Rational Updating at the Crossroads (2024)

with Silvia Milano

Economics and Philosophy (2024), Vol. 40, 190--211.


Earlier version appeared as:

EPICENTER Working Paper No. 23

Abstract: 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

Structure-Preserving Transformations of Epistemic Models (2023)

with Christian W. Bach

Economic Inquiry, Vol. 61, 693-719

DOI: 10.1111/ecin.13136

Previous version available as EPICENTER Working Paper No. 24

Abstract:  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.

Common Belief in Rationality in Games with Unawareness (2022)

Mathematical Social Sciences Vol. 119, 11--30.

Abstract: 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 only limit the possible views that may enter the players' belief hierarchies, and one in which we fix the players' belief hierarchies on views. 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.   

Common Belief in Rationality in Pyschological Games : Belief-Dependent Utility and the Limits of Strategic Reasoning (2022)

with Stephan Jagau

Journal of Mathematical Economics, Vol. 100


Psychological games enable us to study diverse motivations like anger, guilt, and intention-based reciprocity using models of rational strategic choice based on common belief in rationality (aka correlated rationalizability).
This is achieved by letting utility depend not only on outcomes and beliefs about others' behavior but also on higher-order beliefs. It is an open question whether such belief-dependent utilities can be made consistent with common belief in rationality in all empirically relevant cases. In this paper, we use a novel existence condition to show that common belief in rationality is possible for any empirically relevant case of belief dependent utility. In addition, we present a recursive elimination procedure that characterizes common belief in rationality under minimal assumptions on belief-dependent utility functions.

Epistemic Game Theory (2021)

In: Handbook of Rationality (free access)

Editors: Markus Knauff and Wolfgang Spohn

MIT Press


In this chapter, we review some of the most important ideas, concepts, and results in epistemic game theory, with a focus on the central idea of common belief in rationality. We start by showing how belief hierarchies can be encoded by means of epistemic models with types and how this encoding can be used to formally define common belief in rationality. We next indicate how the induced choices can be characterized by a recursive elimination procedure and how the concept relates to Nash equilibrium. Finally, we investigate how the idea of common belief in rationality can be extended to dynamic games by looking at several plausible ways in which players may revise their beliefs.

Incomplete Information and Iterated Strict Dominance (2021)

with Christian Bach

Oxford Economic Papers, Vol. 73, 820--836.


The solution concept of iterated strict dominance for static games with complete information recursively deletes choices that are inferior to some other choice. Here, we devise such an algorithm for the more general case of incomplete information. The ensuing solution concept of generalized iterated strict dominance is characterized in terms of common belief in rationality as well as in terms of best response sets. Besides, we provide doxastic conditions that are necessary and sufficient

for modelling complete information from a one-person perspective.

Two Definitions of Correlated Equilibrium (2020)

with Christian W. Bach

Journal of Mathematical Economics, Vol. 90, 12--24.

Abstract: Correlated equilibrium constitutes one of the basic solution concepts for static games with complete information. Actually two variants of correlated equilibrium are in circulation and have been used interchangeably in the literature. Besides the original notion due to Aumann (1974), there exists a simplified definition typically called canonical correlated equilibrium or correlated equilibrium distribution. It is known that the original and the canonical version of correlated equilibrium are equivalent from an ex-ante perspective. However, we show that they are actually distinct – both doxastically
as well as behaviourally – from an interim perspective. An elucidation of this difference emerges in the reasoning realm: while Aumann’s correlated equilibrium can be epistemically characterized by common belief in rationality and a common prior, canonical correlated equilibrium additionally requires the condition of one-theory-per-choice. Consequently, the application of correlated equilibrium requires a careful choice of the appropriate variant.

Common Belief in Future and Restricted Past Rationality (2020)

with Rubén Becerril-Borja

International Journal of Game Theory Vol. 49, 711--747.

Abstract: 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 it selects exactly those strategies that can be chosen under common belief in future and restricted past rationality.

Generalized Nash Equilibrium without Common Belief in Rationality (2020)

with Christian W. Bach

Economics Letters, Vol. 186, 108526, 1--6. 

Abstract: We provide an existence result for the solution concept of generalized Nash equilibrium, which can be viewed as the direct incomplete information analogue of Nash equilibrium. Intuitively, a tuple consisting of a probability measure for every player on his choices and utility functions is a generalized Nash equilibrium, whenever some mutual optimality property is satisfied. This incomplete information solution concept is then epistemically characterized in a way that common belief in rationality is neither used nor implied. For the special case of complete information, an epistemic characterization

of Nash equilibrium ensues as a corollary.

