SONDERFORSCHUNGSBEREICH 504 Rationalitatsk¨ onzepte, Entscheidungsverhalten und ok¨ onomische Modellierung

No. 03-27

Sequential Two-Player Games with Ambiguity

∗ JÈurgen Eichberger ∗∗ and David Kelsey

October 2003

Financial assistance from the ESRC senior research fellowship scheme, award no. H52427502595, the School of Social Science at the University of Birmingham, the Department of Economics at the Uni- versity of Melbourne and The British Academy is gratefully acknowledged. For comments we would like to thank Simon Grant, Hans Haller, Youngse Kim, Bart Lipman, Frank Milne, Shasi Nandeibam, Hyun Shin, Peter Sinclair, Willy Spanjers, Martin Summer, Peter Wakker and participants in semi- nars at Queen’s university, the universities of Melbourne and Birmingham, the Econometric Society World meetings in Tokyo 1995 and the ESRC workshop, Kenilworth 1997. We thank two anonymous referees of this journal for detailed comments which helped to improve the final version. We ∗ Sonderforschungsbereich 504, email: [email protected] ∗∗ Department of Economics, The University of Birmingham, email: [email protected]

UniversitatÈ Mannheim L 13,15 68131 Mannheim SEQUENTIAL TWO-PLAYER GAMES WITH AMBIGUITY

Jürgen Eichberger and David Kelsey Alfred Weber Institut Department of Economics Universität Heidelberg The University of Birmingham

September 2001

This Version: August 20, 2003

Abstract If players’ beliefs are strictly non-additive, the Dempster-Shafer updating rule can be used to define beliefs off the equilibrium path. We define an equilibrium concept in sequential two-person games where players update their beliefs with the Dempster-Shafer updating rule. We show that in the limit as uncertainty tends to zero, our equilibrium approximates Bayesian . We argue that our equilibrium can be used to define a refine- ment of Bayesian Nash equilibrium by imposing context-dependent constraints on beliefs under uncertainty.

JEL Classification: C72, D81 Keywords: Uncertainty-aversion, capacity, Dempster-Shafer rule, bargaining, sig- nalling game.

Address for correspondence: Professor Dr. Jürgen Eichberger, Wirtschaftstheorie I, Alfred Weber Institut, RUPRECHT-KARLS-UNIVERSITÄT HEIDELBERG, Grabengasse 14, D-69117 HEIDELBERG, GERMANY

* Financial assistance from the ESRC senior research fellowship scheme, award no. H52427502595, the School of Social Science at the University of Birmingham, the Department of Economics at the University of Melbourne and The British Academy is gratefully acknowledged. For comments we would like to thank Simon Grant, Hans Haller, Youngse Kim, Bart Lipman, Frank Milne, Shasi Nandeibam, Hyun Shin, Peter Sinclair, Willy Spanjers, Martin Summer, Peter Wakker and participants in seminars at Queen’s university, the universities of Melbourne and Birmingham, the Econometric Society World meetings in Tokyo 1995 and the ESRC game theory workshop, Kenilworth 1997. We thank two anonymous referees of this journal for detailed comments which helped to improve the the final version. 1. INTRODUCTION

Economists have made a distinction between risk (where probabilities are objectively known) and am- biguity (where probabilities are unknown). Until recently it was not clear how to model this formally. Schmeidler (1989) has proposed an axiomatic decision theory, which is able to model ambigu- ity. In this theory, the decision-maker’s beliefs are represented by a capacity (non-additive subjective probability) and (s)he is modelled as maximising the expected value of utility with respect to the capacity. Ambiguity is represented by strictly non-additive capacities. The expectation is expressed as a Choquet integral (Choquet (1953-4)). Schmeidler’s theory will henceforth be referred to as Choquet expected utility (CEU). A number of researchers have applied CEU (or related theories) to games1. Most of these papers consider strategic (normal) form games. Lo (1999) suggests an equilibrium concept for extensive form games under ambiguity. Since he uses the related multiple-prior expected utility theory to model ambiguity, he discusses a number of conceptual problems which arise in the context of dynamic games if players face strategic ambiguity. The paper contains many instructive examples but no general theorems about existence of equilibrium if players face ambiguity. Approaching the problem of extensive form games in a very general way, Lo (1999) cannot exploit one of the strengths of non-additive probabilities, namely that unlike additive probabilities, they can be updated after events with a capacity value of zero. In the present paper, we apply CEU to sequential two-player games. This class of extensive form games comprises many important game- theoretic models in economics such as signalling games, two-stage industrial organisation models or bargaining problems. Restricting ourself to this class of games allows us to ignore some of the consistency problems encountered in Lo (1999). In extensive form games, updating of beliefs on newly acquired information is important. If beliefs are represented by additive probability distributions, then Bayesian updating is the natural method to incorporate the information obtained from the observed moves of the opponents. Bayesian updating however is possible only at information sets which have a positive probability of being reached. As is well-known, play at information sets off the equilibrium path can have a major effect on the equilibrium itself. Thus it is important to determine players’ beliefs at such information sets. Because

1 See Dow & Werlang (1994), Eichberger & Kelsey (2000), Hendon et al (1994), Klibanoff (1996)), Lo (1996) and Marinacci (2000).

2 Bayesian updating puts no restrictions on such beliefs, a multiplicity of equilibria is compatible with Bayesian beliefs. Games with incomplete information are usually plagued by a large number of Bayes-Nash equilib- ria. Signallinggames in particular havetypically anexcessivelylargenumber of equilibriabecause the signal space is large compared to the type space, which implies that most actions will not be observed in equilibrium. The multiplicity of equilibria depends on the lack of constraints on out-of-equilibrium strategies. There is a huge literature in game-theory which tries to impose further constraints on be- liefs by additional rules about howa player should interpret out-of-equilibrium moves in equilibrium. Such constraints on beliefs refine the set of Bayes-Nash equilibria. Compare Mailath(1992) for a survey of refinements in the context of signalling games. Most refinements have been based on for- ward or arguments. A common criticism of such arguments is that, if the initial situation is indeed an equilibrium, then players should conclude from a deviation that the opponent is not rational or does not understand the structure of the game. In this paper, we propose a definition of equilibrium where players have non-additive beliefs and use an updating rule proposed by Dempster and Shafer in the literature for capacities. This equilib- rium notion comprises Bayes-Nash equilibrium as a special case. The Dempster-Shafer updating rule, which is part of our equilibrium concept, has well-defined updated capacities off the equilibrium path as long as there is ambiguity. Capacities can be further constrained by adequate assumptions about be- liefs without affecting consistency of beliefs in an equilibrium under ambiguity. Hence, there is room to put constraints on beliefs which may be specific to situation one wants to model. For example, one can exogenously determine the degree of ambiguity or one can restrict beliefs to agree with an additive prior distribution, if one wants to model a situation where players are completely confident about the distribution of types but ambiguous about their opponents’ strategic behaviour. It is possi- ble to control for the ambiguity of a situation in experiments in order to see how it affects decision behaviour. For individual decision situations, such experiments have been performed (Camerer & Weber (1992)). There are few experiments so far, which focus on strategic ambiguity, but we are confident that such tests can be conducted. One can parametrise the notion of ambiguity and demonstrate existence of equilibrium for any exogenously determined level of ambiguity. This opens up the possibility to study sequences of equi- libria under ambiguitywhich converge to aBayesian equilibrium as ambiguity vanishes. Assumptions

3 about beliefs under ambiguity will determine which Bayesian equilibrium will be selected. An inter- esting aspect of this approach is, even in a Bayesian equilibrium, beliefs off the equilibrium path may be represented by capacities which are not additive. The greater freedom of modelling beliefs under ambiguity provides a novel and useful modelling device for economic applications. Insection2 we introduce the CEU model and demonstrate someproperties of CEUand the Dempster– Shafer updating rule. Section 3 introduces our for two-stage games under ambiguity and relates it to some existing solution concepts. Section 4 studies limits of sequences of equilibria as ambiguity vanishes. We show that ambiguous beliefs can select among Bayesian equilibria. Section 6 concludes the paper. All proofs are gathered in an appendix.

2. CEU PREFERENCES AND DS-UPDATING

In this section we consider a finite set S of states of nature. A subset of S is referred to as an event. The set of possible outcomes or consequences is denoted by X. An act is a function from S to X. The space of all acts is denoted by A(S) := a a : S X . The decision-maker’s preferences f j ! g over A(S) are denoted by <. A capacity or non-additive probability on S is a real-valued function º on the subsets of S; which satisfies the following properties: (i) A B implies º (A) º (B) ; µ 6 (ii) º (?) = 0; º (S) = 1: The capacity is said to be convex if for all A; B S; º(A B) º(A) + º(B) º(A B): µ [ > ¡ \ Representing beliefs by a convex capacity is compatible with experimental evidence (see Camerer & Weber (1992)) and is commonly used in applications of CEU to model ambiguity averse be- haviour. We shall assume that all capacities are convex. We shall use capacities to represent the beliefs of players. In game-theoretic applications, the opponents’ combinations will be the relevant states for a player. It is possible to define an expected value with respect to a capacity to be a Choquet integral (Choquet (1955)). For any function Á : S R and any x X let B(x Á) := s S : Á (s) x be the ! 2 j f 2 > g event in which Á is greater than or equal to x: Similarly, denote by B(x Á) := s S : Á (s) > x the j f 2 g event in which Á produces a strictly greater outcome than x: The Choquet integral of Á with respect

4 to the capacity º is defined as n Á dº := x [º(B(x Á)) º(B(x Á))]; (1) ¢ j ¡ j x Á(S) Z 2X where the summation is over the range of the act, Á(S) := x X s S; Á(s) = x : f 2 j 9 2 g We shall assume that preferences may be represented by Choquet expected utility (CEU) with respect to a capacity, i.e.

a b u(a(s))dº (s) u(b (s)) dº (s) : < , > Z Z Such preferences have been axiomatised by Gilboa (1987) and Sarin & Wakker (1992).

Definition 2.1 The degree of ambiguity of capacity º is defined by ¸(º) := 1 min(º(A) + º(S A)): A S ¡ µ n This definition is adapted from Dow & Werlang (1992). It has been justified epistemolog- ically by Mukerji (1997). The degree of ambiguity is a measure of the deviation from additivity. For an additive probability ¸(º) = 0, while for complete ambiguity ¸(º) = 1. The following re- sult confirms that the degree of ambiguity is a reasonable measure of deviation from additivity, for convex capacities2.

Lemma 2.2 If a convex capacity º has zero degree of ambiguity then it is additive.

2.1 The support of a capacity

In game theory, players are assumed to maximise their expected payoffs. Strategy choices are consid- ered in equilibrium if beliefs are consistent with actual behaviour. The strongest form of consistency, Nash equilibrium, requires players’ beliefs to coincide with their actual behaviour. In an alternative and equivalent definition of Nash equilibrium the strategies in the support of the opponents’ beliefs about a player’s behaviour must be best responses of that player. In other words, players expect their opponents to play only best-response strategies. If decision makers’ ambiguity is modelledby capacities then there are several concepts of a support which all coincide with the usual notion of support in the case of additive capacities. In this paper we will use the following definition.

2 The lemma is false if convexity is not assumed. A counter-example would be the class of symmetric capacities studied by Gilboa (1989) and Nehring (1994).

