, we model M&A negotiations as a signalling game with two‐sided private information and derive correlations between firm opacity and bid characteristics from equilibrium analysis. Then, we analyze a sample of U.S. deals, using an index based on market measures of adverse selection to quantify firm opacity. We find that the likelihood of stock offers and the bid premium increase with the target's opacity, while more opaque bidders are associated with fewer stock offers and smaller bid premiums.

Keywords: Asymmetric information, bid premium, mergers and acquisitions, method of payment, signalling games.

JEL codes: G34; G14.

DOI: 10.1111/eufm.12474

Abstract

A central aspect of strategic reasoning in sequential games consists in anticipating how co-players would react to information about past play, which in turn depends on how co-players update and revise their beliefs. Several notions of belief system have been used to model how players' beliefs change as they obtain new information, some imposing considerably more discipline than others on how beliefs at different information sets are related. We highlight the differences between these notions of belief system in terms of introspection about one's own conditional beliefs, but we also show that such differences do not affect the essential aspects of rational planning and the behavioral implications of strategic reasoning, as captured by rationalizability.

Keywords: Sequential games, chain rule, partial introspection, rational planning, rationalizability.

JEL codes: C72; C73; D83.

DOI: 10.1016/j.geb.2023.09.003

Abstract

A central aspect of strategic reasoning in sequential games consists in anticipating how co-players would react to information about past play, which in turn depends on how co-players update and revise their beliefs. Several notions of belief system have been used to model how players' beliefs change as they obtain new information, some imposing considerably more discipline than others on how beliefs at different information sets are related. We highlight the differences between these notions of belief system in terms of introspection about one's own conditional beliefs, but we also show that such differences do not affect the essential aspects of rational planning and the behavioral implications of strategic reasoning, as captured by rationalizability.

Keywords: Sequential games, chain rule, partial introspection, rational planning, rationalizability.

JEL codes: C72; C73; D83.

DOI: 10.1016/j.jet.2023.105700

L'approccio matematico della teoria dei giochi psicologici è utile a descrivere molte forme di motivazione in cui le preferenze sugli esiti dipendono dalle credenze proprie o altrui. Essa permette d'incorporare nell'analisi economica, per esempio, le emozioni, la reciprocità, la preoccupazione per la propria immagine e l'autostima. Spieghiamo come e perché, discutendo gli elementi di base della teoria, gli esperimenti, le applicazioni, e la metodologia. 

Parole chiave: teoria dei giochi psicologici; motivazioni che dipendono dalle credenze; immagine; autostima.

Codici JEL: C72; D91

DOI: 10.1257/jel.20201378

Working paper: IGIER Working Paper N. 646

Abstract

This paper provides a general framework for analyzing self-confirming policies. We study self-confirming equilibria in recurrent decision problems with incomplete information about the true stochastic model. We characterize stationary monetary policies in a linear-quadratic setting.

Keywords: Self-confirming equilibrium, partial identification, monetary policy.

DOI: 10.1007/s11238-021-09862-9

Working paper: IGIER Working Paper n. 573

Abstract

We adopt the epistemic framework of Battigalli and Siniscalchi (1999) to model the distinction between a player's behavior at each node, which is part of the external state, and his plan, which is described by his beliefs about his own behavior. This allows us to distinguish between intentional and unintentional behavior, and to explicitly model how players revise their beliefs about the intentions of others upon observing their actions. Rational players plan optimally and their behavior is consistent with their plans. We illustrate our approach with detailed examples and some results. We prove that optimal planning, belief in continuation consistency and common full belief in both imply the backward induction strategies and beliefs in games with perfect information and no relevant ties. More generally, we present within our framework relevant epistemic assumptions about backward and forward-induction reasoning, and relate them to similar ones studied in the previous literature.

Keywords: Epistemic game theory, plans, perceived intentions, backward induction, forward induction.

JEL codes: C72; C73.

DOI: 10.1016/j.jet.2021.105283

Abstract

Two extensive game structures with imperfect information are said to be behaviorally equivalent if they share the same map (up to relabelings) from profiles of structurally reduced strategies to induced terminal paths. We show that this is the case if and only if one can be transformed into the other through a composition of two elementary transformations, commonly known as “Interchanging of Simultaneous Moves” and “Coalescing Moves/Sequential Agent Splitting.”

