EPICENTER Working Paper Series
Joep van Sloun: Rationalizability and monotonicity in games with incomplete information. EPICENTER Working Paper No. 28 (2023)
This paper examines games with strategic complements or substitutes and incomplete information, where players are uncertain about the opponents’ parameters. We assume that the players’ beliefs about the opponent’s parameters are selected from some given set of beliefs. One extreme is the case where these sets only contain a single belief, representing a scenario where the players’ actual beliefs about the parameters are commonly known among the players. Another extreme is the situation where these sets contain all possible beliefs, representing a scenario where the players have no information about the opponents’ beliefs about parameters. But we also allow for intermediate cases, where these sets contain some, but not all, possible beliefs about the parameters. We introduce an assumption of weakly increasing differences that takes both the choice belief and parameter belief of a player into account. Under this assumption, we demonstrate that greater choice-parameter beliefs leads to greater optimal choices. Moreover,
we show that the greatest and least point rationalizable choice of a player is increasing in their parameter, and these can be determined through an iterative procedure. In each round of the iterative procedure, the lowest surviving choice is optimal for the lowest choice-parameter belief, while the greatest surviving choice is optimal for the highest choice-parameter belief.
Martin Meier and Andrés Perea: Forward induction in a backward inductive manner. EPICENTER Working Paper No. 27 (2023)
We propose a new rationalizability concept for dynamic games with imperfect information, forward and backward rationalizability, that combines elements from forward and backward induction reasoning. It proceeds by applying the forward induction concept of extensive-form rationalizability in a backward inductive fashion: It first applies extensive-form rationalizability from the last period onwards, subsequently from the penultimate period onwards, keeping the restrictions from the last period, and so on, until we reach the beginning of the game. We show that it is characterized epistemically by (a) first imposing common strong belief in rationality from the last period onwards, then (b) imposing common strong belief in rationality from the penultimate period onwards, keeping the restrictions imposed by (a), and so on. It turns out that in terms of outcomes, the concept is equivalent to the pure forward induction concept of extensive-form rationalizability, but both concepts may differ in terms of strategies. We argue that the new concept provides a more compelling theory for how players react to surprises. In terms of strategies, the new concept provides a refinement of the pure backward induction reasoning as embodied by backward dominance and backwards rationalizability. In fact, the new concept can be viewed as a backward looking strengthening of the forward looking concept of backwards rationalizability. Combining our results yields that every extensive-form rationalizable outcome is also backwards rationalizable. Finally, it is shown that the concept of forward and backward rationalizability satisfies the principle of supergame monotonicity: If a player learns that the game was actually preceded by some moves he was initially unaware of, then this new information will only refine, but never completely overthrow, his reasoning. Extensive-form rationalizability violates this principle.
Joep van Sloun: Rationalizable behavior in the Hotelling-Downs model of spatial competition. EPICENTER Working Paper No. 26 (2022)
We consider three scenarios of the Hotelling-Downs model of spatial competition. The first scenario is a static setting with fixed prices and an arbitrary number of agents. This setting has typically been explored using Nash equilibrium, but this paper uses rationalizability instead. These findings will be compared to the results of Eaton and Lipsey (1975) and Shaked (1982). We show that as the number of agents increases, the set of point rationalizable choices increases as well. The second variation consists of a sequential Hotelling-Downs model with three agents, which will be solved by backward induction. The third variation is the static case when agents have limited attraction intervals. In this variation, we show that the set of rationalizable choices does not depend on the number of agents, apart from the number of agents being odd or even. It does depend on the size of the attraction interval. More precisely, the set of rationalizable choices shrinks as the attraction interval gets larger.
Stephan Jagau: The fundamental theorem of epistemic game theory: The infinite case. EPICENTER Working Paper No. 25 (2021)
In auction theory, industrial organization, and other applications of games in economics, it is often convenient to let infinite strategy sets stand in for large finite strategy sets. A tacit assumption is that results for infinite games will translate back to their finite counterparts. Transfinite eliminations of
non-best replies pose a radical challenge here, suggesting that common belief in rationality in infinite games may strictly refine up to k-fold belief in rationality for all finite k. I provide two equivalent characterizations of common belief in rationality for general purely measurable beliefs-type spaces.
The first one is the usual transfinite elimination of non-best replies. The second one, elimination of non-best replies and supporting beliefs, entirely avoids transfinite eliminations. Hence, rather than revealing new depths of reasoning, transfinite eliminations signal an inadequacy of eliminating non-best replies as a general description for strategic rationality.