5 Definition 2.3 A support of a capacity º is an event A S such that º(S A) = 0 and º(S B) > 0; for all events B A: µ n n ½

This definition of the support is due to Dow & Werlang (1994). Above we define the support of a capacity to be a minimal set whose complement has a capacity value of zero. This is equivalent to the usual definition of support (i.e. a minimal set of probability one) for an additive capacity but will generally yield a smaller set if the capacity is not additive. With this support notion every capacity has a support. However it has been criticised because the support is not necessarily unique and states outside the support may affect decision making if a bad outcome occurs on them. In Eichberger & Kelsey (2001a) we provide an extensive discussion of various support notions for capacities suggested in the literature3. In particular, we show that the support is unique if and only if one requires in addition º(B) > 0; for all events B in the support. Adding this requirement to Definition 2.3 guarantees a unique support but there are convex capacities for which no such support exists. In game-theoretic applications, the lack of uniqueness poses no problem because we show the existence of an equilibrium in which beliefs have a unique support. Moreover, our results and examples all have unique supports, which satisfy this additional restriction. More substantial is the objection to Definition 2.3 that states outside the support are not Savage- null. An event E is Savage-null if outcomes on E never affect a decision, i.e. if a c b c for all E » E acts a; b; c; where aEc denotes an act which yields a(s) for all states in E and c(s) in all other states. We believe that this argument is not appropriate in game-theoretic applications. We will argue this case below in context with the game-theoretic equilibrium concept, which we advance in this paper. 2.2 CEU preferences and DS-updating

In sequential games players may receive information about the opponents by observing their moves in earlier stages of the game. In particular in signalling games, second-stage players will try to infer informationabout characteristics of their opponentsfrom the signals which they receive. It is therefore necessary to specify a rule for how to revise beliefs represented by capacities in the light of new information. If beliefs are additive, Bayes’ rule is the unique updating rule which maintains additivity. As in the case of the support, with non-additive capacities there are several updating procedures, which all

3 Haller (2000), Marinacci (2000) and in particular Ryan (1999) discuss and argue for other support notions.

6 coincide with Bayesian updating in the case of additivity. Gilboa & Schmeidler (1993) provide an exposition and an axiomatic treatment from a behavioural perspective. In this paper we choose the Dempster-Shafer belief revision rule (see Shafer (1976)).

Definition 2.4 Dempster-Shafer revision The Dempster-Shafer revision of capacity º given event E S is ½ º((F E) (S E)) º(S E) º(F E) := \ [ n ¡ n : (2) j 1 º(S E) ¡ n The axiomatisation by Gilboa & Schmeidler (1993) shows that the Dempster-Shafer rule (DS-rule) may be interpreted as a pessimistic updating rule. If one views capacities as constraints on a set of additive probability distributions then it is equivalent to a maximum likelihood updating procedure. For extensive-form games the DS-rule is particularly interesting, since it may be defined even when º(E) = 0. If the event E; about which the decision maker obtains information, was ambiguous, º(E)+º(S E) < 1; then the DS-rule will be well-defined even if it has a prior capacity value of zero. n Thus, it maybe possible toupdate non-additive beliefs on events with acapacity value of zero. We will argue in Section 4 that this property of DS-updating provides an approach to based on ambiguity of players. Contrary to the refinements of Bayes-Nash equilibrium based on second-order reasoning about out-of-equilibrium moves, which dominate the literature, ambiguity- related refinements can be given a behavioural content which is independent of the equilibrium notion.

3. SEQUENTIAL TWO-PLAYER GAMES

In this paper we will consider two-player games with complete and incomplete information, where players move sequentially. Without loss of generality, we will assume throughout that player 1 moves first and that player 2 knows the move of player 1 when she makes her move. Player 1 may have one of several types which are described by a finite set T: If T contains a single type the game has , otherwise it is of incomplete information. Beliefs about types will be described below. Both players choose actions from finite action sets Ai; i = 1; 2: Their payoffs are described by the utility functions ui(s;a; t); i = 1; 2: Strategies of player 1 coincide with actions, S1 := A1: In contrast, player 2 who observes the action of player 1 can condition her moves on this observation. Hence, S2 := s2 s2 : S1 A2 f j ! g 7 denotes the strategy set of player 2, which is also finite because the action sets of both players are finite. We will denote by s2(s1) A2 the action, which player 2 will choose in response to s1 2 according to her strategy s2: Both players hold beliefs about the opponent’s behaviour which are represented by convex capac- ities. Player 1 has beliefs º2 about the strategies in S2; which player 2 will choose. A belief º2 on S2 of player 1 about player 2’s strategy induces a set of beliefs º2 s1 S1 about the actions in A2; f jj j 2 g 1 1 2 2 2 2 2 1 which player 2 will choose in response to a strategy sj S : ºj(E) := º ( s S s (sj ) E ): 2 e f 2 j 2 g For notational convenience we will state definitions and results in terms of º2 though the respective e statements translate easily into statements about the set of beliefs about actions º2 s1 S1 : f j j j 2 g Player 2 has to form beliefs about strategic behaviour of the possible types of player 1. Beliefs e of player 2 about player 1’s type-contingent strategy choices are represented by the capacity º1 on S1 T: These beliefs represent jointly this player’s ambiguity about type and strategy4. In game- £ theoretic applications it is usually assumed that prior beliefs about types are and additive. We can constrain º1 to be compatible with an exogenous prior distribution over types, which is represented by a (possibly additive) capacity ¹ on the type space T:

1 Definition 3.1 A capacity º on S T agrees with the capacity ¹ on T if, for any subset T 0 of T, º(S1 T ) = ¹(T ): £ £ 0 0 Whether the prior belief of player 2, º1; agrees with a prior distribution on types or not, once player 2 observes the action s1; which player 1 chooses she will have to revise her beliefs in the light of this information. To simplify notation, we will write

1 1 1 1 1 º (T 0 s ) : = º (S T 0 s T) (3) j 1 1 £ jf g £1 1 1 1 1 º (( s T 0) ((S s ) T)) º ((S s ) T ) = f g £ [ nf g £ ¡ nf g £ 1 º1((S1 s1 ) T) ¡ nf g £ 1 to denote the DS-updated capacity of the event T 0 T if the action s has been observed. This ½ DS-update is well-defined if º1((S1 s1 ) T ) < 1 holds. This condition is satisfied if nf g £ either player 2 feels ambiguity about player 1’s choice of the strategy s1, i.e., ² the event s1 T is ambiguous, i.e. if º1( s1 T) + º1((S1 s1 ) T) < 1; f g £ f g £ nf g £ or player 2 is certain that player 1 will play strategy s1 with positive probability, i.e., ² 4 Here we follow the approach of Milgrom & Roberts (1986) for the representation of games under incomplete information.

8 the event s1 T is unambiguous and º1( s1 T ) > 0: f g £ f g £ Finally, let

P 1(s1 t; º2) := u1(s1; s2(s1); t) dº2 (4) j Z and, for a capacity ½ on T;

P 2(a s1; ½) := u2(s1; a; t) d½ (5) j Z be the CEU-payoff of player 1 and 2 respectively. The belief ½ about player 1’s type will either be º1 or º1( s1); depending on whether beliefs are formed by DS-updating or not. ¢j 3.1 Dempster-Shafer equilibrium

In order to see the relationship between the Dempster-Shafer equilibrium concept, which will be proposed in this section, with the familiar notion of a Perfect Bayesian Equilibrium, consider the case of additive capacities ¼1 and ¼2 representing players’ beliefs. A Perfect Bayesian Equilibrium (PBE) with prior distribution p on T is

a probability distribution ¼1 on S1 T such that ¼(s1; t) = p(t) for all t T; ² £ s1 S1 2 2 a probability distribution ¼2 on S2; P ² 1 and a family of probability distributions ¹( s ) s1 S1 on T such that: ² f ¢j g 2 a) (s1; t) supp¼1 then s1 argmax ¼2(s2) u1(s1; s2(s1); t); 2 2 s1 S1 s2 S2 ¢ 2 P2 b) s2 supp ¼2 then s2(s1)e arg max ¹(t s1) ue2(s1; ae; t); 2 2 a A2 t T j ¢ 2 2 (PBE) for all s1 PS1; e 2 e 1 ¼(s1;t) 1 c) ¹(t s ) = ¼(s1;t) if ¼(s ; t) > 0: j t T t T 2 2 P P 1 The standard interpretation is that ¼1(s1 t) := ¼(s ;t); ¼2(a s1) := ¼2( s2 S2 s2(s1) = a ) are j p(t) j f 2 j g behaviour strategies and ¹( s1) are beliefs at the information set reached after move s1: Behaviour ¢j strategies are identified with beliefs of players. Note that the belief interpretation requires ¼1 to be the belief of player 2 about the behaviour of player1 and vice versa. 1 Without the family of probability distributions ¹( s ) s1 S1; conditions PBE-a and PBE-b define f ¢j g 2 a Bayesian equilibrium if one uses ¼(s1; t) instead of ¹(t s1) in PBE-b. Obviously, all PBE are j Bayesian equilibria. All a PBE requires in addition to the conditions of a Bayesian equilibrium is

9 optimality of behaviour at information sets off the equilibrium path, i.e., after moves s1 such that ¼(s1; t) = 0; with respect to some arbitrary additive belief ¹(t s1): Hence, it only rules out strictly t T j 2 dPominated actions at such information sets. Since beliefs at information sets off the equilibrium path are arbitrary, there are usually many PBE depending on the beliefs ¹(t s1) assumed at information j sets off the equilibrium path. This multiplicity of PBE poses a serious problem in games, where there are few types of player 1 and many more strategies of this player, as it is typically the case in signalling games. The literature is therefore particularly rich in refinements for signalling games. Mailath (1992) provides a good survey of the refinements applied in the context of signalling games. We show below in Example 4.2 how with , a quite natural assumption on beliefs can reduce this multiplicity of equilibria.

Games instrategic form wherethebeliefs of players arerepresentedby capacitieshave been studied by Dow & Werlang (1994), Marinacci (2000) and Eichberger & Kelsey (2000). In the strategic form, beliefs (º1; º2) form an equilibria under uncertainty if these beliefs have supports containing only pure strategies which are optimal for the respective player given these beliefs. In this paper we study sequential two-player games where the action of player 1 conveys infor- mation to player 2. This requires a reformulation of the equilibrium concept. In contrast to the equilibrium notion in the strategic form, we will require that player 2’s strategy consists of actions that are optimal at each information set, i.e. after observing the action of player 1.

Definition 3.2 Dempster-Shafer Equilibrium (DSE) A Dempster-Shafer equilibrium consists of capacities º1 on S1 T; º2 on S2 and a family of capac- 1 1 £ 2 ities ½( s ) s1 S1 on T such that there are supports suppº and supp º satisfying: f ¢j g 2 a) (s1; t) supp º1 then s1 arg max P 1(s1 t; º2); 2 2 s1 S1 j 2 b) s2 supp º2 then s2(s1)e arg max eP 2(a s1; ½( s1)) 2 2 a A2 j ¢j (DSE) f2or all s S1; e 2e 1 1 1 1 1 1 c) ½(T 0 s ) = º (T 0 s ) if º ((S s ) T ) < 1: j j nf g £

In general, the support of a capacity need not be unique. Hence, a DSE requires us to specify a

10 set of beliefs and some associated support. In most applications the capacities considered will have a unique support and this seemingly arbitrary choice of support for a given capacity poses no problem5. Condition DSE-a of Definition 3.2 guarantees that only optimal type-contingent strategies of player 1 will be included in the support of player 2’s beliefs. Similarly, by condition DSE-b there are only strategies in the support of player 1’ beliefs which prescribe optimal behaviour after a strategy of player 1. We call the equilibrium a Dempster-Shafer equilibrium (DSE) because, according to DSE- c, beliefs of player 2 about the type of player 1 are obtained by the DS-updating rule. Remark: A Dempster-Shafer equilibrium is defined as a set of beliefs over type-contingent strate- 1 2 1 gies (º ; º ) and a set of updated beliefs after strategy choices of player 1, ½( s ) s1 S1: By Con- f ¢j g 2 1 dition DSE-c the updated beliefs ½( s ) s1 S1 are however a derived concept. When referring to a f ¢j g 2 DSE we therefore often mention only (º1; º2); if there is no danger of confusion. The degree of ambiguity ¸(ºi); formally defined in the previous section, is a property of the equi- librium beliefs (º1; º2). We will demonstrate below in Proposition 3.5 that one can take this degree of ambiguity as an exogenous parameter and deduce equilibrium beliefs. The DSE concept is useful for economic applications because one can study games under different degrees of ambiguity. Nash equilibrium is a special case of an equilibrium under no ambiguity. Since the DS-updated capacity ½( s1) is a derived concept its degree of ambiguity ¸(º1( s1)) is also a derived property6. ¢j ¢j We define the degree of ambiguity of a game to be the maximal degree of ambiguity of the equi- librium beliefs. Formally, we will say that

a Dempster-Shafer equilibrium (º1; º2) has degree of ambiguity ¸ [0; 1] if ¸ := max ¸(º1); ¸(º2) ; ² 2 f g and a Dempster-Shafer equilibrium (º1; º2) agreeswith the additive prior distribution p onT if º1(S1 ² £ t ) = p(t) for all t T: f g 2 If the beliefs of a DSE are additive, i.e., if there is no ambiguity, and if there is a common prior distribution over types then a DSE is a Bayesian equilibrium. The following proposition relates the Dempster-Shafer equilibrium concept to the Bayesian equi-

5 There are, however, interesting cases where supports are not unique, yet where there is only one support for each capacity which is consistent with the optimality conditions DSE-a and DSE-b. 6 For specific types of capacities one can prove implications for the degree of ambiguity of their DS-update. Compare Eichberger & Kelsey (1999).