An Epistemic Approach to Stochastic Games (2019)

with Arkadi Predtetchinski

International Journal of Game Theory, Vol. 48, 181--203.

Abstract: 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 are always possible in every stochastic game, and always allow for stationary optimal strategies. That is, for both versions we can always find belief hierarchies that express common belief in future rationality, and that have stationary optimal strategies. We also provide an epistemic characterization of subgame perfect equilibrium for two-player stochastic games, showing that it is equivalent to mutual belief in future rationality together with some "correct beliefs assumption".

Algorithms for Cautious Reasoning in Games (2019)

with Geir Asheim

International Journal of Game Theory, Vol. 48, 1241--1275.

Abstract: We provide comparable algorithms for the Dekel-Fudenberg procedure, iterated admissibility, proper rationalizability and full permissibility by means of the concepts of preference restrictions and likelihood orderings. We apply the algorithms for comparing iterated admissibility, proper rationalizability and full permissibility, and provide a sufficient condition under which iterated admissibility does not rule out properly rationalizable strategies. Finally, we use the algorithms to examine an economically relevant strategic situation, namely a bilateral commitment bargaining game.

Limited Focus in Dynamic Games (2019)

with Elias Tsakas

International Journal of Game Theory, Vol. 48, 571--607.

A previous version appeared as EPICENTER Working Paper No. 4

Abstract: In this paper we introduce a novel framework that allows us to model games with players who reason about the opponents' rationality only in some part of the game tree. We refer to this type of bounded rationality as limited focus. In particular, players try to rationalize their opponents' moves only at the histories they focus on, i.e., formally, they strongly believe in their opponents' rationality in these particular histories only. Our main result characterizes the strategy profiles that can be  played under rationality and common strong belief in rationality by means of a simple elimination procedure, for every specification of the players' focus. Finally, we present several special cases and applications of our framework, and we discuss how it differs from other forms of bounded perception such as unawareness.

Why Forward Induction Leads to the Backward Induction Outcome: A New Proof for Battigalli's Theorem (2018)

Games and Economic Behavior, Vol. 110, 120--138

Abstract: Battigalli (1997) has shown that in dynamic games with perfect information and without relevant ties, the forward induction concept of extensive-form rationalizability yields the backward induction outcome. In this paper we provide a new proof for this remarkable result, based on four steps. We first show that extensive-form rationalizability can be characterized by the iterated application of a special reduction operator, the strong belief reduction operator. We next prove that this operator satisfies a mild version of monotonicity, which we call monotonicity on reachable histories. This property is used to show that for this operator, every possible order of elimination leads to the same set of outcomes. We finally show that backward induction yields a possible order of elimination for the strong belief reduction operator. These four properties together imply Battigalli's theorem.

Common belief in approximate rationality (2018)

with Angie Mounir and Elias Tsakas

Mathematical Social Sciences, Vol. 91, 6--16

Abstract: This paper substitutes the standard rationality assumption with approximate rationality in normal form

games. We assume that players believe that their opponents might be ε-rational, i.e. willing to settle for a

suboptimal choice, and so give up an amount ε of expected utility, in response to the belief they hold. For

every player i and every opponents’ degree of rationality ε, we require player i to attach at least probability

Fi(ε) to his opponent being ε-rational, where the functions Fi are assumed to be common knowledge

amongst the players. We refer to this event as belief in F -rationality. The notion of Common Belief in

F -Rationality (CBFR) is then introduced as an approximate rationality counterpart of the established

Common Belief in Rationality. Finally, a corresponding recursive procedure is designed that characterizes

those beliefs players can hold under CBFR.

Forward Induction Reasoning and Correct Beliefs (2017)

Journal of Economic Theory, Vol. 169, 489--516.

Abstract: All equilibrium concepts implicitly make a correct beliefs assumption, stating that a player believes that his opponents are correct about his first-order beliefs. In this paper we show that in many dynamic games of interest, this correct beliefs assumption may be incompatible with a very basic form of forward induction reasoning: the first two layers of extensive-form rationalizability (Pearce (1984), Battigalli (1997), epistemically characterized by Battigalli and Siniscalchi (2002)). Hence, forward induction reasoning naturally leads us away from equilibrium reasoning. In the second part we classify the games for which equilibrium reasoning is consistent with this type of forward induction reasoning, and find that this class is very small. 