Christian W. Bach and Andrés Perea: Structure preserving transformations of epistemic models. EPICENTER Working Paper No. 24 (2021)
The prevailing approaches to modelling interactive uncertainty with epistemic models in economics are state-based and type-based. We explicitly formulate two general procedures that transform state models into type models and vice versa. Both transformation procedures preserve the belief hierarchies as well as the common prior assumption. By means of counterexamples it is shown that the two procedures are not inverse to each other. However, if attention is restricted to maximally reduced epistemic models, then isomorphisms can be constructed and an inverse relationship emerges.
Silvia Milano and Andrés Perea: Rational updating at the crossroads. EPICENTER Working Paper No. 23 (2021)
In this paper we explore the absentminded driver problem using two different scenarios. In the first scenario we assume that the driver is capable of reasoning about his degree of absentmindedness before he hits the highway. This leads to a Savage-style model where the states are mutually exclusive and the act-state independence is in place. In the second we employ centred possibilities, by modelling the states (i.e. the events about which the driver is uncertain) as the possible final destinations indexed by a time period. The optimal probability we find for continuing at an exit is different from almost all papers in the literature. In this scenario, act-state independence is still violated, but states are mutually exclusive and the driver arrives at his optimal choice probability via Bayesian updating. We show that our solution is the only one guaranteeing immunity from sure loss via a Dutch strategy, and that — despite initial appearances — it is time consistent.
Andrés Perea: A foundation for expected utility in decision problems and games, EPICENTER Working Paper No. 22 (2020)
In a decision problem or game we typically fix the person’s utilities but not his beliefs. What, then, do these utilities represent? To explore this question we assume, like Gilboa and Schmeidler (2003), that the decision maker holds a conditional preference relation — a mapping that assigns to every possible probabilistic belief a preference relation over his choices. We impose a list of axioms on such conditional preference relations, and show that it singles out precisely those conditional preference relations that admit an expected utility representation. The key axiom is the existence of a uniform preference increase, stating that there must be an alternative conditional preference relation that, for a given choice, uniformly increases the preference for that choice by a constant degree. We also present a procedure that can be used to construct, for a given conditional preference relation satisfying the axioms, a utility function that represents it. If there are no weakly dominated choices, the existence of a uniform preference increase is equivalent to two easily verifiable conditions: strong transitivity and the line property.
Martin Meier and Andrés Perea: Reasoning about your own future mistakes, EPICENTER Working Paper No. 21 (2020)
We propose a model of reasoning in dynamic games in which a player, at each information set, holds a conditional belief about his own future choices and the opponents’ future choices. These conditional beliefs are assumed to be cautious, that is, the player never completely rules out any feasible future choice by himself or the opponents. We impose the following key conditions: (a) a player always believes that he will choose rationally in the future, (b) a player always believes that his opponents will choose rationally in the future, and (c) a player deems his own mistakes infinitely less likely than the opponents’ mistakes. Common belief in these conditions leads to the new concept of perfect quasi-perfect rationalizability. We show that perfectly quasi-perfectly rationalizable strategies exist in every finite dynamic game. We prove, moreover, that perfect quasi-perfect rationalizability constitutes a refinement of both perfect rationalizability (a rationalizability analogue to Selten’s (1975) perfect equilibrium) and quasi-perfect rationalizability (a rationalizability analogue to van Damme’s (1984) quasi-perfect equilibrium).
Niels Mourmans: Reasoning in psychological games: When is iterated elimination of choices enough?, EPICENTER Working Paper No. 20 (2019)
The framework of psychological game theory has allowed for the modelling of a wide range of belief-dependent motivations. At the same time, analysing psychological games can get complex rather quickly due to the fact that higher-order beliefs may enter the utility functions. As a result, some nice properties of traditional games fail to carry over to psychological games in general. This includes the failure of the iterated elimination of strictly dominated choices (IESDC) to always exactly characterize the choices that are rationally played under belief hierarchies expressing common belief in rationality. In this paper we characterize the families of two-player expectation-based psychological games for which IESDC yields exactly the choices that are rationally played under common belief in rationality. We characterize these games based on which orders of beliefs are directly utility-relevant for a decision-maker. In total we identify three cases. Two of these are relatively trivial: (i) the decision-maker’s utility depends on a single, even order of belief and (ii) the decision-maker’s utility and her opponent’s utility depend on a single order of belief. We also identify a third, non-trivial case. Our novel notion of causality diagrams, which capture those orders of beliefs that are (indirectly) utility-relevant, is used to obtain our results.