11 librium and the perfect Bayesian equilibrium notions.

Proposition 3.3 a) A Dempster-Shafer Equilibrium (º1; º2) with a degree of ambiguity ¸ = 0; for which the belief of player 2, º1; agrees with the additive prior distribution p on T; is a Bayesian Equilibrium. b) Consider a Dempster-Shafer Equilibrium (º1; º2) with a degree of ambiguity ¸ = 0; for which the belief of player 2, º1; agrees with the additive prior distribution p on T: If for each strategy s1 S1 there exists a type t T such that (s1; t) suppº; then the Dempster-Shafer Equilibrium 2 2 2 (º1; º2) is a perfect Bayesian equilibrium.

Discussion of the DSE concept. Proposition 3.3 shows that Bayesian equilibrium and perfect Bayesian equilibrium are special cases of a DSE if there is no ambiguity. There is mounting exper- imental evidence that Nash equilibrium and its refinements do not yield good predictions of actual behaviour in all games. Situations in which one finds consistent deviations from the Nash equilib- rium hypothesis include bargaining, e.g. the ultimatum bargaining experiments (Roth (1995)), coordination problems (Ochs (1995)), public goods provision (Ledyard (1995)), and signalling games (Brandts & Holt (1992,1993)). These findings pose a challenge to theory and call for the investigation of modified equilibrium concepts. Some of these anomalies can be explained by altruistic preferences, e.g. in the cases of public goods and bargaining. Even in these cases however, it is difficult to account for all the observed phenomena by modifying preferences alone. The approach we propose in this paper focuses on ambiguity about the behaviour of the opponent players. We do not give up the idea that players maximise their expected payoff but we investigate howambiguity about the strategic behaviour of opponents affects the equilibria of games. Ambiguity about the opponents’ behaviour may arise by a number of reasons. Traditional game theory maintains that players deduce beliefs about the opponents’ behaviour from firm knowledge about the preferences of opponents. In games with incomplete information one re- places knowledge about other players’ payoffs, with knowledge about the probability distribution over possible types of payoffs. Though logically sound and completely consistent, modelling games of incomplete information by probability distributions over type spaces assumes that players have extremely high computational abilities. Ambiguity about the strategic behaviour of the opponent players reflects the difficulty of settling

12 one’s beliefs firmly on a particular probability distribution over types and their behaviour. This does not imply that players do not care about the motivation of the opponents or that they do not consider the possibility of different types of opponents. It means however that their behaviour may be influenced by the fact that they do not feel certain about such inferences. Ambiguous beliefs represented by capacities, allow us to model players who hold and process information about their opponents in order to predict their behaviour but who, depending on the situation, may feel more or less certain about these predictions. If ambiguity is about two or more possible characteristics of an opponent then beliefs should be modelled by a capacity over the relevant type space, if ambiguity concerns the correct description of the situation in general it is best modelled by ambiguity about the opponents’ strategy choices. Equilibrium concepts for ambiguous players, like the DSE suggested here, provide also a unified framework, in which the completely consistent beliefs of Nash equilibrium analysis as well as behaviour influenced by ambiguity can be analysed.

3.1.1 Ambiguity about the strategy choice of the opponents

In traditional game-theoretic reasoning, players trust completely their reasoning about the rationality of their opponents. If players believe that an opponent’s strategy is strictly dominated, then they will act on the presumption that this player will never choose this strategy. Similarly, strategies of the opponents which are not in the support of the capacity representing a player’s belief should not influence this player’s behaviour. The following example illustrates that these properties are not true for DS-equilibria. In Example 3.1 there are two DS-equilibria, one which describes behaviour similar to the backward-induction Nash equilibrium. The second DSE shows that strategies of the opponent with bad outcomes may influence the decision of a player, even if the opponents strategy is strictly dominated and not in the support of this player’s beliefs.

Example 3.1 Frivolous lawsuits7

Bebchuk (1988) studies legal disputes where the plaintiff threatens to go to court even if the ex- pected value of the court case is negative in order to extract a settlement offer from the defendant. Figure 1 represents a stylised version of this situation. Once the potential plaintiff has threatened to go to court the defendant, D; can make an settlement offer o which will be accepted or refuse to

7 The structure of this game corresponds to the well-known entry game in Industrial Organisation.

13 1; 1 d r P

r no c 1; 1 ¡ ¡ D r r o 0; 3 r

Figure 1: Frivolous suit make an offer, no; in which case the plaintiff, P; has to decide whether to drop the case, d; or to go to court, c: The payoffs reflect the incentives of the players. If the defendant makes no offer, no; and the potential plaintiff decides not to file the suit, d; both players receive a payoff of 1: If the plaintiff goes to court, c; both players obtain 1, which reflects the negative expected value of the court case. ¡ A settlement offer, which is accepted, yields the plaintiff a payoff of 3 and the defendant a payoff of 0: The settlement yields the plaintiff a higher payoff than not going to court, the incentive for the frivolous suit8. This is a game with complete information where player 1, the defendant, has a single type t, the no- tation of which is suppressed. Hence, (½(t no); ½(t o)) = (1; 1) in any DSE. Equilibrium beliefs for j j the two types of DS-equilibria of this game are given in the following table. DS-equilibria (ºD; ºP ); (ºD; ºP ) f 1 1 2 2 g D D P P event º1 ( ) º2 ( ) event º1 ( ) º2 ( ) D ¢ ¢ P ¢ 1 P ¢ 1 no ® > 0 0 d ® 2 ® < 2 fo g 0 ®D > 0 fcg 0¸ 0 Sf Dg 1 1 Sf Pg 1 1 0 0 0 0 s;upport: no o s;upport: d d f g f g f g f g

D P The first DSE (º1 ; º1 ) describes behaviour which is similar to the backward induction equilibrium in

8 This example is a slightly modified and parametrised version of the model in Bebchuk (1988), p. 441.

14 the analysis without ambiguity. The equilibrium beliefs need however not be additive. The plaintiff decides not to go to trial. Whether the defendant D will refuse a settlement or not depends on the strength of the belief ®P that the plaintiff will drop the case. For ®P 1 the defendant will make ¸ 2 no offer, otherwise a settlement offer is made. Filing the suit becomes weakly optimal only if the plaintiff is completely certain that the defendant will not refuse a settlement, ºD( no ) = 0: i f g D P The second DSE (º2 ; º2 ) shows behaviour which cannot occur in a Nash equilibrium. In this case, the plaintiff plans to drop the case if challenged. The defendant, on the other hand, feels sufficiently ambiguous, ºP ( d ) = ®P < 1; about the prediction that the plaintiff will not file a suit, suppºP = 2 f g 2 2 d ; and will offer settlement. Such behaviour cannot be supported by a Nash equilibrium. Yet it f g does not appear unreasonable. After careful consideration of the situation, the defendant may well recognise the incentives of the plaintiff to drop the action. This conclusion depends however on the correct perception of the situation as modelled by the game. If the defendant feels sufficiently uncertain about this information she may be justified in offering settlement. Moreover, since the defendant does not challenge the potential plaintiff, no information about the actual behaviour of the plaintiff is generated. So one is also justified to call such a situation an equilibrium. From Part (b) of Definition 3.2 it is clear that no strictly dominated strategy will be chosen in a DSE. Therefore, one may be led to conclude that all DS-equilibria are backward induction equilibria. D P This conclusion is false however, as the DSE (º2 ; º2 ) in Example 3.1 demonstrates. Ambiguity may prevent players from choosing strategies which expose them to situations where they might be hurt by a strictly dominated choice of the opponents. This may be so, even if they do not expect the opponents to play strictly dominated strategies, if they do not trust this conclusion sufficiently. Example 3.1 shows also that the DSE concept can describe behaviour, which is inconsistent with the strict consistency requirements of a Nash equilibrium. Indeed, behaviour, as in equilibrium D P (º2 ; º2 ); can occur only if the worst outcome of an interaction can influence the decision of a player even if it is on events which are outside the support. If one would constrain the support notion to make events outside the support Savage-null, i.e., irrelevant for the player, then this equilibrium would dis- appear. This is obvious from the following lemma which has been proved in Ryan (1998) (Lemma 1, p.34).

Lemma 3.4 Let º be a capacity ona set S: Anevent E S is Savage-null if and only if º(S E) = 1: ½ n 15 Applying Lemma 3.4 to the equilibrium in Example 3.1 would imply ®D = ®P = 1: Hence, DSE D P D P (º2 ; º2 ) wouldno longer be possible and only the DSE (º1 ; º1 ) corresponding to a Nash equilibrium would survive. Indeed, Lo (1996) (Corollary of Proposition 4, p. 468) shows that this is true for all two-player games. Adopting such a strong notion of support therefore defeats the objective of modelling ambiguity of players. DSE offers more possibilities to model economic situations than traditional Bayesian analysis be- cause consistency requirements on beliefs are weaker. In our opinion this additional freedom is useful for modelling economic situations since it allows us to include aspects of the economic environment, which are precluded by Bayesian analysis, but which are supported by experimental evidence or other robust findings. It is beyond the scope of this paper to investigate these applications in depth.

3.1.2 Ambiguity and Pessimism

With convex capacities, as we assume throughout this paper, ambiguity aversion is built into the concept of the Choquet integral. This pessimism concerns however only events on which there is ambiguity. DSE leaves us with more modelling options. DSE allows us to distinguish between the of players for unambiguous choices and their pessimism in the face of ambiguity. For example, if one would like to restrict ambiguity to an opponent’s strategic behaviour and considers information about types as hard, one can model this by a capacity which agrees with an additive prior distribution. In this case, pessimism is restricted to the behaviour of the opponent but not to the probability over types. The following example which is due to Ryan (2002) illustrates such a modelling option9.

Example 3.2 (Ryan 2002)

Consider a signalling game with two players, i = 1; 2; where player 1 can be one of two types 1 T = t1; t2 : It is known that each type occurs with probability : Action sets for the two players are f g 2 A1 = R; L and A2 = U; D : Figure 2 represents the game. f g f g

9 This game has been advanced in Ryan (2002) as an argument for a stronger support notion called ‘‘robust support’’. In Eichberger & Kelsey (2001) we provide a detailed and more formal discussion of this approach.