A New Epistemic Characterization of Epsilon-Proper Rationalizability (2017)

with Souvik Roy

Games and Economic Behavior, Vol. 104, 304--328.

Abstract: For a given ε>0, the concept of ε-proper rationalizability (Schuhmacher (1999)) is based on two assumptions: (1) every player is cautious, i.e., does not exclude any opponent's choice from consideration, and (2) every player satisfies the ε-proper trembling condition, i.e., the probability he assigns to an opponent's choice a is at most ε times the probability he assigns to b whenever he believes the opponent to prefer b to a. In this paper we provide a new epistemic foundation for ε-proper rationalizability within an incomplete information framework, where players are uncertain about the opponent's utilities. We show that a belief hierarchy is ε-properly rationalizable in the complete information framework, if and only if, there is an equivalent belief hierarchy within the incomplete information framework that expresses common belief in the events that (1) players are cautious, (2) the players' beliefs about the opponent's utilities are "centered around the original utilities" in some specific way parametrized by ε, and (3) players rationalize each opponent's choice by a utility function that is as close as possible to the original utility function.

Book review on "The Theory of Extensive Form Games" by Carlos-Alós Ferrer and Klaus Ritzberger (2017)

Journal of Economic Literature 2017, Vol. 55, 217--218.

When do types induce the same belief hierarchy? (2016)

with Willemien Kets

Games, 7, 28; doi: 10.3390/g7040028.

Abstract: Type structures are a simple device to describe higher-order beliefs. But how can we check whether two types generate the same belief hierarchy? This paper generalizes the concept of a type morphism and shows that one type structure is contained in another if and only if the former can be mapped into the other using a generalized type morphism. Hence, every generalized type morphism is a hierarchy morphism and vice versa. Importantly, generalized type morphisms do not make reference to belief hierarchies. We use our results to characterize the conditions under which types generate the same belief hierarchy.

Finite Reasoning Procedures for Dynamic Games (2015)

Book chapter in: Johan van Benthem, Sujata Ghosh and Rineke Verbrugge (eds.), Models of Strategic Reasoning: Logics, Games and Communities. FoLLI-LNAI State-of-the-Art Survey, LNCS 8972, Springer, Heidelberg, 2015.

Abstract: In this chapter we focus on the epistemic concept of common belief in future rationality (Perea (2014)), which describes a backward induction type of reasoning for general dynamic games. It states that a player always believes that his opponents will choose rationally now and in the future, always believes that his opponents always believe that their opponents choose rationally now and in the future, and so on,  ad infinitum. It thus involves infinitely many conditions, which might suggest that this concept is too demanding for real players in a game. In this chapter we show, however, that this is not true. For finite dynamic games we present a finite reasoning procedure that a player can use to reason his way towards common belief in future rationality.

On the Outcome Equivalence of Backward Induction and Extensive Form Rationalizability (2015)

with Aviad Heifetz

International Journal of Game Theory, Vol. 44, 37--59.

Abstract: Pearce’s (1984) extensive-form rationalizablity (EFR) is a solution concept embodying a best-rationalization principle (Battigalli 1996, Battigalli and Siniscalchi 2002)) for forward-induction reasoning. EFR strategies may hence be distinct from backward-induction (BI) strategies. We provide a direct and transparent proof that in perfect-information games with no relevant ties, the unique BI outcome is nevertheless identical to the unique EFR outcome, even when the EFR strategy profile and the BI strategy profile are distinct.

Commitment in Alternating Offers Bargaining (2015)

with Topi Miettinen

Mathematical Social Sciences, Vol. 76, 12--18.

Abstract: We extend the Ståhl-Rubinstein alternating-offer bargaining procedure to allow players to simultaneously and visibly commit to some share of the pie prior to, and for the duration of, each bargaining round. If commitment costs are small but increasing in the committed share, then the unique subgame perfect equilibrium outcome exhibits a second mover advantage. In particular, as the horizon approaches infinity, and commitment costs approach zero, the unique bargaining outcome corresponds to the reversed Rubinstein outcome (δ/(1+δ),1/(1+δ)), where δ is the common discount factor.

Plausibility Orderings in Dynamic Games (2014)

Economics and Philosophy, Vol. 30, 331--364.