Keywords: Extensive game structure, Behavioral equivalence, Invariant transformations.

• article (391 Kb)

Abstract

In social dilemmas, choices may depend on belief-dependent motivations enhancing the credibility of promises or threats at odds with personal gain maximization. We address this issue theoretically and experimentally in the context of the Ultimatum Minigame, assuming that the choice of accepting or rejecting a greedy proposal is affected by a combination of frustration, due to unfulfilled expectations, and inequity aversion. We increase the responder’s payoff from the default allocation (the proposer’s outside option) with the purpose of increasing the responder’s frustration due to the greedy proposal, and thus his willingness to reject it. In addition, we manipulate the method of play, with the purpose of switching on (direct response method) and off (strategy method) the responder’s experience of anger. Our behavioral predictions across and within treatments are derived from the theoretical model complemented by explicit auxiliary assumptions, without relying on equilibrium analysis.

Keywords: Experiments, Psychological games, Ultimatum minigame, Frustration, Anger, Non-equilibrium analysis.

• article (512 Kb)

Abstract

We consider multi-stage games with incomplete information, and we analyze strategic reasoning by means of epistemic events within a “total” state space made of all the profiles of behaviors (paths of play) and possibly incoherent infinite hierarchies of conditional beliefs. Thus, we do not rely on types structures, or similar epistemic models. Subjective rationality is defined by the conjunction of coherence of belief hierarchies, rational planning, and consistency between plan and on-path behavior. Since consistent hierarchies uniquely induce beliefs about behavior and belief hierarchies of others, we can define rationality and common strong belief in rationality, and analyze their behavioral and low-order beliefs implications, which are characterized by strong rationalizability. Our approach allows to extend known techniques to the epistemic analysis of psychological games where the utilities of outcomes depend on beliefs of order k or lower. This covers almost all applications of psychological game theory.

Keywords: Epistemic game theory, Hierarchies of beliefs, Consistency, Subjective rationality, Strong rationalizability, Psychological games.

• article (762 Kb)

Abstract

Psychological game theory (PGT), introduced by Geanakoplos et al.(1989) and significantly generalized by Battigalli and Dufwenberg (2009), extends the standard game theoretic framework by letting players’ utility at endnodes depend on their interactive beliefs. While it is understood that a host of applications that model and/or test the role of emotional and other psychological forces find their home in PGT, the framework is abstract and comprises complex mathematical objects, such as players’ infinite hierarchies of beliefs.Thus, PGT provides little guidance on how to model specific belief dependent motivations and use them in game theoretic analysis.This paper takes steps to fill this gap. Some aspects are simplified–e.g., which beliefs matter–but others are refined and brought closer to applications by providing more structure. We start with belief-dependent motivations and show how to embed them in game forms to obtain psychological games. We emphasize the role of time and of the perception of players’ intentions. We take advantage of progress made on the foundations of game theory to expand and improve on PGT solution concepts.

Keywords: Psychological game theory, Belief-dependent motivation, Intentions, Time, Rationalizability, Self-confirming equilibrium, Bayesian sequential equilibrium.

• article (1.154 Kb)

Abstract

Frustration, anger, and blame have important consequences for economic and social behavior, concerning for example monopoly pricing, contracting, bargaining, violence, and politics. Drawing on insights from psychology, we develop a formal approach to exploring how frustration and anger, via blame and aggression, shape interaction and outcomes in a class of two-stage games.

Keywords: Frustration, Anger, Blame, Belief-dependent preferences, Psychological games, Threats.

• article (734 Kb)

Abstract

We extend the epistemic analysis of dynamic games of Battigalli and Siniscalchi (J Econ Theory 88:188–230, 1999, J Econ Theory 106:356–391, 2002, Res Econ 61:165–184, 2007) from finite dynamic games to all simple games, that is, finite and infinite-horizon multistage games with finite action sets at nonterminal stages and compact action sets at terminal stages. We prove a generalization of Lubin’s (Proc Am Math Soc 43:118-122, 1974) extension result to deal with conditional probability systems and strong belief. With this, we can provide a short proof of the following result: in every simple dynamic game, strong rationalizability characterizes the behavioral implications of rationality and common strong belief in rationality.

Keywords: Epistemic game theory, Simple infinite dynamic game, Strong belief, Strong rationalizability.