Christian W. Bach and Jérémie Cabessa: Agreeing to Disagree and Lexicographic Probability Systems, EPICENTER Working Paper No. 19 (2019)
In this note we explore agreeing to disagree with lexicographic probability systems. By means of a counterexample, it is shown that agents can agree to lexicographically disagree on their posteriors. Based on this observation, we propose the same excluding condition which essentially states that agents synchronically either neglect or consider their private information. A lexicographic agreement theorem ensues with equal posteriors at every level.
Christian W. Bach and Andrés Perea: Two Definitions of Correlated Equilibrium, EPICENTER Working Paper No. 18 (2018)
Correlated equilibrium has been introduced by Aumann (1974). Often, in the literature, correlated equilibrium is defined in a simplified as well as more direct way, and sometimes called canonical correlated equilibrium or correlated equilibrium distribution. In fact, we show that the simplified notion of correlated equilibrium is not equivalent – neither doxastically nor behaviourally — to the original from an ex post perspective. We then compare both solution concepts in terms of reasoning. While correlated equilibrium can be characterized by common belief in rationality and a common prior, the simplified variant additionally requires the one-theory-per-choice condition. Since this condition features a correctness of beliefs property, the latter solution concept exhibits a larger degree of Nash equilibrium flavour than the former.
Rubén Becerril-Borja and Andrés Perea: Common Belief in Future and Restricted Past Rationality, EPICENTER Working Paper No. 17 (2018)
We introduce the idea that a player believes at every stage of a dynamic game that his opponents will choose rationally in the future and have chosen rationally in a restricted way in the past. This is summarized by the concept of common belief in future and restricted past rationality, which is defined epistemically. Moreover, it is shown that every properly rationalizable strategy of the normal form of a dynamic game can be chosen in the dynamic game under common belief in future and restricted past rationality. We also present an algorithm that uses strict dominance, and show that its full implementation selects exactly those strategies that can be chosen under common belief in future and restricted past rationality.
Niels Mourmans: Cautious Reasoning in Psychological Games, EPICENTER Working Paper No. 16 (2018)
Caution is an integral part of many solution concepts in traditional game theory and is commonly modeled using lexicographic beliefs. We show here that lexicographic beliefs lack the expressive power to model caution once we extend traditional games to psychological games. Quantification of the relation of ‘deeming an event infinitely more likely than another event’ is necessary, which can be accomplished by using non-standard beliefs.
Shuige Liu: Characterizing Assumption of Rationality by Incomplete Information, EPICENTER Working Paper No. 15 (2018)
We characterize common assumption of rationality of 2-person games within an incomplete information framework. We use the lexicographic model with incomplete information and show that a belief hierarchy expresses common assumption of rationality within a complete information framework if and only if there is a belief hierarchy within the corresponding incomplete information framework that expresses common full belief in caution, rationality, every good choice is supported, and prior belief in the original utility functions.
Shuige Liu: Characterizing Permissibility and Proper Rationalizability by Incomplete Information, EPICENTER Working Paper No. 14 (2018)
We characterize permissibility and proper rationalizability within an incomplete information framework. We define the lexicographic epistemic model for a game with incomplete information, and show that a choice is permissible (properly rationalizable) within a complete information framework if and only if it is optimal for a belief hierarchy within the corresponding incomplete information framework that expresses common full belief in caution, primary belief in the opponent’s utilities nearest to the original utilities (the opponent’s utilities are centered around the original utilities), and a best (better) choice is supported by utilities nearest (nearer) to the original ones.
Andrés Perea: Common Belief in Rationality in Games with Unawareness, EPICENTER Working Paper No. 13 (2017)
This paper investigates static games with unawareness, where players may be unaware of some of the choices that can be made by other players. That is, different players may have different views on the game. We propose an epistemic model that encodes players’ belief hierarchies on choices and views, and use it to formulate the basic reasoning concept of common belief in rationality. We do so for two scenarios: one in which we do not fix the players’ belief hierarchies on views, and one in which we do. For both scenarios we design a recursive elimination procedure that yields for every possible view the choices that can rationally be made under common belief in rationality.
Niels Mourmans: Reasoning about the Surprise Exam Paradox: An Application of Psychological Game Theory, EPICENTER Working Paper No. 12 (2017)
In many real-life scenarios, decision-makers do not exclusively care for materialized outcomes from decisions they and their co-players make but also display other-regarding preferences such as reciprocity and surprise. Psychological game theory is able to model such belief-dependent motivations. In this paper we discuss the reasoning concepts of common belief in rationality and common belief in future rationality in a psychological game-theoretic setting and use them to provide an explanation for the puzzle of the Surprise Exam Paradox. We consider two versions of the surprise exam game, both in a static and dynamic scenario. In the version that best captures the actual crux of the paradox, we show that, as long as no cautious reasoning is imposed, full surprise is always possible. This contrasts the previous game-theoretic literature on the Surprise Exam Paradox, which relied on equilibrium concepts for traditional and psychological games alike and showed that at most partial surprise is possible under these concepts.