Abstract: In this paper we explore game-theoretic reasoning in dynamic games within the framework of belief revision theory. More precisely, we focus on the forward induction concept of “common strong belief in rationality” (Battigalli and Siniscalchi (2002)) and the backward induction concept of “common belief in future rationality” (Perea (2012)) and Baltag, Smets and Zvesper (2009)) and see whether the belief revision policies involved in these two concepts can be represented by plausibility orderings. We find that belief revision in “common strong belief in rationality” can always be represented by suitably chosen plausibility orderings, whereas this is impossible in some games for “common belief in future rationality”.

Belief in the opponents' future rationality (2014)

Games and Economic Behavior, Vol. 83, 231--254.

Abstract:  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.

Utility Proportional Beliefs (2014)

with Christian W. Bach

International Journal of Game Theory, Vol. 43, 881--902.

Abstract. In game theory, basic solution concepts often conflict with experimental findings or intuitive reasoning. This fact is possibly due to the requirement that zero probability is assigned to irrational choices in these concepts. Here, we introduce the epistemic notion of common belief in utility proportional beliefs which also attributes positive probability to irrational choices, restricted however by the natural postulate that the probabilities should be proportional to the utilities the respective choices generate. Besides, we propose an algorithmic characterization of our epistemic concept. With regards to experimental findings common belief in utility proportional beliefs fares well in explaining observed behavior.

From Classical to Epistemic Game Theory (2014)

International Game Theory Review, Vol. 16, No. 1, 144001, 22 pages.

Special issue on LOFT 2012 Conference in Sevilla.

Invited talk at LOFT conference in Sevilla, June 2012

Abstract: 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.

Agreeing to Disagree with Lexicographic Prior Beliefs (2013)

with Christian W. Bach

Mathematical Social Sciences, Vol. 66, 129-133.

Abstract: The robustness of Aumann’s seminal agreement theorem with respect to the common prior assumption is considered. More precisely, we show by means of an example that two Bayesian agents with almost identical prior beliefs can agree to completely disagree on their posterior beliefs. Secondly, a more detailed agent model is introduced where posterior beliefs are formed on the basis of lexicographic prior beliefs. We then generalize Aumann’s agreement theorem to lexicographic prior beliefs and show that only a slight perturbation of the common lexicographic prior assumption at some – even arbitrarily deep – level is already compatible with common knowledge of completely opposed posterior beliefs. Hence, agents can actually agree to disagree even if there is only a slight deviation from the common prior assumption.

A Simple Bargaining Procedure for the Myerson Value (2013)

with Noemí Navarro

B.E. Journal of Theoretical Economics, Vol. 13, 131-150.

Abstract: We consider situations where the cooperation and negotiation possibilities between pairs of agents are given by an undirected graph. Every connected component of agents has a value, which is the total surplus the agents can generate by working together. We present a simple, sequential, bilateral bargaining procedure, in which at every stage the two agents in a link (i,j) bargain about their share from cooperation in the connected component they are part of. We show that, if the marginal value of a link is increasing in the number of links in the connected component it belongs to, then this procedure yields exactly the Myerson value payoff (Myerson, 1977) for every player.

An Algorithm for Proper Rationalizability (2011)

Games and Economic Behavior, Vol. 72, 510-525.

Abstract: Proper rationalizability (Schuhmacher (1999), Asheim (2001)) is a concept in epistemic game theory based on the following two conditions: (a) a player should be cautious, that is, should not exclude any opponent’s strategy from consideration; and (b) a player should respect the opponents’ preferences, that is, should deem an opponent’s strategy a infinitely more likely than b if he believes the opponent to prefer a to b. A strategy is properly rationalizable if it can optimally be chosen under common belief in the events (a) and (b). In this paper we present an algorithm that for every finite game computes the set of all properly rationalizable strategies. The algorithm is based on the new idea of a preference restriction, which is a pair (s,A) consisting of a strategy s, and a subset of strategies A, for player i. The interpretation is that player i prefers some strategy in A to s. The algorithm proceeds by successively adding preference restrictions to the game.

Strategic Disclosure of Random Variables (2011)

with János Flesch

European Journal of Operations Research, Vol. 209, 73-82.