• article (603 Kb - english version)

Abstract

We study in a theoretical and experimental setting the interaction between belief-dependent preferences and reputation building in a finitely repeated trust game. We focus mainly on the effect of guilt aversion. In a simple two-type model, we analyze the effect of reputation building in the presence of guilt-averse trustees and derive behavioral predictions. We test these predictions in a laboratory experiment where we elicit information on trustees’ belief-dependent preferences and disclose it to the paired trustor before the repeated game.

Keywords: Repeated psychological game, reputation, guilt, almost complete information.

• article (1.990 Kb)

Abstract 
We consider an ambiguity averse, sophisticated decision maker facing a recurrent decision problem where information is generated endogenously. In this context, we study self-confirming actions as the outcome of a process of active experimentation. We provide inter alia a learning foundation for self-confirming equilibrium with model uncertainty (Battigalli et al., 2015), and we analyze the impact of changes in ambiguity attitudes on convergence to self-confirming equilibria. We identify conditions under which the set of self-confirming equilibrium actions is invariant to changes in ambiguity attitudes, and yet ambiguity aversion may affect the dynamics. Indeed, we argue that ambiguity aversion tends to stifle experimentation, increasing the likelihood that the decision maker gets stuck into suboptimal “certainty traps.”

Keywords: Learning, Stochastic control, Ambiguity aversion, Self-confirming equilibrium.

• article (531 Kb)

Abstract

We consider a game in extensive form recurrently played by agents who are randomly drawn from large populations and matched. We assume that preferences over actions at any information set admit a smooth-ambiguity representation in the sense of Klibanoff et al. (2005), which may induce dynamic inconsistencies. We take this into account in our analysis of self-confirming equilibrium (SCE) given players’ feedback about the path of play. Battigalli et al. (2015) show that the set of SCE’s of a simultaneous-move game with feedback expands as ambiguity aversion increases. We show by example that SCE in a sequential game is not equivalent to SCE applied to the strategic form of such game, and that the previous monotonicity result does not extend to general sequential games. Still, we provide sufficient conditions under which the monotonicity result holds for SCE.

Keywords: Sequential games with feedback, smooth ambiguity, self-confirming equilibrium.
 

• article (710 Kb)

Abstract

This article unpacks the concept of cooperation in management. To stimulate future research, we illustrate the different conceptualizations of cooperation in strategic management, organization theory, and behavioral economics, and the unique insights that each discipline offers. Based on this account, we discuss overlaps and differences in how the concept has been used in these fields and across their different levels of theory and analysis. By taking strategic alliances and other interfirm relations as an exemplary illustration, we elaborate a research agenda on how the understanding of cooperation at each level of analysis – firm/alliance, work group/team, individual and inter-individual – would benefit from greater integration of knowledge from other levels. Borrowing from the different disciplines, we explicate the social mechanisms – and related research directions – determining cooperation and its outcomes across levels: from macro-to-micro (e.g., explaining how cultural issues, different alliance structures, and organizational designs affect the cooperative behavior of inter-firm team members); micro-to-micro (e.g., how individual motivation to cooperate or defect, or how different leadership styles, affect inter-firm team members’ collaboration); micro-to-macro (e.g., how self-interested individuals who identify with their firm can be aggregated into collaborative inter-firm bureaucracies). The emerging framework contributes to the understanding of the microfoundations of management phenomena by placing cooperative relationships between and among persons at center stage in explaining how organizational outcomes are generated. 

Abstract

In a decision problem under uncertainty, a decision maker considers a set of alternative actions whose consequences depend on uncertain factors beyond his control. Following Luce and Raiffa (Games and decisions: introduction and critical survey. Wiley, New York, 1957), we adopt a natural representation of such a situation which takes as primitives a set of conceivable actions A, a set of states S and a consequence function ρ: A× S → C. Each action induces a map from states to consequences, a Savage act, and each mixed action induces a map from states to probability distributions over consequences, an Anscombe–Aumann act. Under an axiom of consequentialism, preferences over pure or mixed actions yield corresponding preferences over the induced acts. This observation allows us to relate the Luce–Raiffa description of a decision problem to the most common framework of modern decision theory which directly takes as primitive a preference relation over the set of all Anscombe– Aumann acts. The key advantage of the latter framework is the possibility of applying powerful convex analysis techniques as in the seminal work of Schmeidler (Econometrica 57:571–587, 1989) and the vast literature that followed. This paper shows that we can maintain the mathematical convenience of the Anscombe–Aumann framework within a description of decision problems which is closer to many applications and experiments. We argue that our framework is more expressive as it allows us to be both explicit and parsimonious about the assumed richness of the set of conceivable actions, and to directly capture preference for randomization as an expression of uncertainty aversion.