Christian Bach and Andrés Perea: Generalized Nash Equilibrium without Common Belief in Rationality, EPICENTER Working Paper No. 11 (2017)
This note considers generalized Nash equilibrium as an incomplete information analogue of Nash equilibrium and provides an epistemic characterization of it. It is shown that the epistemic conditions do not imply common belief in rationality. For the special case of complete information, an epistemic characterization of Nash equilibrium ensues as a corollary.
Stephan Jagau and Andrés Perea: Common belief in rationality in psychological games, EPICENTER Working Paper No. 10 (2017)
Belief-dependent motivations and emotional mechanisms such as surprise, anxiety, anger, guilt, and intention-based reciprocity pervade real-life human interaction. At the same time, traditional game theory has experienced huge difficulties trying to capture them adequately. Psychological game theory, initially introduced by Geanakoplos et al. (1989), has proven to be a useful modeling framework for these and many more psychological phenomena. In this paper, we use the epistemic approach to psychological games to systematically study common belief in rationality, also known as correlated rationalizability. We show that common belief in rationality is possible in any game that preserves rationality at infinity, a mild requirement that is considerably weaker than the previously known continuity conditions from Geanakoplos et al. (1989) and Battigalli and Dufwenberg (2009). Also, we provide an example showing that common belief in rationality might be impossible in games where rationality is not preserved at infinity. We then develop an iterative procedure that, for a given psychological game, determines all rationalizable choices. In addition, we explore classes of psychological games that allow for a simplified procedure.
Christian Bach and Andrés Perea: Incomplete Information and Equilibrium, EPICENTER Working Paper No. 9 (2017)
In games with incomplete information Bayesian equilibrium constitutes the prevailing solution concept. We show that Bayesian equilibrium generalizes correlated equilibrium from complete to incomplete information. In particular, we provide an epistemic characterization of Bayesian equilibrium as well as of correlated equilibrium in terms of common belief in rationality and a common prior. Bayesian equilibrium is thus not the incomplete information counterpart of Nash equilibrium. To fill the resulting gap, we introduce the solution concept of generalized Nash equilibrium as the incomplete information analogue to Nash equilibrium, and show that it is more restrictive than Bayesian equilibrium. Besides, we propose a simplified tool to compute Bayesian equilibria.
Andrés Perea: Order Independence in Dynamic Games, EPICENTER Working Paper No. 8 (2017)
In this paper we investigate the order independence of iterated reduction procedures in dynamic games. We distinguish between two types of order independence: with respect to strategies and with respect to outcomes. The first states that the specific order of elimination chosen should not affect the final set of strategy combinations, whereas the second states that it should not affect the final set of reachable outcomes in the game. We provide sufficient conditions for both types of order independence: monotonicity, and monotonicity on reachable histories, respectively.
We use these sufficient conditions to explore the order independence properties of various reduction procedures in dynamic games: the extensive-form rationalizability procedure (Pearce (1984), Battigalli (1997)), the backward dominance procedure (Perea (2014)) and Battigalli and Siniscalchi’s (1999) procedure for jointly rational belief systems (Reny (1993)). We finally exploit these results to prove that every outcome that is reachable under the extensive-form rationalizability procedure is also reachable under the backward dominance procedure.
Christian Bach and Andrés Perea: Incomplete Information and Generalized Iterated Strict Dominance, EPICENTER Working Paper No. 7 (2016)
In games with incomplete information, players face uncertainty about the opponents’ utility functions. We follow Harsanyi’s (1967-68) one-person perspective approach to modelling incomplete information. Moreover, our formal framework is kept as basic and parsimonious as possible, to render the theory of incomplete information accessible to a broad spectrum of potential applications. In particular, we formalize common belief in rationality and provide an algorithmic characterization of it in terms of decision problems, which gives rise to the non-equilibrium solution concept of generalized iterated strict dominance.