16 1; 1 0; 1 r U U r

2 L t1 R 2 r r r D 1 D 1; 0 2 0; 4 r r 0 r 1 0; 1 2 1; 1 r U U r

2 L t2 R 2 r r r D D 0; 4 1; 0 r r

Figure 2: Ryan (2002)

It is easy to check that this game has a unique perfect Bayesian equilibrium where

player 1 of type t1 chooses L; ² player 1 of type t2 chooses R; ² player 2 chooses ² U in response to L and R: Ryan (2002) (p.12) argues that this equilibrium describes the only sensible behaviour in this game because each type of player 1 has a strictly dominant strategy and player 2, knowing the strategy of both types of player 1, maximises her payoff by choosing U: Moreover, the move U is also recom- mended if player 2 is ambiguity averse since it guarantees the certain payoff of 1 no matter what type player 1 turns out to be. 1 1 2 For a degree of ambiguity ® < 2; the following beliefs (º ; º ) are a DSE which agrees with the 1 1 additive prior distribution (p(t1); p(t2)) = (2 ; 2): 1 1 1 º ( (t1; L) ) = º ( (t2; R) ) = ® 2 ; 1 f g f 1 g ¢ 1 º ( (t1; L); (t2; L) ) = º ( (t1; L); (t2; L) ) = ® 2 ; 1 f g 1 f g 1 ¢ º ( (t1; L); (t1; R) ) = º ( (t2; R); (t2; L) ) = 2; 1 f g f 1 g 1 º ( (t1; L); (t2; L); (t2; R) ) = º ( (t1; L); (t2; L); (t2; R) ) = (1 + ®) 2; º1(Ef ) = 0 for all othergE T f S1; g ¢ ½ £ and º2 is an additive probability distribution with º2( (D; D) ) = 1: f g

17 1 One checks easily that, for ® > 0, supp º = (t1; L); (t2; R) : The DS-updates are additive and f g 1 1 1 1 ® always well-defined: ½(t1 L) := º (t1 L) = 2 ® and ½(t1 R) := º (t1 R) = 2¡®: j j ¡ j j ¡ 1 2 1 2 1 2 Computing the CEU payoffsfor these beliefs yields P (L t1; º ) = P (R t2; º ) = 1; P (R t1; º ) = j j j 1 2 P (L t2; º ) = 0 and j

1 for a = U 1 1 ® for s = L 4 2¡® for a = D 2 1 1 8 ¢ ¡ P (a s ; º ) = > e : j > > 1 for ea = U 1 < 1 ® for s = R 4 ¡ for a = D e ¢ 2 ® > ¡ > e 2 : For ® < 3; D is the best response of player 2 no maetter which strategy player 1 chooses, and L 1 and R are the best strategies for player 1 of type t1 and t2 respectively. Since the support of º is 2 (t1; L); (t2; R) and of º is D;D ; playing D is a Dempster-Shafer equilibrium. f g f g For ® 2 ; (U; U) is the of player 2 and the associated DSE yields the same behaviour ¸ 3 as the perfect Bayesian equilibrium. Notice that the DS-updates º1(t s1) equal the prior distribution for ® = 0; the case of complete 1j strategic ambiguity ¸(º1) = 1. For ® = 1; that is complete strategic certainty ¸(º1) = 0; the DS- updates correspond to the Bayesian updates, º1(t L) = 1 and º1(t R) = 0: 1j 1j If there is no strategic ambiguity, then player 1’s moves provide a perfect signal for the type of 1 player 1, i.e. suppº = (t1; L); (t2; R) ; and player 2 will respond by choosing U in response. f g With complete ambiguity about player 1’s strategy choice, player 2 will assess the likelihood of the 1 two types with the prior probability 2 : Based on the expected payoff with respect to the unambiguous prior distribution player 2 will find action D optimal, and not U: Whether the action U or D is chosen depends therefore on the degree of ambiguity which player 2 feels about the deduced equilibrium behaviour. If ambiguity is low, ¸(º1) = 1 ® < 1; then player 2 will choose U and if ambiguity ¡ 3 is high she will choose D: The critical level which determines when the behaviour of player 2 will change depends, of course, on the payoff of actions. If a player feels great ambiguity regarding the strategy of the opponent but not with respect to the prior type distribution, then DS-updating on the observed actions leads the player to disregard the ambiguous strategy and to decide based on the unambiguous prior. Faced with strategic informa- tion, a player who is extremely ambiguous about strategic information and unambiguous about type

18 information will revert to the unambiguous information of the prior distribution10. One can, of course, question the assumption about the unambiguous prior distribution. Indeed, we do not require that beliefs do in general agree with unambiguous priors. It is an easy exercise to check that, with complete ambiguity about the prior distribution, the argument that pessimism commends to play U in order to secure the constant payoff of 1 is correct. 3.2 Existence and properties of DSE

Since Bayesian equilibria are DS-equilibria with a degree of ambiguity ¸ = 0, existence of a DSE is guaranteed under the usual conditions. It is not clear however whether there exist DS-equilibria for arbitrary degrees of ambiguity ¸ and arbitrary prior beliefs about types. Proposition 3.5 shows that DS-equilibria exist under the usual assumptions for any degree of ambiguity.

Proposition 3.5 For any degree of ambiguity ¸ (0; 1) and any additive prior probability distribu- tion p on T; there exists a Dempster-Shafer equilib2rium with this degree of ambiguity ¸ which agrees with the distribution p on T:

Proposition 3.5 shows that the DSE concept can be applied in all cases, in which standard Nash equilibria exist. Moreover, it shows that one can choose the degree of ambiguity ¸ exogenously as a characteristic of a situation and still obtain DS-equilibria. This property is particularly important in economic applications where one wants to study the impact of ambiguity on the behaviour of agents11 Games with complete information, i.e., with a type space containing a single type, T = 1; form j j D P an important special case to which one can apply DSE. The DSE (º2 ; º2 ) in Example 3.1 shows that behaviour in DS-equilibria does not necessarily correspond to behaviour in backward induction equilibria. Backward induction in the presence of Knightian uncertainty has also been discussed by Dow & Werlang (1994). This paper shows that if there is ambiguity, there are non-backward induction equilibria in the finitely repeated prisoner’s dilemma game. These equilibria arise with large degrees of ambiguity, which is compatible with our analysis. Dow & Werlang (1994) analyse games in normal form. Our theory confirms their analysis with an extensive form solution concept based on Knightian uncertainty. We believe that an extensive form solution concept is preferable, since DSE

10 1 Ryan (2002) considers the case ® = 2 : In this case, the DSE predicts player 2 to play D: He sees a tension between the interpretation of Choquet preferences as pessimistic and the preference for the action D which is risky rather than playing U; an action yielding a constant outcome of 1. 11 Eichberger & Kelsey (2001, 2001b) study applications to economic problems.

19 requires that equilibrium strategies are optimal when each move is made. A solution defined on the normal form cannot do this.

4. DSEL AS A NASH EQUILIBRIUM REFINEMENT

In Example 3.1 the DSE with little or no ambiguity selects the backward induction equilibrium. With no ambiguity ¸ = 0; Dempster-Shafer equilibria, where each strategy of player 1 is played by some type, are perfect Bayesian equilibria (Proposition 3.3). These results suggest that DSE equilibria with ambiguity may provide reasonable restrictions on beliefs, which in the limit as ambiguity vanishes se- lect Bayesian equilibria which are robust with respect to ambiguity. In order to explore this possibility more formally we define a Dempster-Shafer Equilibrium Limit (DSEL). Ambiguity may affect beliefs over types and beliefs over strategy choice. We do not want to exclude the possibility that ambiguity extends also to ambiguity about types, but we will require only ambiguity about strategies in order to allow for capacities which agree with an additive prior distribution.

Condition A A DSE (º1; º2) is subject to strategic ambiguity if

º1( s1 T) + º1((S1 s1 ) T) < 1 f g £ nf g £ holds for all s1 S1: 2 If a DSE is subject to strategic ambiguity, then the degree of ambiguity is strictly positive, even if the equilibrium beliefs agree with an additive prior distribution over types. Thus, Condition A allows us to consider sequences of Dempster-Shafer equilibria with positive degree of ambiguity which agree with a given additive prior distribution over types.

Definition 4.1 Dempster-Shafer Equilibrium Limit (DSEL) 1 2 1 A set of beliefs (º ; º ) and updated beliefs ½( s ) s1 S1 is a Dempster-Shafer Equilibrium Limit f ¢j g 2 1 2 (DSEL) if it is the limit of asequenceof strategically ambiguous Dempster-Shafer equilibria((ºn; ºn); 1 ½n( s ) s1 S1 ) such that the degree of ambiguity ¸ tends to zero as n tends to infinity. f ¢j g 2

1 2 1 By Proposition 3.5 there exists a DSE ((ºn; ºn); ½n( s ) s1 S1) for any degree of ambiguity f ¢j g 2 1 1 1 1 1 ¸n > 0; where, for all s S ; ½ ( s ) is well-defined by the DS-updates º ( s ): By convexity of 2 n ¢j n ¢j the capacities, ºi (E) [0; 1]; for all events E Si; i = 1; 2; and for all n: Since we consider finite n 2 µ games, the sequence (º1; º2) is contained in [0; 1]m; where m = T + S1 + S2 : Hence, for any n n j j j j j j 20 1 2 1 2 1 1 sequence ¸n 0 theremust be a converging subsequence (º ; º ) (º ; º )and½ ( s ) ½( s ) ! n n ! n ¢j ! ¢j for all s1 S1: Thus, DSEL is always well-defined. 2 1 Notice that a DSEL requires also to specify a sequence of updated capacities ½n( s ) s1 S1 : f ¢j g 2 Since we impose strategic ambiguity there exists always a supporting sequence of Dempster-Shafer 1 equilibria for which the updated beliefs ½n( s ) s1 S1 are well defined by the DS-updates. Even if f ¢j g 2 1 2 1 DS-updates are well-defined along the sequence of DS-equilibria, a DSEL ((º ; º ); ½( s ) s1 S1) f ¢j g 2 may have non-additive DS-updates ½( s1) for strategies s1; which are not played in equilibrium. ¢j The following example shows that beliefs off the equilibrium path need not be additive. In partic- ular, the sequence of DS-equilibria supporting a DSEL

can be strategically ambiguous, ² agree with an additive prior distribution and ² have well-defined DS-updates. ² Notice that a DSEL also requires the specification of a sequence of updated capacities. Yet, as the degree of ambiguity converges to zero, additive beliefs (º1; º2) obtain in the limit, but DS-updates ½( s1) may remain non-additive if strategy s1 is not played in the DS-equilibria of the supporting ¢j sequence.

Example 4.1

Consider the signalling game, in Figure 3. The strategy set of player 1 is A1 = L; R and of player f g 2, A2 = u; m; d : There are two types of player 1, T = t ; t , which occur with probability p f g f 1 2g 1 and p2; p1 > p2 for concreteness. In any perfect Bayesian equilibrium both types of player 1 choose R; since any belief ¹( R) makes d ¢j strictly dominated for player 2. But if player 2 plays d with probability zero, strategy R strictly dom- inates strategy L for player 1. Hence, there is a unique perfect Bayesian equilibrium where player 2 chooses u:

There are two types of DSEL agreeing with the additive prior distribution (p1; p2): The completely ad- 1 2 1 1 ditive DSEL (º ; º ); (½( R); ½( L)) ; º (t1; R) = p1; º (t2; R) = p2; ½(t1 R) = p1; ½(t2 R) = ¢j ¢j j j 2 p2; º (u) = 1¡; is behaviourally equiva¢lent to the perfect Bayesian equilibrium. This DSEL is sup- e e e e e e e e ported by a sequences of strategically ambiguous E-capacities which agree with the prior distribution. e For details about the construction of such a sequence see Eichberger, J. & Kelsey, D. (1999).