• article (540 Kb)

• Article (472 Kb)

Abstract

• article (118 Kb)

Abstract

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. But 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 called for. 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 organize experimental results in the Trust Game.

Keywords: Psychological games, Trust Game, guilt, incomplete information.

• article (764 Kb)

Abstract

We analyze a notion of self-confirming equilibrium with non-neutral ambiguity attitudes that generalizes the traditional concept. We show that the set of equilibria expands as ambiguity aversion increases. The intuition is quite simple: By playing the same strategy in a stationary environment, an agent learns the implied distribution of payoffs, but alternative strategies yield payoffs with unknown distributions; increased aversion to ambiguity makes such strategies less appealing. In sum, a kind of "status quo bias" emerges: In the long run, the uncertainty related to tested strategies disappears, but the uncertainty implied by the untested ones does not.

 Keywords: Selfconfirming equilibrium, conjectural equilibrium, model uncertainty, smooth ambiguity.

• article (751 Kb)

Abstract

We analyze forward-induction reasoning in games with asymmetric information assuming some commonly understood restrictions on beliefs. Specifically, we assume that some given restrictions Δ on players' initial or conditional first-order beliefs are transparent, that is, not only the restrictions Δ hold, but there is common belief in Δ at every node. Most applied models of asymmetric information are covered as special cases whereby Δ pins down the probabilities initially assigned to states of nature. But the abstract analysis also allows for transparent restrictions on beliefs about behavior, e.g. independence restrictions or restrictions induced by the context behind the game. Our contribution is twofold. First, we use dynamic interactive epistemology to formalize assumptions that capture foward-induction reasoning given the transparency of Δ, and show that the behavioral implications of these assumptions are characterized by the Δ-rationalizability solution procedure of Battigalli (1999, 2003). Second, we study the differences and similarities between this solution concept and a simpler solution procedure put forward by Battigalli and Siniscalchi (2003). We show that the two procedures are equivalent if Δ is "closed under compositions", a property that holds in all the applications considered by Battigalli and Siniscalchi (2003). We also show that when Δ is not closed under compositions the simpler solution procedure may fail to characterize the behavioral implications of forward induction reasoning.

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Slides presented at the SAET 2013 conference in Paris.

• final draft of 2013 (479 Kb)
• Slides (206 Kb)

Abstract

Evidence suggests that whether or not people dislike lying is situation-dependent. We argue that the theory of simple guilt can accommodate this well.

Keywords: Deception, Guilt aversion, Experiments, Psychological games.

• article (498 Kb)

Abstract

Battigalli and Siniscalchi (2002) formalize the idea of forward induction reasoning as “rationality and common strong belief of rationality” (RCSBR). Here we study the behavioral implications of RCSBR across all type structures. Formally, we show that RCSBR is characterized by a solution concept we call extensive form best response sets (EFBRS’s). It turns out that the EFBRS concept is equivalent to a concept already proposed in the literature, namely directed rationalizability (Battigalli and Siniscalchi 2003). We conclude by applying the EFBRS concept to games of interest.

Keywords: Epistemic game theory, forward induction, extensive form best response set, directed rationalizability.

• article (519 Kb)

Abstract

We use an interactive epistemology framework to provide a systematic analysis of some solution concepts for games with asymmetric information. We characterize solution concepts using expressible epistemic assumptions, represented as events in the canonical space generated by primitive uncertainty about the payoff relevant state, payoff irrelevant information, and actions. In most of the paper, we adopt an interim perspective, which is appropriate to analyze genuine incomplete information. We relate Delta-rationalizability (Battigalli and Siniscalchi, Advances in Theoretical Economics 3, 2003) to interim correlated rationalizability (Dekel, Fudenberg and Morris, Theoretical Economics 2, 2007) and to rationalizability in the interim strategic form. We also consider the ex ante perspective, which is appropriate to analyze asymmetric information about an initial chance move. We prove the equivalence between interim correlated rationalizability and an ex ante notion of correlated rationalizability.