Abstract: We consider a game G(n) played by two players. There are n independent random variables Z(1),…,Z(n), each of which is uniformly distributed on [0,1]. Both players know n, the independence and the distribution of these random variables, but only player 1 knows the vector of realizations z:=(z(1),…,z(n)) of them. Player 1 begins by choosing an order z(k(1)),…,z(k(n)) of the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns z(k(1)). If player 2 accepts, then player 1 pays z(k(1)) euros to player 2 and play ends. Otherwise, if player 2 rejects, play continues similarly at period 2 with player 1 offering z(k(2)) euros to player 2. Play continues until player 2 accepts an offer. If player 2 has rejected n-1 times, player 2 has to accept the last offer at period n. This model extends Moser’s (1956) problem, which assumes a non-strategic player 1. We examine different types of strategies for the players and determine their guarantee-levels. Although we do not find the exact value v(n) of the game G(n) in general, we provide an interval I(n)=[a(n),b(n)] containing v(n) such that the length of I(n) is at most 0.07 and converges to 0 as n tends to infinity. We also point out strategies, with a relatively simple structure, which guarantee that player 1 has to pay at most b(n) and player 2 receives at least a(n). In addition, we solve the special case G(2) where there are only two random variables. We mention a number of intriguing open questions and conjectures, which may initiate further research on this subject.

Backward Induction versus Forward Induction Reasoning (2010)

Games, Vol. 1, Issue 3, 168-188.

Special issue on Epistemic Game Theory and Modal Logic.

Abstract: In this paper we want to shed some further light on what we mean by backward induction and forward induction reasoning in dynamic games. To that purpose, we take the concepts of common belief in future rationality (Perea (2010)) and extensive form rationalizability (Pearce (1984), Battigalli (1997), Battigalli and Siniscalchi (2002)) as possible representatives for backward induction and forward induction reasoning. We compare both concepts on an epistemic and an algorithm level, thereby highlighting some of the crucial differences between backward and forward induction reasoning in dynamic games.

The Kalai-Smorodinsky Bargaining Solution with Loss Aversion (2011)

with Bram Driesen and Hans Peters

Mathematical Social Sciences, Vol. 61, 58-64.

Abstract: We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as reference point, yields a transformed problem in which the solution is z. We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.

On Loss Aversion in Bimatrix Games (2010)

with Bram Driesen and Hans Peters

Theory and Decision, Vol. 68, 367-391.

Abstract: In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points—points below which they consider payoffs to be losses—are endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29(2):269) under the name of ‘myopic loss aversion equilibrium.’ There, the players’ reference points depend on the beliefs about their opponents’ strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2×2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.

A Model of Minimal Probabilistic Belief Revision (2009)

Theory and Decision, Vol. 67, 163-222.

Abstract: In the literature there are at least two models for probabilistic belief revision: Bayesian updating and imaging (Lewis (1973, 1976), Gärdenfors (1988)). In this paper we focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules.

Repeated Games with Voluntary Information Purchase (2009)

with János Flesch

Games and Economic Behavior, Vol. 66, 126-145.

Abstract: We consider discounted repeated games in which players can voluntarily purchase information about the opponents’ actions at past stages. Information about a stage can be bought at a fixed but arbitrary cost. Opponents cannot observe the information purchase by a player. For our main result, we make the usual assumption that the dimension of the set FIR of feasible and individually rational payoff vectors is equal to the number of players. We show that, if there are at least three players and each player has at least four actions, then every payoff vector in the interior of the set FIR can be achieved by a Nash equilibrium of the discounted repeated game if the discount factor is sufficiently close to 1. Therefore, nearly efficient payoffs can be achieved even if the cost of monitoring is high. We show that the same result holds if there are at least four players and at least three actions for each player. Finally, we indicate how the result can be extended to sequential equilibrium. 

Optimal Search for a Moving Target with the Option to Wait (2009)

with János Flesch and Emin Karagozoglu

Naval Research Logistics, Vol. 56, 526-539.

Abstract: We investigate the problem in which an agent has to find an object that moves between two locations according to a discrete Markov process (see Pollock, 1970). At every period, the agent has three options: searching left, searching right, and waiting. We assume that waiting is costless whereas searching is costly. Waiting can be useful because it could induce a more favorable probability distribution over the two locations next period. We find an essentially unique (nearly) optimal strategy, and prove that it is characterized by two thresholds (as conjectured by Weber, 1986). We show, moreover, that it can never be optimal to search the location with the lower probability of containing the object. The latter result is far from obvious and is in clear contrast with the example in Ross (1983) for the model without waiting. We also analyze the case of multiple agents. This makes the problem a more strategic one, since now the agents not only compete against time but also against each other in finding the object. We find different kinds of subgame perfect equilibria, possibly containing strategies that are not optimal in the one-agent case. We compare the various equilibria in terms of cost-effectiveness.