21 2; 6 u q 1; 6 L t1 R 2 m 2; 3 q q q q d p1 0; 4 q 0 q 2; 3 p2 u q 1; 6 m 2; 6 L t2 R 2 q q q q d 0; 4 q Figure 3: Non-additive beliefs off the equilibrium path

There is however another DSEL (º1; º2) where the updated beliefs of player 1 are not additive. For any ® [0; 1); consider the following sequence of DS-equilibria: 2 3

1 E ºn(E) (t1; L) (1 ¸n) p1 f(t ; R)g ¡ 0 ¢ f 1 g (t2; L) (1 ¸n) p2 f(t ; R)g ¡ 0 ¢ E º2 (E) f 2 g n (t1; L); (t1; R) p1 u 0 f g f g (t1; L); (t2; L) 1 ¸n m 0 f g ¡ f g (t1; L); (t2; R) (1 ¸n) p1 d (1 ¸n) f g ¡ ¢ f g ¡ (t1; R); (t2; L) (1 ¸n) p2 u; m 0 f g ¡ ¢ f g (t1; R); (t2; R) 0 u; d (1 ¸n) f g f g ¡ (t2; R); (t2; L) p2 m; d (1 ¸n) f(t ; L); (t ; L);g(t ; R) 1 ¸ + ¸ ® Sf g ¡1 f 1 2 2 g ¡ n n ¢ (t1; R); (t2; L); (t2; R) p2 0 f g ; 2 (t2; L); (t1; L); (t1; R) 1 ¸n + ¸n ® supp º d f g ¡ ¢ f g (t2; R); (t1; L); (t1; R) p1 Sf g 1 0 ; 1 supp º (t1; L); (t2; L) f g The DS-updated capacities ½ (t R) = º1 (t R) = ® and ½ (t R) = º1 (t s ) = ® are well- n 1j n 1j n 2j g n 2j 1 defined, but strictly non-additive:

22 1 2 1 2 Computing the CEU payoffs for the DSE (º ; º ); one easily checks that P (L ti; º ) = 1 > 0 = n n j n 1 2 1 2 2 2 P (R ti; º ) and, for ® < ; P (d R; ½ ) = 4 > 3 + 3 ® = P (u R; ½ ) = P (m R; ½ ) for all j n 3 j n ¢ j n j n n: Hence, the DSEL ((º1; º2); (½( R); ½( L))); 1 ¢j ¢j 1 º (t1; L) = p1; º (t2; L) = p2; additive ½(t1 R) = ®; ½(t2 R) = ®; non-additive º2(dj) = 1; j additive follows from this sequence of DS-equilibria as ¸ 0. n ! The DSEL (º1; º2) is interesting since beliefs off the equilibrium path are non-additive, even though beliefs on the equilibrium path are additive. Since perfect Bayesian equilibrium requires that beliefs be additive at all information sets, the expected payoff from u dominates the payoff from d. DSEL, however, allows strict non-additivity off the equilibrium path, so that the certain payoff of 4 obtained from strategy d becomes more attractive. It is plausible that a player who has observed an out-of-equilibrium move will have some doubts about his original theory of how the game is played. This could cause him to become ambiguity-averse as represented by the non-additivity of the updated beliefs. DSEL allows us to model ambiguity of a player as a consequence of having to update beliefs on events with a capacity weight of zero. Example 4.1 shows also that there are few constraints on the DS-updates. Indeed, DSE, and there- fore DSEL, allow us to impose constraints on players’ beliefs directly and to deduce equilibrium beliefs satisfying these constraints. This opens the opportunity to design experiments where ambigu- ity is manipulated independently from the equilibrium play which one wants to test. 4.1 Properties of DSEL

In this section, we will compare the concept of a DSEL with Bayesian and perfect Bayesian equilib- rium. Since Bayesian and perfect Bayesian equilibria have an additive prior distribution over types as a defining criterion we will restrict attention to Dempster-Shafer equilibria which agree with an additive prior distribution throughout this section. The capacities (º1; º2) of a DSEL are additive. So it is not difficult to prove that a DSEL is a Bayesian equilibrium.

Proposition 4.2 A DSEL which agrees with an additive prior distribution over types is a Bayesian equilibrium.

All DSEL are Bayesian equilibria. The potential of strategic ambiguity to select among the set

23 of Bayesian equilibrium lies in the updated beliefs. Beliefs are generated by the DS-updating rule in combination with constraining assumptions about equilibrium beliefs. A DSE does not tie down the equilibrium beliefs as much as Nash equilibrium does. Hence, there is room for game-specific constraints on beliefs and attitudes towards ambiguity. Depending on the application one can focus on the consequences of the degree of ambiguity aversion, of ambiguity about types or of other character- istics of beliefs. The DS-updates inherit their properties from these fundamental assumptions. To the extent that one can control for the degree of ambiguity, ambiguity aversion and other characteristics of an environment one may be able to test equilibrium properties in experiments. One of the weakest refinements of Bayesian equilibria is a perfect Bayesian equilibrium. By mak- ing updated beliefs part of the equilibrium concept it guarantees optimising behaviour at all informa- tion sets whether or not they will be reached in equilibrium. Since perfect Bayesian equilibrium puts no constraint on out-of-equilibrium beliefs, it eliminates only equilibria relying on strictly dominated strategies at information sets off the equilibrium path. A DSEL allows for beliefs off the equilibrium path which are strictly non-additive. Hence, Exam- ple 4.1 shows that a DSEL need not be a perfect Bayesian equilibrium. One may however conjecture 1 that a DSEL with additive updates ½( s ) s1 S1 at all information sets is a perfect Bayesian equi- f ¢j g 2 librium. We will show below in Proposition 4.3 that this is the case, indeed. One may also conjecture that the restrictions on beliefs induced by the sequence of stategically 1 ambiguous Dempster-Shafer equilibria would rule out DSEL with additive updates ½( s ) s1 S1 at f ¢j g 2 all information sets where a player uses a weakly dominated action. This is however not true. For every perfect Bayesian equilibrium it is possible to construct a sequence of strategically ambiguous DS-equilibria, which agree on the additive prior over types and converges to this perfect Bayesian equilibrium. This is almost obvious if all information sets will be reached in the perfect Bayesian equi- librium. If there are information sets following actions which are not played in a perfect Bayesian equilibrium, then one can find a sequence of DS-equilibria in which the DS-updates are not defined at these information sets. Hence, one can assign the off-the-equilibrium-path beliefs of the perfect Bayesian equilibrium to those DS-equilibria. Thus, one can obtain even a perfect Bayesian equilib- rium where player 2 chooses weakly dominated strategies off the equilibrium path as a DSEL.

Proposition 4.3 Perfect Bayesian equilibrium and DSEL (i) Every perfect Baysian equilibrium is a DSEL.

24 (ii) A DSEL which agrees with an additive prior distribution is a perfect Bayesian equilibrium if all updates are additive.

DS-updates of a DSEL can, but need not, be additive. Proposition 4.3 shows that additive limits of the DS-updates is the crucial condition for the two concepts to coincide. If DS-updates do not con- verge to additive probability distributions off the equilibrium path, then strategic ambiguity, modelled by the DS equilibrium concept, provides a refinement of Nash equilibrium based on other principles than standard refinements in the literature. Mailath (1992) provides an excellent survey of the refinements most commonly used in sig- nalling games. They all operate by restricting out-of-equilibrium beliefs. Justification for such re- strictions is obtained by forward or backward induction arguments. There is an obvious tension in such arguments because out-of-equilibrium behaviour is constrained by reasoning about behaviour which will never be observed. The DSEL provides an alternative approach to equilibrium selection. Modelling ambiguity about the equilibrium strategy choices directly avoids the tension in the interpretation of out-of-equilibrium beliefs. Moreover, there are behavioural theories behind the DS-updating rule (Gilboa & Schmei- dler (1993)) and the Choquet expected utility model (Gilboa (1987), Sarin & Wakker (1993)). Assumptions about the behavioural foundations of this decision and updating model can and have been tested independently from the equilibrium notion (Camerer, C. & Weber, M. (1992)). 4.2 Out-of-equilibrium beliefs

Refinement of the set of equilibria can be obtained by imposing additional restrictions on the players’ non-additive beliefs. As in the standard refinement literature, one can strengthen or weaken the ro- bustness requirement imposed on Bayesian equilibrium by putting further constraints on the sequence of ambiguous DS-equilibria which support it. In contrast to this literature such assumptions are in principle testable. DS-equilibria which are not perfect Bayesian equilibria are plausible, since they correspond to cases in which player 2 is ambiguity-averse after observing an unexpected move. Example 4.1 il- lustrates the potential of DSEL to select among Bayesian equilibria based on ambiguity about the behaviour in case of an unexpected out-of-equilibrium move. Yet, even if we do not want to rely on

25 non-additive beliefs off the equilibrium path, DSEL offers quite intuitive out-of-equilibrium beliefs. It is impossible to develop here a complete theory of reasonable refinements based on ambiguity, but the following example may provide some intuition. It is a simplified version of the education- signalling model introduced by Spence (1973). It shows how ambiguity imposes plausible restric- tions on the out-of-equilibrium beliefs, which select the pooling equilibrium as the a unique DSEL12. The intuition about beliefs is as follows. A DSEL which agrees with an additive prior distribution over types models a situation where a player feels ambiguity about the opponents’ behaviour but not about the prior distribution over types. This is a natural assumption if past experience has provided in- formation about the frequency of types but if there is no well-established way of signalling private in- formation. In such a situation signalling is endogenous equilibrium behaviour. An out-of-equilibrium move indicates a break-down of the implicit understanding of equilibrium behaviour. In such a case, it appears quite reasonable to return to the ‘‘firm’’ information about the prior distribution over types.

Example 4.2 education signalling13

Consider two workers with different productivity, ÁH > ÁL: A worker’s productivity is private infor- mation but it is commonknowledge that the proportion of high-productivity workers is p: Workers will

apply for a position in the firm with a wage proposal. A worker can ask for a high wage wH = ÁH; a

low wage wL = Á ; or the average wage w = p Á +(1 p) Á : The firm can only choose to accept L ¢ H ¡ ¢ L the application, a; or to reject it, r. In order to qualify for a high wage wH; a worker must present an

education certificate. The strategy wH implies that the worker has obtained this education certificate. High-productivity workers can obtain the certificate at no cost, while low-productivity workers incur a cost of 2: We will assume throughout that the education costs of the low-productivity worker are ¡ not justified by the productivity and wage difference, 0 < wH wL = Á Á < 2. ¡ H ¡ L Figure 4 illustrates the situation. 1 2 2 1 In the notation of Section 3, the game is described by S = wH; wL; w and S = s (s ) f g f j

12 Notice that Example 4.2 is no contradiction to the result in Proposition 4.3. There are other DSELs corresponding to the typical perfect Bayesian equilibria of the Spence signalling model. 13 This is a highly stylised version of the education signalling model by Spence (1973). For simplicity of the ex- position, we have assumed that the education level is not a choice variable. A more general treatment of the education signalling model can be found in Eichberger & Kelsey (1999a).

26 w¹; Á w¹ H ¡ a r F

wH; 0 r wL; Á wL w¹ H ¡ r a r a r 0; 0 F wH H wL r F r r r r p r 0; 0 0; 0 w¹; Á w¹ r L ¡ r a r 0 r Fr wH 2; Á wH 1 p wL; 0 ¡ L ¡ ¡ r a w¹ r a r 0; 0 r F wH L wL F r r r r r 0; 0 0; 0 r r

Figure 4: Signalling game s1 S1 and T = H; L : It is easy to see that there are exactly two perfect Bayesian equilibria: 2 g f g Worker Firm out-of-equilibrium beliefs 2 2 2 (i) ((wH; H); (wL; L)) (s (wH); s (wL); s (w)) = (a; a; r); ¹(H w) = 0: 2 2 2 j (ii) ((w; H); (w; L)) (s (wH); s (wL); s (w)) = (r; a; a); ¹(H wH) = ¹(H wL) = 0: j j For notational convenience, we have only noted the equilibrium strategies. In terms of beliefs, a per- fect Bayesian equilibrium (¼1; ¼2; ¹( s1) ) is described by ¼1(wH; H) = 1; ¼1(wL; L) = 1 and f ¢j g ¼2(a; a; r) = 1 in case (i) and ¼1(w; H) = p; ¼1(w; L) = 1 p and ¼2(r; a; a) = 1 in case (ii). ¡ Beliefs about all other strategies are zero. Equilibrium (i) is the Pareto-optimal selected by the intuitive criterion. Equi- librium (ii) is the Pareto-optimal pooling equilibrium, which does not satisfy the ‘‘intuitive’’ belief condition that wH > w could only come from the high-productivity type since only this player would gain from such a deviation relative to the equilibrium payout. If we assume that the prior distribution of types is hard knowledge, while equilibrium inferences about

27 behaviour are ambiguous, then only behaviour of the pooling equilibrium (ii) can arise in a DSEL. We will formalise this assumption about the beliefs by an E-capacity14. E-capacities are a modification of simple capacities (or distorted probabilities) which have a constant degree of ambiguity and which allow for marginal distributions which are additive. t Fix a common degree of ambiguity ¸n for both players. Denote by ¹o(E) the capacity which equals 1 for E = S1 T and 0 otherwise and by ¹ (F) the capacity equalling 1 for F = S2 and 0 otherwise. £ o A compact way to write the E-capacities based on additive probability distributions ¼1 and ¼2 is 1 1 1 ºn(E) := ¸n [p ¹H(E) + (1 p) ¹L(E)] + (1 ¸n) ¼ (E) for E S T; ² 2 ¢ ¢ ¡ 2 ¢ ¡ ¢ µ 2 £ (6) º (F) := ¸n ¹(F) + (1 ¸n) ¼ (F) for F S : ² n ¢ ¡ ¢ µ E-capacities are a convex combination between an additive probability distribution ¼i and a weighted t average of the capacities ¹o with weights equal to the probabilities of the prior distribution. Notice 1 1 1 2 2 2 that º (E) + º ((S T ) E) = 1 ¸n and º (F) + º (S F ) = 1 ¸n holds for all events n n £ n ¡ n n n ¡ 1 2 E = S T and F = S : Thus, there is strict ambiguity if ¸n > 0 holds. E-capacities have also the 6 £ 6 property that suppºi = supp¼i. The strategies in the support of the capacity, i.e., the strategies of the opponents which must be optimal, are the strategies in the additive part of the E-capacity. Using