Keywords: asymmetric information, type spaces, Bayesian games, rationalizability.

• article (333 Kb)

Abstract

The motivation of decision makers who care for various emotions, intentions-based reciprocity, or the opinions of others may depend directly on beliefs (about choices, beliefs, or information). Geanakoplos, Pearce and Stacchetti [J. Geanakoplos, D. Pearce, E. Stacchetti, Psychological games and sequential rationality, Games Econ. Behav. 1 (1989) 60–79] point out that traditional game theory is ill-equipped to address such matters, and they pioneer a new framework which does. However, their toolbox – psychological game theory – incorporates several restrictions that rule out plausible forms of belief-dependent motivation. Building on recent work on dynamic interactive epistemology, we propose a more general framework. Updated higher-order beliefs, beliefs of others, and plans of action may influence motivation, and we can capture dynamic psychological effects (such as sequential reciprocity, psychological forward induction, and regret) that were previously ruled out. We develop solution concepts, provide examples, explore properties, and suggest avenues for future research.

Keywords: Psychological games, Belief-dependent motivation, Extensive-form solution concepts, Dynamic interactive epistemology.

• article (367 Kb)

Abstract

We examine a model of contracting where parties interact repeatedly and can contract at any point in time, but writing formal contracts is costly. A contract can describe the external environment and the parties’ behavior in a more or less detailed way, and the cost of writing a contract is proportional to the amount of detail. We consider both formal (externally enforced) and informal (self-enforcing) contracts. The presence of writing costs has important implications both for the optimal structure of formal contracts, particularly the tradeoff between contingent and spot contracting, and for the interaction between formal and informal contracting. Our model sheds light on these implications and generates a rich set of predictions about the determinants of the optimal mode of contracting.

• article (257 Kb)

Abstract

We adopt an interactive epistemology perspective to analyze dynamic games with partially unknown payoff functions. We consider solution procedures that iteratively delete strategies conditional on private information about the state of nature. In particular we focus on a weak and a strong version of the ∆-rationalizability solution concept, where ∆ represents given restrictions on players’ beliefs about state of nature and strategies (Battigalli, 2003, Battigalli and Siniscalchi, 2003). We first show that weak ∆-rationalizability is characterized by initial common certainty of rationality and of the restrictions ∆, whereas strong ∆-rationalizability is characterized by common strong belief in rationality and the restrictions ∆ (cf. Battigalli and Siniscalchi, 2002). The latter result allows us to obtain an epistemic characterization of the iterated intuitive criterion. Then we use the framework to analyze the robustness of complete-information rationalizability solution concepts to the introduction of “slight” uncertainty about payoffs. If the set of conceivable payoff functions is sufficiently large, the set of strongly rationalizable strategies with slight payoff uncertainty coincides with the set of complete-information, weakly rationalizable strategies.

• final draft (286 Kb)

• article (248 Kb)

Abstract

This paper analyses the sources of buyer power and its effect on sellers' investment in quality improvements. In our model retailers make take-it-or-leave-it offers to a producer and each of them in equilibrium obtains its marginal contribution to total profits (gross of sunk costs). In turn, the individual marginal contribution depends on the rivalry between retailers in the bargaining process. Rivalry increases when retailers are less differentiated and when decreasing returns to scale in production are larger. The allocation of total surplus affects the incentives of the producer to invest in product quality, an instance of the hold-up problem. An increase in buyer power not only makes the supplier and consumers worse off, but it may even harm retailers, that obtain a larger share of a smaller surplus.

Keywords: Buyer power, Non-cooperative Bargaining, Hold-up.

• final draft (233 Kb)

Abstract

Focusing on signaling games, I illustrate the relevance of the rationalizability approach for the analysis multistage games with incomplete information. I define a class of iterative solution procedures, featuring a notion of “forward induction”: the Receiver tries to explain the Sender’s message in a way which is consistent with the Sender’s strategic sophistication and certain given restrictions on beliefs. The approach is applied to some numerical examples and economic models. In a standard model with verifiable messages a full disclosure result is obtained. In a model of job market signaling the best separating equilibrium emerges as the unique rationalizable outcome only when the high and low types are sufficiently different. Otherwise, rationalizability only puts bounds on the education choices of different types.

Keywords: Incomplete information, signaling, rationalization.