Minimal Belief Revision leads to Backward Induction (2008)

Mathematical Social Sciences , Vol. 56, 1-26.

Abstract: We present an epistemic model for games with perfect information in which players, upon observing an unexpected move, may revise their belief about the opponents’ preferences over outcomes. For a given profile P of preference relations over outcomes, we impose the following conditions: (1) players initially believe that opponents have preference relations as specified by P; (2) players believe at every instance of the game that each opponent is carrying out an optimal strategy; (3) if a player revises his belief about an opponent’s type, he must search for a “new” type that disagrees with the “old” type on a minimal number of elementary statements; (4) if a player revises his belief about an opponent’s preference relation over outcomes, he must search for a “new” preference relation that disagrees with the “old” preference relation on a minimal number of pairwise rankings. It is shown that every player whose preference relation is given by P, and who throughout the game respects common belief in the events (1) – (4), has a unique sequentially rational strategy, namely his backward induction strategy in the game induced by P.

Proper Belief Revision and Equilibrium in Dynamic Games (2007)

Journal of Economic Theory, Vol. 136, 572-586.

Abstract: We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies, and (3) a player at information set h should not change his belief about an opponent’s ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.

A One-Person Doxastic Characterization of Nash Strategies (2007)

Synthese, Vol. 158, 251-271. (Knowledge, Rationality and Action 341-361).

Abstract: Within a formal epistemic model for simultaneous-move games, we present the following conditions: (1) belief in the opponents’ rationality (BOR), stating that a player believes that every opponent chooses an optimal strategy, (2) self-referential beliefs (SRB), stating that a player believes that his opponents hold correct beliefs about his own beliefs, (3) projective beliefs (PB), stating that i believes that j’s belief about k’s choice is the same as i’s belief about k’s choice, and (4) conditionally independent beliefs (CIB), stating that a player believes that opponents’ types choose their strategies independently. We show that, if a player satisfies BOR, SRB and CIB, and believes that every opponent satisfies BOR, SRB, PB and CIB, then he will choose a Nash strategy (that is, a strategy that is optimal in some Nash equilibrium). We thus provide a sufficient collection of one-person conditions for Nash strategy choice. We also show that none of these seven conditions can be dropped.

Epistemic Foundations for Backward Induction: An Overview (2007)

Interactive Logic Proceedings of the 7th Augustus de Morgan Workshop, London. Texts in Logic and Games 1 (Johan van Benthem, Dov Gabbay, Benedikt Löwe (eds.)), Amsterdam University Press, 159-193.

Abstract: In this survey we analyze and compare various sufficient epistemic conditions for backward induction that have been proposed in the literature. To this purpose we present a simple epistemic base model for games with perfect information, and express the conditions of the different models in terms of our base model. This will enable us to explictly analyze the differences and similarities between the various sufficient conditions for backward induction.

Weak Monotonicity and Bayes-Nash Incentive Compatibility (2007)

with Rudolf Müller and Sascha Wolf

Games and Economic Behavior Vol. 61, 344-358.

Abstract: An allocation rule is called Bayes-Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents’ types form a Bayes-Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides characterizations of Bayes-Nash incentive compatible allocation rules in social choice settings where agents have one-dimensional or multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterizations are derived by constructing complete directed graphs on agents’ type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For one-dimensional types and agents’ valuation functions satisfying non-decreasing expected differences, we show that weak monotonicity of the allocation rule is a necessary and sufficient condition for the rule to be Bayes-Nash incentive compatibile. In the case where types are multi-dimensional and the valuation for each outcome is a linear function in the agent’s type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes-Nash incentive compatibility.

Revision of Conjectures about the Opponent’s Utilities in Signaling Games (2007)

with Tim Schulteis, Hans Peters and Dries Vermeulen

Economic Theory, Vol. 30, 373-384.

Abstract: In this paper we apply the concept of preference conjecture equilibrium introduced in Perea (2007) to signaling games and show its relation to sequential equilibrium. We introduce the concept of minimum revision equilibrium and show how this can be interpreted as a refinement of sequential equilibrium.

Proper Belief Revision and Rationalizability in Dynamic Games (2006)

International Journal of Game Theory, Vol. 34, 529-559.

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

Stochastic Dominance Equilibria in Two-Person Noncooperative Games (2006)

with Hans Peters, Tim Schulteis and Dries Vermeulen

International Journal of Game Theory, Vol. 34, 457-473.