15 1 Equation 3, one can compute the DS-update of ºn as ¸ p + (1 ¸ ) ¼1(s1; H) º1 (H s1) = n n : n ¢ ¡1 1 ¢ 1 1 j ¸n + (1 ¸n) [¼ (s ; H) + ¼ (s ; L)] 1 1 ¡ 1 1¢ 1 1 For ¸n > 0; º (H s ) is well defined even if ¼ (s ; H) + ¼ (s ;L) = 0 holds, i.e., if no type plays n j strategy s1 in equilibrium. Notice also that for ¼1(s1; H) + ¼1(s1; L) = 0; º1 (t s1) = p(t) coincides n j with the prior distribution. This means that a player who observes an out-of-equilibrium move s1 will update her beliefs to the prior distribution. This property of an E-capacity which agrees with a prior distribution appears sensible if one views the knowledge about the type distribution as firm compared to the beliefs about strategy choices which represent just a consistency requirement for beliefs and optimal actions. It is easy to check that º1 ¼1 and º2 ¼2: Notice, however, that º1 (H s1) p for all s1: Hence, n ! n ! n j ! in the limit, we have out-of-equilibrium beliefs ¹(H wH) = ¹(H wL) = p. j j It remains to show that the beliefs in Equation 6 based on the additive probability distributions ¼1(w; H) = p; ¼1(w; L) = 1 p and ¼2(r; a; a) = 1 form a DSE. This is easily established since ¡

14 Eichberger & Kelsey (1999) provide a thorough study of the properties of E-capacities and an axiomatisation. 15 An explicit computation is in Eichberger & Kelsey (1999).

28 suppº1 = ((w; H); (w; L)) ; supp º2 = (r; a; a) and the expected payoff functions are: n f g n f g 1 2 1 wH ºn(s (wH) = a) = 0 for s = wH 1 1 2 ¢ 1 2 1 P (s H; ºn) = wL ºn(s (wL) = a) = (1 ¸n) wL for s = wL ; j 8 ¢ 1 2 ¡ ¢ 1 w º (s (w) = a) = (1 ¸n) w for s = w < ¢ n ¡ ¢ 1 2 1 (wH 2) ºn(s (wH) = a) = 0 for s = wH 1 1 2 : ¡1 2¢ 1 P (s L; ºn) = wL ºn(s (wL) = a) = (1 ¸n) wL for s = wL j 8 w º¢ 1(s2(w) = a) = (1 ¡ ¸ ) ¢ w for s1 = w < ¢ n ¡ n ¢ and : [Á w ] [1 p] for a = a L H if s1 = w 0 ¡ ¢ ¡ for a = r H 8 2 1 1 [ÁH wL] p for a = ae 1 P (a s ; ºn) = > ¡ ¢ if s = wL : j > 0 for a = re <> p Á + (1 p) Á w = 0 for a = a H L e if s1 = w e 0 ¢ ¡ ¢ ¡ for a = r > e Hence, choosing w is op>timal for the worker of either type and accepting a wage offer w is optimal :> e e for the firm. This establishes that the beliefs in Equation 6 form a DSE for any n: The resulting DSEL is therefore Worker Firm out-of-equilibrium beliefs 2 2 2 (iii) ((w; H); (w; L)) (s (wH); s (wL); s (w)) = (r; a; a); ¹(H wH) = ¹(H wL) = p: j j

The equilibrium selection in the DSEL of Example 4.2 depends on the joint assumptions of an

unambiguous prior distribution over types and a degree of ambiguity aversion ¸n which, for each step n; is the same for all events E S1 T: Constant ambiguity aversion controls for distorted beliefs16. ½ £ The result that out-of-equilibrium beliefs coincide with the additive prior distribution is driven by the assumption that the prior distribution is unambiguously known. If this is the case, it makes sense for a player to revert to the unambiguous information as implied by DS-updating, whenever an out- of-equilibrium move occurs which invalidates the equilibrium behaviour prediction. In the game of Example 4.2, this assumption about the prior distribution rules out the separating equilibrium. The separating equilibrium would require complete trust in the equilibrium behaviour because a low- productivity worker has an incentive to break away from the separating equilibrium and to propose the average wage rather than the low wage. To argue that the firm should assume that an average wage offer could only come from the low-productivity type would mean that the firm feels no ambiguity about the behaviour of the workers.

16 To establish the result of Proposition 4.3 that every perfect Bayesian equilibrium is a DSEL we had to relax this assumption of a constant degree of ambiguity in each step of the belief sequence supporting the DSEL.

29 Incontrast to Bayesian equilibrium, DSE hasa updating rule which works also for out-of-equilibrium beliefs if there is some ambiguity about strategy choices. Reasonable assumptions about beliefs can be imposed directly. For example, partial information can be assumed as in the case of a well-known prior distribution in Example 4.2. Whether this is an appropriate assumption or not can be assessed independent from the equilibrium, which is an advantage in economic applications.

5. CONCLUSION

We have applied the theory of Knightian uncertainty to sequential games. The evidence suggests that there are occasions in which individuals have large degrees of ambiguity. Despite this we believe that an interesting case is when the degree of ambiguity is small. Under this assumption, we have shown that our definition of equilibrium is a refinement of Bayesian equilibrium, which is similar in spirit to perfect Bayesian equilibrium but does not exactly coincide with it. Since DSEL is a special case of Bayesian equilibrium, no irrational behaviour is introduced by considering non-additive beliefs. As we have shown, even in the limit as beliefs converge to additive probabilities, significant deviations from behaviour under subjective expected utility are possible off the equilibrium path. We believe this is one of our main innovations.

Appendix A

Lemma 2.2: If a convex capacity º has zero degree of ambiguity then it is additive. Proof. Suppose that º is not additive, then there exist A; B S, such that A B = and º(A B) > ½ \ ; [ º(A) + º(B): Let C = S (A B). Then since the degree of ambiguity is zero: n [ 1 = º(A B) + º(C) > º(A) + º(B) + º(C): (A-1) [ By convexity, 1 = º((A B) (A C)) º(A B) + º(A C) º(A): Since the degree [ [ [ > [ [ ¡ of ambiguity is zero, º(A B) = 1 º(C) and º(A C) = 1 º(B): Substituting, we obtain [ ¡ [ ¡ 1 1 º(C) + 1 º(B) º(A); but this contradicts A-1. > ¡ ¡ ¡

Proposition 3.3:

a) A Dempster-Shafer Equilibrium (º1; º2) with a degree of ambiguity ¸ = 0; for which the

30 belief of player 2, º1; agrees with the additive prior distribution p onT is a Bayesian Equilibrium. b) Consider a Dempster-Shafer Equilibrium (º1; º2) with a degree of ambiguity ¸ = 0; for which the belief of player 2, º1; agrees with the additive prior distribution p on T: If for each strategy s1 S1 there exists a type t T such that (s1; t) supp º; then the Dempster-Shafer 2 2 2 Equilibrium (º1; º2) is a perfect Bayesian equilibrium. Proof. Part (a): Since the DSE (¼1; ¼2) has a degree of ambiguity ¸ = 0, by Lemma 2.2, ¼1 and ¼2 must be additive probability distributions. Since the DSE agrees with an additive prior distribution 1 1 p on T; s1 S1 ¼ (s ; t) = p(t) for all t T: Hence, condition DSE-a of Definition 3.2 can be 2 2 written aPs s1 argmax ¼2(s2) u1(s1; s2(s1); t) 2 1 1 ¢ s S s2 S2 2 2 1 1 1 1 X for all s S with ¼ (s ; t) > 0 eand all t T: ConditioneDSE-b requires 2 2 s2(s1) arg max ½(t s1) u2(s1; a; t) 2 2 j ¢ a A t T 2 X2 for all s2 S2 with ¼2(s2) > 0: By Condition DSE-c and Equation 3, ½(t s1) := ¼1(S1 t s1 2 e e j £ f gjf g £ ¼1(s1;t) 1 1 1 T) = 1 1 provided ¼ (s ; t) = 0: Note that beliefs off the equilibrium path ½(t s ) are t T ¼ (s ;t) t T 2 2 6 j 1 1 1 1 arbitraPry and need not even Pbe additive. For actions s S such that t T ¼ (s ; t) = 0 all actions 2 2 a A2 are optimal. Hence, Part (a) of Proposition 3.3 defines a Bayesian equilibrium with mixed 2 P strategies (¼1; ¼2): Part (b): If, in addition, for each strategy s1 S1 there exists a type t T such that (s1; t) supp¼1; 2 2 2 1 1 1 1 1 ¼1(s1;t) then ¼ (s ; t) = 0 for all s S and ½(t s ) = 1 1 is defined at all information sets. t T t T ¼ (s ;t) 2 6 2 j 2 1 2 In thiPs case, (¼ ; ¼ ) is a perfect Bayesian equilibrium.P

Lemma 3.4:Let º be a capacity on a set S: An event E S is Savage-null if and only if º(S E) = 1: ½ n Proof. An event E is Savage-null if for any three outcomes x; y; z X the CEU value of the acts 2 xEy and zEy are equal, i.e. u(x) º(E) + u(y) [1 º(E)] = u(y) º(S E) + u(z) [1 º(S E)] ¢ ¢ ¡ ¢ n ¢ ¡ n where we assume, without loss of generality, u(x) > u(y) > u(z): This equality can hold for arbitrary outcomes x; y; z X with this order if and only if º(S E) = 1 and º(E) = 0: 2 n

31 Proposition 3.5: For any degree of ambiguity ¸ (0;1) and any additive prior probability distri- 2 bution p on T; there exists a Dempster-Shafer equilibrium with this degree of ambiguity ¸; which agrees with the distribution p on T:

Proof. The proof uses the special form of an E-capacity, which is extensively discussed in Eich- berger & Kelsey (1999). E-capacities are modifications of an additive probability distribution with a constant degree of ambiguity and, possibly, some additive marginal distributions. If there are no additive marginals then E-capacities are simple capacities. Moreover, the support of an E-capacity coincides with the support of its additive part. Hence, for given prior distributions of types and given degrees of ambiguity, E-capacities are completely described by their additive part. Given beliefs modelled by E-capacities, one can use standard arguments to show that there is a Nash-equilibrium in mixed strategies for the modified game where the Choquet payoff functions are viewed as functions of the additive part of the E-capacities. Fix ¸1; ¸2 (0; 1) and any additive probability distribution p on T: For any finite set X denote by 2 1 ¢(X) the set of additive probability distributions on X: Let ¢p(S T) be the set of additive prob- £ ability distributions on S1 T with marginal distribution p on T: This set is non-empty, compact and £ convex. For any E S1 T let the capacity º be defined as ½ £ t 1 if S1 t E º (E) := ; t 0 otherwi£sef g ½ 1 ½ and for any ¼ ¢p(S T) consider the capacity 2 £ 1 1 1 º (E) := ¸ p(t) ºt(E) + (1 ¸ ) ¼(E): (A-2) ¼ ¢ ¢ ¡ ¢ t T X2 For any set T 0 T; the DS-update of the capacity defined in Equation A-2 is (see Eichberger & ½ Kelsey (1999),Lemma 4.2, p. 132) ¸1 p(t) + (1 ¸1) ¼(s1; t) 1 1 ¢ t T0 ¡ ¢ t T0 º (T 0 s ) = 2 2 : (A-3) ¼ j P P ¼(s1; t) + ¸1 1 ¼(s1; t) t T ¢ ¡ t T 2 · 2 ¸ 1 1 1 P P For ¸ > 0, º (T 0 s ) is a continuous function of ¼: ¼ j