• article (383 Kb)

Abstract

We analyze the consequences of strategically sophisticated bidding without assuming equilibrium behavior. In particular, we characterize interim rationalizable bids in symmetric first-price auctions with interdependent values and affiliated signals. We show that (1) every nonzero bid below the equilibrium is rationalizable, (2) some bids above the equilibrium are rationalizable, (3) the upper bound on rationalizable bids of a given player is a nondecreasing function of her signal. In the special case of independent signals and quasi-linear valuation functions, (i) the least upper bound on rationalizable bids is concave; hence (ii) rationalizability implies substantial proportional shading for high valuations, but is consistent with negligible proportional shading for low valuations. We argue that our theoretical analysis may shed some light on experimental findings about deviations from the risk-neutral Nash equilibrium.

 Keywords: Auctions, Rationalizability, Overbidding, Proportional shading.

• article (367 Kb)

Abstract

We analyze a family of extensive-form solution procedures for games with incomplete information that do not require the specification of an epistemic type space a la Harsanyi, but can accommodate a (commonly known) collection of explicit restrictions D on firstorder beliefs. For any fixed D we obtain a solution called D-rationalizability. In static games, D-rationalizability characterizes the set of outcomes (combinations of payoff types and strategies) that may occur in any Bayesian equilibrium model consistent with D; these are precisely the outcomes consistent with common certainty of rationality and of the restrictions D. Hence, our approach to the analysis of incomplete-information games is consistent with Harsanyi’s, and it may be viewed as capturing the robust implications of Bayesian equilibrium analysis. In dynamic games, D-rationalizability yields a forward-induction refinement of this set of Bayesian equilibrium outcomes. Focusing on the restriction that first-order beliefs be consistent with a given distribution on terminal nodes, we obtain a refinement of selfconfirming equilibrium. In signalling games, this refinement coincides with the Iterated Intuitive Criterion.

Keywords: Incomplete Information, Rationalizability, Bayesian Equilibrium, Selfconfirming Equilibrium, Iterated Intuitive Criterion.

• article (519 Kb)

Abstract

In this paper, we analyze two nested iterative solution procedures for infinite, dynamic games of incomplete information. These procedures do not rely on the specification of a type space a` la Harsanyi. Weak rationalizability is characterized by common certainty of rationality at the beginning of the game. Strong rationalizability also incorporates a notion of forward induction. The solutions may take as given some exogenous restrictions on players’ conditional beliefs. In dynamic games, strong rationalizability is a refinement of weak rationalizability. Existence, regularity properties, and equivalence with the set of iteratively interim undominated strategies are proved under standard assumptions. The analysis mainly focus on two-player games with observable actions, but we show how to extend it to n-player games with imperfectly observable actions. Finally, we briefly survey some applications of the proposed approach.

Keywords: Incomplete information, Rationalizability, Forward induction.

• article (344 Kb)

Abstract

We provide a unified epistemic analysis of some forward-induction solution concepts in games with complete and incomplete information. We suggest that forward induction reasoning may be usefully interpreted as a set of assumptions governing the players’ belief revision processes, and define a notion of strong belief to formalize these assumptions. Building on the notion of strong belief, we provide an epistemic characterization of extensive-form rationalizability and the intuitive criterion, as well as sufficient epistemic conditions for the backward induction outcome in generic games with perfect information.

 Keywords: conditional belief, strong belief, forward induction, rationalizability, intuitive criterion.

• article (257 Kb)

Abstract

In this paper we model contract incompleteness "from the ground up," as arising endogenously from the costs of describing the environment and the parties' behavior. Optimal contracts may exhibit two forms of incompleteness: discretion, meaning that the contract does not specify the parties' behavior with sufficient detail; and rigidity, meaning that the parties' obligations are not sufficiently contingent on the external state. The model sheds light on the determinants of rigidity and discretion in contracts, and yields rich predictions regarding the impact of changes in the exogenous parameters on the degree and form of contract incompleteness.

• article (2.662 Kb)

• article (515 Kb)

Abstract 

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, 1992) and is outcome-equivalent to backward induction in generic games of perfect information.

Keywords: Interactive epistemology, forward induction, rationalizability, independence, explicability.

