Published and Accepted Papers

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 (with János Flesch) (2011), 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 (with Bram Driesen and Hans Peters) (2010), Accepted for publication in Mathematical Social Sciences.

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 (with Bram Driesen and Hans Peters) (2010), 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 (with János Flesch) (2009), 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 (with János Flesch and Emin Karagozoglu) (2009), 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 (with Rudolf Müller and Sascha Wolf) (2007), 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 (with Tim Schulteis, Hans Peters and Dries Vermeulen) (2007), Economic Theory, Vol. 30, 373-384.

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 (with Hans Peters, Tim Schulteis and Dries Vermeulen) (2006), International Journal of Game Theory, Vol. 34, 457-473.

The Weak Sequential Core for Two-Period Economies (with Jean-Jacques Herings and Arkadi Predtetchinski) (2006), International Journal of Game Theory, Vol. 34, 55-65.

Monotonicity and equal-opportunity equivalence in bargaining (with Antonio Nicolò) (2005), 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 (with Geir Asheim) (2005), 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 (with Laurence Kranich and Hans Peters) (2005), 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 (with Salvador Barberà) (2002), 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 (with Mathijs Jansen and Dries Vermeulen) (2000), 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 (with János Flesch) (1999), 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 (with Hans Peters) (1998), Journal of Optimization Theory and Applications, Vol 98, No. 1, July 1998.

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 (with Mathijs Jansen and Hans Peters) (1997), 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 (with Mathijs Jansen and Hans Peters) (1997),  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.