Abstract: Two-person noncooperative games with finitely many pure strategies are considered, in which the players have linear orderings over sure outcomes but incomplete preferences over probability distributions resulting from mixed strategies. These probability distributions are evaluated according to t-degree stochastic dominance. A t-best reply is a strategy that induces a t-degree stochastically undominated distribution, and a t-equilibrium is a pair of t-best replies. The paper provides a characterization and an existence proof of t-equilibria in terms of representing utility functions, and shows that for large t behavior converges to a form of max–min play. Specifically, increased aversion to bad outcomes makes each player put all weight on a strategy that maximizes the worst outcome for the opponent, within the supports of the strategies in the limiting sequence of t-equilibria.

The Weak Sequential Core for Two-Period Economies (2006)

with Jean-Jacques Herings and Arkadi Predtetchinski

International Journal of Game Theory, Vol. 34, 55-65.

Abstract: We adapt the classical core concept to deal with situations involving time and uncertainty. We define the weak sequential core as the set of allocations that are stable against coalitional deviations ex ante, and moreover cannot be improved upon by any coalition after the resolution of uncertainty. We restrict ourselves to credible deviations, where a coalitional deviation cannot be counterblocked by some subcoalition. We study the relationship of the resulting core concept with other sequential core concepts, give sufficient conditions under which the weak sequential core is non-empty, but show that it is possible to give reasonable examples where it is empty.

Monotonicity and equal-opportunity equivalence in bargaining (2005)

with Antonio Nicolò

Mathematical Social Sciences, Vol. 49, Issue 2, 221-243.

Abstract: In this paper we study two-person bargaining problems represented by a space of alternatives, a status quo point, and the agents’ preference relations on the alternatives. The notion of a family of increasing sets is introduced, which reflects a particular way of gradually expanding the set of alternatives. For any given family of increasing sets, we present a solution which is Pareto optimal and monotonic with respect to this family, that is, makes each agent weakly better off if the set of alternatives is expanded within this family. The solution may be viewed as an expression of equal-opportunity equivalence as defined in Thomson (1994). It is shown to be the unique solution that, in addition to Pareto optimality and the monotonicity property mentioned above, satisfies a uniqueness axiom and unchanged contour independence. A non-cooperative bargaining procedure is provided for which the unique backward induction outcome coincides with the solution.

Sequential and quasi-perfect rationalizability in extensive games (2005)

with Geir Asheim

Games and Economic Behavior, Vol. 53, Issue 1, 15-42.

Abstract: Within an epistemic model for two-player extensive games, we formalize the event that each player believes that his opponent chooses rationally at all information sets. Letting this event be common certain belief yields the concept of sequential rationalizability. Adding preference for cautious behavior to this event likewise yields the concept of quasi-perfect rationalizability. These concepts are shown to (a) imply backward induction in generic perfect information games, and (b) be non-equilibrium analogues to sequential and quasi-perfect equilibrium, leading to epistemic characterizations of the latter concepts. Conditional beliefs are described by the novel concept of a system of conditional lexicographic probabilities.

Core concepts for dynamic TU-games (2005)

with Laurence Kranich and Hans Peters

International Game Theory Review, Vol. 7, 43-61.

Abstract: This paper is concerned with the question how to define the core when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face a finite sequence of exogenously specified TU-games. Three different core concepts are presented: the classical core, the strong sequential core and the weak sequential core. The differences between the concepts arise from different interpretations of profitable deviations by coalitions. Sufficient conditions are given for nonemptiness of the classical core in general and the weak sequential core for the case of two players. Simplifying characterizations of the weak and strong sequential core are provided. Examples highlight the essential differences between these core concepts.

A note on the one-deviation property in extensive form games (2002)

Games and Economic Behavior, Vol 40, 322-338.

Abstract: In an extensive form game, an assessment is said to satisfy the one-deviation property if for all possible payoffs at the terminal nodes the following holds: if a player at each of his information sets cannot improve upon his expected payoff by deviating unilaterally at this information set only, he cannot do so by deviating at any arbitrary collection of information sets. Hendon et al. (1996) have shown that pre-consistency of assessments implies the one-deviation property. In this note, it is shown that an appropriate weakening of pre-consistency, termed updating consistency, is both a sufficient and necessary condition for the one-deviation property. The result is extended to the context of rationalizability.

Supporting others and the evolution of influence (2002)

with Salvador Barberà

Journal of Economic Dynamics and Control, Vol 26, 2051-2092.