32 1 1 1 For any s S and any ¼ ¢p(S T); let 2 2 £ 1 2 1 1 ½(¼; s ) =argmax ¾(a) P (a s ; º¼) (A-4) ¾ ¢(A2) ¢ j 2 a A2 X2 2 1 1 be the set of best behaviour strategies on A for given history s and given belief º¼ based on ¼: From the definition of the Choquet integral in Equation 1, it is clear that P 2(a s1; º1) is a continuous j ¼ 1 1 function of the capacity º¼: From the DS-update in Equation A-3 we know that º¼ is a continu- 2 1 1 ous function of ¼: Thus, ¾(a) P (a s ;º¼) is a continuous function of ¼: Hence, by Berge’s a A2 ¢ j maximum theorem (e.g., TP2akayama (1985), p. 254), ½(¼; s1) is a upper-hemi-continuous cor- 2 2 1 1 respondence. Since ¢(A ) is a convex set and since ¾(a) P (a s ; º¼) is linear in ¾; the a A2 ¢ j correspondence ½(¼; s1) is also convex-valued. P2 1 1 2 For any t T and a vector ¹ = (¹( s ))s1 S1 of additive probability distributions ¹( s ) ¢(A ) 2 ¢j 2 ¢j 2 define the capacities º2(E s1) := (1 ¸2) ¹(E s1): The capacity º2( s1) is continuous in ¹: Let ¹ j ¡ ¢ j ¹ ¢j 1 1 1 2 1 Ã(¹; t) =arg max ¾(s ) P (s t; º¹( s )) (A-5) ¾ ¢(S1) ¢ j ¢j 2 s1 S1 X2 be the best-response correspondence for player 1. Finally, let

1 1 1 Á(¹) := ¼ ¢p(S T ) ¼(s ; t) = p(t) ¾t(s ); ¾t Ã(¹; t) : (A-6) f 2 £ j ¢ 2 g t T X2 Since the Choquet integral P 1(s1 t; º2( s1)) is continuous in º2( s1); which in turn is continuous in j ¹ ¢j ¹ ¢j 1 1 1 2 1 ¹; we can conclude that ¾(s ) P (s t; º¹( s )) is continuous in ¹: Moreover, the objective s1 S1 ¢ j ¢j function is linear in ¾: ApPp2lying Berge’s maximum theorem again, we conclude that the correspon- dences Ã(¹; t) are upper-hemi-continuous and convex-valued. The correspondence Á(¹) defined in Equation A-6 is a convex set for each ¹ and clearly also upper-hemi-continuous. Consider the mapping £ : ¢ (S1 T) ¢(A2) S1 ¢ (S1 T) ¢(A2) S1 defined by p £ £ j j ! p £ £ j j (¼; ¹) £(¼; ¹) := ½(¼; s1) Á(¹): 7! s1£S1 £ 2 As the Cartesian product of upper-hemi-continuous and convex-valued correspondences £ is itself 1 a upper-hemi-continuous and convex-valued correspondence. Moreover, the set ¢p(S T) £ £ 2 S1 ¢(A )j j is compact and convex. Hence one can apply Kakutani’s fixed-point theorem to estab- lish the existence of (¼¤; ¹¤) £(¼¤; ¹¤): 2 2 2 Define the additive probability distribution ¿ ¤ onS ; the set of strategies of player 2, by ¿ ¤(s ) := s1 S1 2 1 1 2 2 2 2 2 2 ¹¤(s (s ) s ) for all s S : For all E S ; let º (E) := (1 ¸ ) ¿ ¤(E): Q j 2 ½ ¿¤ ¡ ¢

33 We claim that the capacities (º1 ; º2 ) are a DSE. To see this, note first that suppº1 = s1 ¼¤ ¿ ¤ ¼¤ f 2 1 1 2 2 2 2 S ¼¤(s ) > 0 and suppº = s S ¿¤(s ) > 0 (Eichberger & Kelsey (1999), j g ¿¤ f 2 j g 1 1 1 1 Lemma 2.2, p. 121). Hence, (s ; t) supp º implies ¼¤(s ; t) = p(t) ¾t(s ) > 0 and ¾t 2 ¼¤ ¢ 2 1 1 1 2 1 2 2 2 Ã(¹¤; t): Hence, s must maximise P (s t; º ( s )): Similarly, s suppº implies ¿¤(s ) > 0: j ¹¤ ¢j 2 ¿¤ 2 1 1 1 1 2 1 2 1 1 Hence, ¹¤(s (s ) s ) > 0 for all s S : Therefore, s (s ) must be a maximiser of P (a s ; º ): j 2 j ¼¤ Finally, by construction, 1 1 1 and 2 2 2. Since 1 and 2 were chosen arbitrarily, ¸ (º¼¤) = ¸ ¸ (º¿¤ ) = ¸ ¸ ¸ the existence result follows for any ¸ (0; 1): 2

Proposition 4.2:A DSEL which agrees with an additive prior distribution over types is a Bayesian equilibrium. Proof. Note first that ¸ = 0 in the limit implies that the limit capacities (º1; º2) are additive proba- bility distributions. Hence, suppº1 = (s1; t) S1 T º1(s1; t) > 0 and suppº2 = s2 S2 f 2 £ j g f 2 j º2(s2) > 0 : Moreover, º1(s1; t) º1(s1; t) > 0 and º2 (s2) º2(s2) > 0; imply º1(s1; t) > 0 g n ! n ! n and º2 (s2) > 0 for n large enough. Note also that P 1(s1 t; º2) is continuous in º2 and P 2(a s1; ½) is n j j continuous in ½: If º1( s1 T ) > 0; i.e., if the DS-update º1(t s1) is defined, then ½( s1) = º1( s1). f g£ j ¢j ¢j Suppose now that (º1; º2) is not a Bayesian equilibrium. Then there exists (s1; t) suppº1 such 2 that P 1(s1 t; º2) < P 1(s1 t; º2) for some s1 S1 and/or s2 supp º2 such that P 2(s2(s1) s1; ½) < j j 2 2 j P 2(a s1; ½) for some a A2: By continuity of P 1(s1 t; º2) and P 2(a s1; ½) in ½ and º2; respectively, j 2 j j e1 1 2 1e 1 2 1 1 2 2 1 1 we can conclude that P (s t; ºn) < P (s t; ºn) for some s S and/or P (s (s ) s ; ½n) < e e j j 2 j P 2(a s1; ½ ) for some a A2: Since (s1; t) supp º1 and s2 suppº2 imply that º1(s1; t) > 0 and j n 2 2 2 n 2 2 1 1 e 2 2 e 1 2 ºn(s ) > 0 for n large, (s ; t) suppºn and s suppºn follows. Hence, (ºn; ºn) is not a DSE. e e 2 2

Proposition 4.3: Perfect Bayesian equilibrium and DSEL (i) Every perfect Baysian equilibrium is a DSEL. (ii) A DSEL which agrees with an additive prior distribution is a perfect Bayesian equilibrium if all updates are additive.

1 2 1 Proof. Part (i). The proof is constructive. For agivenperfect Baysianequilibrium (¼ ; ¼ ; ¹( s ) s1 S1 ) f ¢j g 2 17 1 2 1 we show that there is a sequence of appropriately modified E-capacites (ºn; ºn; ½n( s ) s1 S1) f ¢j g 2 which converges to the perfect Bayesian equilibrium. The trick is to construct this sequence such

17 For more on E-capacities, their properties and updates, see Eichberger & Kelsey (1999).

34 that the Dempster-Shafer-update of any strategy s1 which is not played by any type in the perfect Bayesian equilibrium, ¼1( s1 T) = 0; is not defined, i.e., º1(S1 s1 T) = 1. Hence, one can f g £ n nf g £ choose the update ½ ( s1) arbitrarily, in particular equal to the update ¹( s1) of the perfect Bayesian n ¢j ¢j 1 2 1 equilibrium. This sequence of capacities (ºn; ºn; ½n( s ) s1 S1 ) is a DSE by standard continuity f ¢j g 2 arguments. 1 1 1 1 Suppose there is a perfect Bayesian equilibrium ¼ ¢(S ) ¢(T) with ¼ (S T 0) = p(t) for 2 £ £ t T0 2 2 1 2 1 1 all T 0 T; ¼ ¢(S ) with (additive) out-of-equilibrium beliefs ¹( s ) ¢(T) for aPll s S : µ 2 ¢j 2 2 1 1 Define a sequence of DS-equilibria ((ºn; ºn); ½n( s ) s1 S1) as follows: f ¢j g 2 Consider sequences ¸1 > 0 and ¸2 > 0 which converge to zero. Denote by S1 S1 the set of n n + µ 1 1 1 strategies with t T ¼ (s ; t) = 0: For any non-empty E S and F T let 2 6 µ µ 1 P 1 if S+ T E F º1(E F ) = 1 1 1 £ µ £ ; n ¸n p(t) ºt(E F) + (1 ¸n) ¼ (E F) otherwise £ ( ¢ t T ¢ £ ¡ ¢ £ 2 where P 1 if S1 t E F ºt(E F) := £ f g ½ £ : £ 0 otherwise 1 ½ It is easy to check that ºn is a capacity which, by construction, agrees with the prior distribution p(t): Moreover, º1 ¼1: n ! 1 1 1 1 1 1 1 1 1 1 For s S ; S T * S s T and º (S s T) < 1: Hence, the DS-updates º (T 0 s ) 2 + + £ nf g £ n nf g £ n j are well-defined and converge to the Bayesian updates . The updates ½ ( s1) = º1 ( s1) are well- n ¢j n ¢j defined and converge to ¹( s1): ¢j On the other hand, for s1 = S1 S1 T S1 s1 T and the DS-updates are not defined. Hence, 2 + + £ µ nf g £ we can choose ½ ( s1) = ¹( s1) for all n in this case. n ¢j ¢j 2 2 2 2 Finally, for any subset E S ; ºn(E) = (1 ¸n) s2 E ¼ (s ): ½ ¡ ¢ 2 It is easytocheck that the capacities º1; º2 arestrategically ambiguousand that suppº1 = (s1; t) ¼1(s1; t) n n P n f j 2 2 2 2 1 2 2 1 0 andsuppºn = s ¼ (s ) > 0 : By continuityof P andP inºn and ½n; respectively, s arg max g f j g 2 s1 S1 1 1 2 1 1 2 1 2 1 1 2 2 2 P (s t; ºn) for all (s ; t) supp ºn and s (s ) arg max P (a s ; ½n( s )) for all s suppºn: j 2 2 a A2 j ¢j 2 e 1 2 1 2 Hence, ((ºn; ºn); ½n( s ) s1 S1) is a DSE: e f ¢j g 2 e e Part (ii). Consider a sequence of strategically ambiguous DSE (º1; º2) (º1;º2) and, for all n n ! 1 1 1 1 1 s1 S1; ½ ( s ) converges to an additive update ½( s ). Since º is additive, (s ; t) supp º 2 n ¢j ¢j 2 implies º1(s1; t) > 0: Suppose there is a s1 supp º1 such that P 1(s1 t; º2) > P 1(s1 t; º2). By 2 j j 1 2 1 1 2 1 1 2 1 1 continuity of P in º ; P (s t; ºn) > P (s t; ºn) for large n: Moreover, for large n; ºn(s ; t) > j e j e e 35 e 0 and, therefore, s1 supp º1. This contradicts the assumption that (º1; º2) is a DSE. Hence, 2 n n n 1 2 1 ((º ; º ); ½( s ) s1 S1) satisfies Condition PBE-a. f ¢j eg 2 Since ½( s1) is additive, an analogous argument shows that Condition PBE-b must hold for ((º1; º2); ¢j 1 1 1 1 1 1 1 1 1 ½( s ) s1 S1): Finally, if º ( s T) = 1 º ((S s ) T) > 0; then ºn((S s ) T) < 1 f ¢j g 2 f g£ ¡ nf g £ nf g £ 1 1 1 for n large enough. Hence, all DS-updates are well-defined for large n and ½ (T 0 s ) = º (T 0 s ) n j n j ! 1 1 1 ½(T 0 s ) = º (T 0 s ): Thus, Condition PBE-c is satisfied. Given the assumption that the DSEL j j agrees with an additive prior distribution over types it is therefore a perfect Bayesian equilibrium.