Christian Nauerz and Andrés Perea: Local Prior Expected Utility: a basis for utility representations under uncertainty, EPICENTER Working Paper No. 6 (2015)
Abstract Models of decision-making under ambiguity are widely used in economics. One stream of such models results from weakening the independence axiom in Anscombe et al. (1963). We identify necessary assumptions on independence to represent the decision maker’s preferences such that he acts as if he maximizes expected utility with respect to a possibly local prior. We call the resulting representation Local Prior Expected Utility, and show that the prior used to evaluate a certain act can be obtained by computing the gradient of some appropriately defined utility mapping. The numbers in the gradient, moreover, can naturally be interpreted as the subjective likelihoods the decision maker assigns to the various states. Building on this result we provide a unified approach to the representation results of Maximin Expected Utility and Choquet Expected Utility and characterize the respective sets of priors.
Andrés Perea: Forward Induction Reasoning versus Equilibrium Reasoning EPICENTER Working Paper No. 5 (2015)
In the literature on static and dynamic games, most rationalizability concepts have an equilibrium counterpart. In two-player games, the equilibrium counterpart is obtained by taking the associated rationalizability concept and adding the following correct beliefs assumption: (a) a player believes that the opponent is correct about his beliefs, and (b) a player believes that the opponent believes that he is correct about the opponent’s beliefs. This paper shows that there is no equilibrium counterpart to the forward induction concept of extensive-form rationalizability (Pearce (1984), Battigalli (1997)), epistemically characterized by common strong belief in rationality (Battigalli and Siniscalchi (2002)). The reason is that there are games where the epistemic conditions of common strong belief in rationality are logically inconsistent with the correct beliefs assumption. In fact, we show that this inconsistency holds for “most” dynamic games of interest.
Andrés Perea and Elias Tsakas: Local Reasoning in Dynamic Games EPICENTER Working Paper No.4 (2015)
In this paper we introduce a novel framework for modeling the players’ reasoning in a dynamic game: at each history each active player reasons about her opponents’ rationality at certain histories only. As a result we obtain a generalized solution concept, called local common strong belief in rationality, and we characterize the strategy profiles that can be rationally played under our concept by means of a simple elimination procedure. Finally, we show that standard models of reasoning can be embedded as special cases on our framework. In particular, the forward induction concept of common strong belief in rationality (Battigalli and Siniscalchi, 2002) is a special case of our model with the players reasoning about all histories, whereas the backward induction concept of common belief in future rationality (Perea, 2014) is a special case of our model with the players reasoning about future histories only.
Andrés Perea and Arkadi Predtetchinski: An Epistemic Approach to Stochastic Games EPICENTER Working Paper No.3 (2014)
In this paper we focus on stochastic games with finitely many states and actions. For this setting we study the epistemic concept of common belief in future rationality, which is based on the condition that players always believe that their opponents will choose rationally in the future. We distinguish two different versions of the concept — one for the discounted case with a fixed discount factor δ, and one for the case of uniform optimality, where optimality is required for “all discount factors close enough to 1”. We show that both versions of common belief in future rationality always yield non-empty predictions for every stochastic game. This is in sharp contrast with the non-existence of subgame perfect equilibrium in many stochastic games under the uniform optimality criterion. We also provide an epistemic characterization of subgame perfect equilibrium for 2-player stochastic games, showing that it is essentially equivalent to common belief in future rationality together with some “correct beliefs assumption”. We finally present a recursive procedure to compute the set of stationary strategies that can be chosen by certain simple types under common belief in future rationality.
Christian Nauerz: Understanding reasoning in games using utility proportional beliefs EPICENTER Working Paper No.2 (2014)
Traditionally very little attention has been paid to the reasoning process that underlies a game theoretic solution concept. When modeling bounded rationality in one-shot games, however, the reasoning process can be a great source of insight. The reasoning process itself can provide testable assertions, which provide more insight than the fit to experimental data. Based on Bach and Perea’s (2014) concept of utility proportional beliefs, I analyze the players’ reasoning process and find three testable implications: (1) Players form an initial belief that is the basis for further reasoning. (2) Players reason by alternatingly considering their own and their opponent’s incentives. (3) Players perform only several rounds of deliberate reasoning.
Andrés Perea: When Do Types Induce the Same Belief Hierarchy? EPICENTER Working Paper No.1 (2014)
Harsanyi (1967–1968) showed how infinite belief hierarchies can be encoded by means of type structures. Such encodings, however, are far from unique: Two different types — possibly from two different type structures — may generate exactly the same belief hierarchy. In this paper we present a finite recursive procedure, the Type Partitioning Procedure, which verifies whether two types, from two potentially different finite type structures, induce the same belief hierarchy or not. Important is that the procedure does not make explicit reference to belief hierarchies, but operates entirely within the “language” of type structures. In the second part of this paper we relate the procedure to the notion of type morphisms and hierarchy morphisms between type structures.