Abstract: In this paper we study environments in which agents can transfer influence to others by supporting them. When planning whom to support, they should take into account the future effect of this, since the receiving agent might use this influence so support others in the future. We show that in the presence of a finite horizon there is an essentially unique optimal support behavior which can be characterized in terms of associated marginal value functions. The analysis of these marginal value functions allows us to derive qualitative properties of optimal support strategies under different specific environments and to explicitly compute the optimal support behavior in some numerical examples. We also investigate the case of an infinite horizon. Examples show that multiple equilibria may appear in this setting, some of which sustaining a degree of cooperation that would not be possible under a finite horizon.

Player splitting in extensive form games (2000)

with Mathijs Jansen and Dries Vermeulen

International Journal of Game Theory, Vol 29, 433-450.

Abstract: By a player splitting we mean a mechanism that distributes the information sets of a player among so-called agents. A player splitting is called independent if each path in the game tree contains at most one agent of every player. Following Mertens (1989), a solution is said to have the player splitting property if, roughly speaking, the solution of an extensive form game does not change by applying independent player splittings. We show that Nash equilibria, perfect equilibria, Kohlberg-Mertens stable sets and Mertens stable sets have the player splitting property. An example is given to show that the proper equilibrium concept does not satisfy the player splitting property. Next, we give a definition of invariance under (general) player splittings which is an extension of the player splitting property to the situation where we also allow for dependent player splittings. We come to the conclusion that, for any given dependent player splitting, each of the above solutions is not invariant under this player splitting. The results are used to give several characterizations of the class of independent player splittings and the class of single appearance structures by means of invariance of solution concepts under player splittings.

Repeated games with endogenous choice of information mechanisms (1999)

with János Flesch

Mathematics of Operations Research, Vol 24, 785-794.

Abstract: We consider two-player repeated games with non-observable actions (cf. Lehrer, 1989). An information mechanism for a player is a function which assigns a private signal to every action-pair of the one-shot game. In this paper, we extend the model to a situation in which both players can buy an information mechanism before playing the repeated game. Within this model, we provide a characterization of the lower equilibrium payoffs in terms of the one-shot game for the case that both players choose a non-trivial information mechanism with probability one. Moreover, we construct a lower equilibrium in a repeated game in which one of the players strictly randomizes between information mechanisms. It is shown that the corresponding payoffs can not be induced by a lower equilibrium in which players choose a particular information mechanism with probability one.

Limit consistent solutions in non-cooperative games (1998)

with Hans Peters

Journal of Optimization Theory and Applications, Vol 98, No. 1, 109-129.

Abstract: Strong and limit consistency in finite noncooperative games are studied. A solution is called strongly consistent if it is both consistent and conversely consistent (Peleg and Tijs, 1996). We provide sufficient conditions on one-person behavior such that a strongly consistent solution is nonempty. We introduce limit consistency for normal form and extensive form games. Roughly, this means that the solution can be approximated by strongly consistent solutions. We then show that the perfect and proper equilibrium correspondences in normal form games, as well as the weakly perfect and sequential equilibrium correspondences for extensive form games are limit consistent.

Characterization of consistent assessments in extensive form games (1997)

with Mathijs Jansen and Hans Peters

Games and Economic Behavior, Vol 21, 238-252.

Abstract: In this paper an algebraic characterization of consistent assessments in extensive form games (in the sense of Kreps and Wilson, 1982) is given. As a corollary, we show that consistency can be characterized by so-called `simple’ sequences of assessments. The algebraic characterization is used to develop an algorithm which computes the set of consistent assessments. Moreover, the geometrical structure of the set of consistent assessments is described: It turns out to be the intersection of a finite product of simplices with a finite number of logarithmic cones. Finally, the class of extensive form games for which Bayesian consistency implies consistency is characterized.

Consistency of assessments in infinite signaling games (1997)

with Mathijs Jansen and Hans Peters

Journal of Mathematical Economics, Vol 27, 425-449.

Abstract: In this paper we investigate possible ways to define consistency of assessments in infinite signaling games, i.e., signaling games in which the sets of types, messages and answers are complete, separable metric spaces. Roughly speaking, a consistency concept is called appropriate if it implies Bayesian consistency and copies the original idea of consistency in finite extensive form games as introduced by Kreps and Wilson (1982). We present a particular appropriate consistency concept which we call strong consistency and give a characterization of strongly consistent assessments. It turns out that all appropriate consistency concepts are refinements of strong consistency. Finally, we define and characterize structurally consistent assessments in infinite signaling games.