• article (247 Kb)

Abstract

The epistemic analysis of solution concepts for dynamic games involves statements about the players' beliefs conditional upon different histories of play, their conditional beliefs about each other's conditional beliefs, etc. To represent such statements, we construct a space of infinite (coherent) hierarchies of conditional probability systems, defined with respect to a fixed collection of relevant hypotheses concerning an external state (e.g., the strategy profile being played.) As an application, we derive results about common certainty of the opponent's rationality conditonal on an arbitrary collection of histories in multistage games with observed actions and (possibly) incomplete information.

• article (319 Kb)

Abstract

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, ratonalizability in dynamic games, forward and backward induction.

Keywords: Games, belief, knowledge, interactive epistemology, solution concepts.

• survey article (9.007 Kb)

Abstract

Restricting attention to the class of extensive games defined by von Neumann and Morgenstern (1944) with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of ‘‘maximum information’’ we are able to provide an alternative characterization of the meet of the information partitions.

Keywords: Information partition, knowledge, common knowledge, extensive game.

• article (757 Kb)

Abstract

We argue that a notion of constrained time consistency is appropriate to evaluate the interim incentives to deviate from a plan in a decision problem with imperfect recall. Under perfect recall, constrained time consistency is equivalent to the standard notion of time consistency. It turns out that a behavioral strategy ß is constrained time consistent if and only if every realization equivalent strategy ß' is a modified multiselves equilibrium and this implies that every optimal strategy is constrained time consistent. Furthermore, every constrained time consistent strategy is equivalent to a modified multiselves sequential equilibrium.

• article (245 Kb)

Abstract

This paper analyzes Pearce's notion of extensive form rationalizability (EFR). Although EFR was originally defined as a reduction procedure, this paper shows that it can be characterized in terms of restrictions on players' updating systems. These restrictions correspond to a common hierarchy of nested hypotheses. Next the relationship of EFR to more familiar reduction procedures is examined. In generic games of perfect information, EFR is realization-equivalent to iterated weak dominance and backward induction. Equivalence with iterated weak dominance is complete in the subset of games with "iterated perfect information.''

• article (397 Kb)

• article (536 Kb)

• scanned article (1.100 Kb)

Abstract

This paper evaluates different refinements of subgame perfection, which rely on different restrictions on players' assessments, using a simple and intuitive independence property for conditional probability systems on the space of strategy profiles. This independence property is necessary for full consistency of assessments, and it is equivalent to full consistency in games with observable deviators. Furthermore, while every conditional system on the strategies satisfying the independence property corresponds to a generally reasonable extended assessment as defined by Fudenberg and Tirole [J. Econ. Theory 53 (1991), 236-260], such extended assessments may violate independence, full consistency, and invariance with respect to interchanging of essentially simultaneous moves.

• article (952 Kb)

Abstract

We provide two axiomatic characterizations of a stochastic independence property for conditional probability systems, previously proposed by Hammond. One characterization relies on the theory of lexicographic expected utility due to L. Blume et al. [Econometrica 59 (1991), 61-79]; the other relies on the theory of conditional expected utility maximization due to Myerson [``Game Theory: Analysis of Conflict,'' Harvard Univ. Press, Cambridge, MA, 1991]. 

• article (572 Kb)

Abstract

In a finite game fix a space of extended probabilities over strategies and a profile of best response correspondences. A profile of rationality orderings is then given by an ordered partition of the set of strategies of each player, representing different degrees of rationality, where at-least k + 1-rational strategies are best responses against extended probabilities reflecting at least k degrees of rationality. This solution can be constructed inductively, providing a Bayesian foundation for controversial deletion procedures such as extensive form rationalizability and iterated weak dominance. Focusing on extensive games, this approach formalizes the best rationalization principle.

• article (243 Kb)

Abstract

In this note I provide a formulation of the joint principle of structural consistency and strategic independence, which is used to model players' expectations in finite extensive ganes. I compare updating systems of conjectures and conditional probability systems, showing that they represent equivalent formalizations of structural consistency. The notion of strategic independence cannot be adequately formalized by properties of updating systems of conjectures. However, it can be naturally translated in an intuitive stochastic independence property for conditional probability systems.

Keywords: Extensive games, structural consistency, strategic independence, conditional probability systems.

• article (882 Kb)

nr. 23, pp. 19-25

nr. 46, pp. 335-378

• scanned article (1.941 Kb)

nr. 35, pp- 705-733.

• scanned article (1.161 Kb)

nr. 47, pp. 99-118