36 References

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39 SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

03-32 Siegfried K. Berninghaus Reciprocity - an indirect evolutionary analysis 03-21 Thomas Gschwend Stimmensplitting und Koalitionswahl Christian Korth Franz Urban Pappi Stefan Napel 03-20 Thomas Langer Does Binding or Feeback Influence Myopic Loss 03-31 Peter Albrecht Random Walk oder Mean Reversion? Eine Martin Weber Aversion - An Experimental Analysis Cemil Kantar statistische Analyse des Kurs/Gewinn-Verh¨altnisses f¨ur den deutschen Aktienmarkt 03-19 Peter Albrecht Asset/Liability Management of German Life Carsten Weber Insurance Companies: A Value-at-Risk Approach 03-30 J¨urgen Eichberger Ambiguity and Social Interaction in the Presence of Interest Rate Guarantees David Kelsey Burkhard Schipper 03-18 Markus Glaser Online Broker Investors: Demographic Information, Strategy, Portfolio 03-29 Ulrich Schmidt Security And Potential Level Preferences With Positions, and Trading Activity Alexander Zimper Thresholds 03-17 Markus Glaser September 11 and Stock Return Expectations of 03-28 Alexander Zimper Uniqueness Conditions for Point-Rationalizable Martin Weber Individual Investors Solutions of Games with Metrizable Strategy Sets 03-16 Siegfried K. Berninghaus Network Formation and Coordination Games 03-27 J¨urgen Eichberger Sequential Two-Player Games with Ambiguity Bodo Vogt David Kelsey 03-15 Johannes Keller When negative expectancies turn into negative 03-26 Alain Chateauneuf A Simple Axiomatization and Constructive Herbert Bless performance: The role of ease of retrieval. J¨urgen Eichberger Representation Proof for Choquet Expected Utility Simon Grant 03-14 Markus Glaser Behavioral Finance Markus N¨oth 03-25 Volker Stock´e Informationsverf¨ugbarkeit und Response-Effects: Martin Weber Die Prognose von Einfl¨ussen unterschiedlich kategorisierter Antwortskalen durch 03-13 Hendrik Hakenes Banks as Delegated Risk Managers Antwortsicherheiten und Antwortlatenzen 03-12 Elena Carletti The Structure of Bank Relationships, Endogenous 03-24 Volker Stock´e Entstehungsbedingungen von Antwortverzerrungen Monitoring and Loan Rates durch soziale Erw¨unschtheit. Ein Vergleich der Prognosen der Rational-Choice Theorie und des 03-11 Isabel Schnabel The Great Banks‘ Depression - Deposit Modells der Frame-Selektion Withdrawals in the German Crisis of 1931

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00-34 Markus N¨oth Information Aggregation with Random Ordering: 00-19 Henning Plessner Sequential effects in important sport-decisions: The Martin Weber Cascades and Overconfidence Tilmann Betsch case of penalties in soccer

00-33 Tone Dieckmann Dynamic Coalition Formation and the 00-18 Susanne Haberstroh Is the interdependent self a better communicator Ulrich Schwalbe Ulrich K¨uhnen than the independent self? Self-construal and the Daphna Oyserman observation of conversational norms 00-32 Martin Hellwig Corporate Governance and the Financing of Norbert Schwarz Investment for Structural Change 00-17 Tilmann Betsch Explaining and Predicting Routinized Decision 00-31 Peter Albrecht Rentenversicherung versus Fondsentnahmepl¨ane, Susanne Haberstroh Making: A Review of Theories Thorsten G¨obel oder: Wie groß ist die Gefahr, den Verzehr des Connie H¨ohle eigenen Verm¨ogens zuuberleben? ¨ 00-16 Susanne Haberstroh When guessing is better than thinking: Multiple 00-30 Roman Inderst Influence Costs and Hierarchy Tilmann Betsch bases for frequency judgments Holger M. M¨uller Henk Aarts Karl W¨arneryd 00-15 Axel B¨orsch-Supan Household Portfolios in Germany 00-29 Dezs¨o Szalay Optimal Delegation Angelika Eymann

00-28 Dezs¨o Szalay Financial Contracting, R&D and Growth 00-14 Annette Reil-Held Einkommen und Sterblichkeit in Deutschland: Leben Reiche l¨anger? 00-27 Axel B¨orsch-Supan Rentabilit¨atsvergleiche im Umlage- und Kapitaldeckungsverfahren: Konzepte, empirische 00-13 Nikolaus Beck Comparing Rule Histories in the U.S. and in Ergebnisse, sozialpolitische Konsequenzen Martin Schulz Germany: Searching for General Principles of Organizational Rules 00-26 Axel B¨orsch-Supan How much is transfer and how much insurance in a Annette Reil-Held pay-as-you-go system? The German Case. 00-12 Volker Stock´e Framing ist nicht gleich Framing. Eine Typologie unterschiedlicher Framing-Effekte und Theorien zu 00-25 Axel B¨orsch-Supan Rentenreform und die Bereitschaft zur deren Erkl¨arung Eigenvorsorge: Umfrageergebnisse in Deutschland 00-11 Oliver Kirchkamp Local and Group Interaction in Prisoners‘ 00-24 Christian Ewerhart Chess-like games are dominancesolvable in at most Rosemarie Nagel Dilemmas two steps 00-10 Oliver Kirchkamp An experimental analysis of auctions with 00-23 Christian Ewerhart An Alternative Proof of Marshall´sRule Benny Moldovanu interdependent valuations

00-22 Christian Ewerhart Market Risks, Internal Models, and Optimal 00-09 Oliver Kirchkamp WWW Experiments for Economists, a Technical Regulation: Does Backtesting Induce Banks to Introduction Report Their True Risks? 00-08 Alain Chateauneuf Organizational Learning through Rule Adaptation: 00-21 Axel B¨orsch-Supan A Blue Print for Germany’s Pension Reform Alain Chateauneuf From the Behavioral Theory to Transactive Organizational Learning 00-20 Axel B¨orsch-Supan Data and Research on Retirement in Germany

SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

00-07 Raimond Maurer Inflation Risk Analysis of European Real Estate 99-82 Holger M. M¨uller Inside vs Outside Ownership - A Political Theory of Steffen Sebastian Securities Karl W¨arneryd the Firm

00-06 Martin Hellwig Costly State Verification: The Choice Between Ex 99-81 Ralf Rodepeter Rules of thumb in life-cycle savings models Ante and Ex Post Verification Mechanisms Joachim Winter

00-05 Peter Albrecht 100% Aktien zur Altersvorsorge - Uber¨ die 99-80 Michael Adam Risk Value Analysis of Covered Short Call and Raimond Maurer Langfristrisiken einer Aktienanlage Raimond Maurer Protective Put Portfolio Strategies

00-04 Douglas Gale Aging and the Pension Crisis: Flexibilization 99-79 Peter Albrecht Rendite oder Sicherheit in der Altersversorgung - through Capital Markets unvereinbare Gegens¨atze?

00-03 Axel B¨orsch-Supan Data and Research on Saving in Germany 99-78 Karsten Fieseler The Efficient Bilateral Trade of an Indivisible Good: Successively Arriving Information 00-02 Raimond Maurer Internationale Diversifikation von Aktien- und Alexander Mertz Anleiheportfolios aus der Perspektive deutscher 99-77 Karsten Fieseler Optimal Leasing Durations: Options for Extension Investoren 99-76 Peter Albrecht Zur Bedeutung einer Ausfallbedrohtheit von 00-01 Office SFB504 Jahresbericht 1999 Raimond Maurer Versicherungskontrakten - ein Beitrag zur Behavioral Insurance 99-89 Holger M. M¨uller Project Bundling, Liquidity Spillovers, and Capital Roman Inderst Market Discipline 99-75 Benny Moldovanu The Optimal Allocation of Prizes in Contests Aner Sela 99-88 Raimond Maurer Efficient Risk Reducing Strategies by International Gy¨ongyi Bug`ar Diversification: Evidence from a Central European 99-74 Phillipe Jehiel Efficient Design with Interdependent Valuations Emerging Market Benny Moldovanu

99-87 Berit Ernst In Search of Explanations for the Consulting 99-73 Phillipe Jehiel A Note on Revenue Maximization and Efficiency in Alfred Kieser Explosion. A Critical Perspective on Managers’ Benny Moldovanu Multi-Object Auctions Decisions to Contract a Consultancy 99-72 Eva Brit Kramer Uber¨ kurz oder lang - Spielt der Anlagehorizont 99-86 Martin Hellwig Wage Growth, Productivity Growth, and the Martin Weber eine berechtigte Rolle bei der Beurteilung von Andreas Irmen Evolution of Employment ?

99-85 Siegfried K. Berninghaus Decentralized or Collective Bargaining in a Strategy 99-71 Karsten Fieseler Partnerships, Lemons and Efficient Trade Werner Gueth Experiment Thomas Kittsteiner Claudia Keser Benny Moldovanu

99-84 Jan Vleugels Bidding Against an Unknown Number of 99-70 Dagmar Stahlberg Exculpating Victims and the Reversal of Hindsight Competitors With Affiliated Information Sabine Sczesny Bias Stefan Schwarz 99-83 Stefan Schwarz Drop-out wegen JavaScript: Ulf-Dietrich Reips SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

99-69 Karl-Martin Ehrhart Mobility and cooperation: on the run 99-56 Roman Inderst Efficiency Wages under Adverse Selection and the Claudia Keser Role of Rigid Wages

99-68 Roman Inderst Delegation of Control Rights, Ownership 99-55 Daniel Probst Evolution, Automata and Repeated Games Holger M. M¨uller Concentration, and the Decline of External Finance 99-54 Christian Laux The Ambiguous Effects of Rankings: Strategically 99-67 Eric Igou Ursachen der Verw¨asserung oder: Daniel Probst Biased Forecasting by Advisers Herbert Bless Konversationslogische Aspekte des Michaela W¨anke ”Dilution-Effektes” 99-53 Martin Hellwig Endogenous Technical Change in a Competitive Andreas Irmen Economy 99-66 Stefan Schwarz Auswirkungen des Hindsight Bias aufokonomische ¨ Dagmar Stahlberg Entscheidungen 99-52 Roman Inderst Competitive Search Markets with Adverse Selection Holger M. M¨uller 99-65 Susanne Abele Why Timing Matters: Differential Effects of Karl-Martin Ehrhart Uncertainty about the Outcome of Past versus 99-51 Abdolkarim Sadrieh Is there evidence for an adaptive toolbox? Current Events Werner Gueth Peter Hammerstein 99-64 Thomas Langer Prospect-Theory, Mental Accounting and Stevan Harnard Martin Weber Differences in Aggregated and Segregated Ulrich Hoffrage Evaluation of Lottery Portfolios Bettina Kuon Betrand R. Munier 99-63 Andreas Laschke Der ”Overconfidence Bias” und seine Peter M. Todd Martin Weber Konsequenzen in Finanzm¨arkten Massimo Warglien Martin Weber 99-62 Nikolaus Beck From Statistical Quality Control, over Quality Peter Walgenbach Systems to Total Quality Management - The 99-50 Ulrich Hoffrage How to Foster Diagnostic Insight in Experts Institutionalization of a New Management Gerd Gigerenzer Approach 99-49 Martin Lages Intransitivity of fast and frugal 99-61 Paul Povel Endogenous Debt Contracts With Undistorted Ulrich Hoffrage Michael Raith Incentives Gerd Gigerenzer

99-60 Nikolaus Beck Unspectacular Organizational Change in Normal 99-48 Axel B¨orsch-Supan Pension reform, savings behavior and corporate Alfred Kieser Times: Rule Change as a Routine Activity Joachim Winter governance

99-59 Roman Inderst Why Peaches Must Circulate Longer than Lemons 99-47 Craig R. Fox Ambiguity Aversion, Comparative Ignorance, and Holger M. M¨uller Martin Weber the Role of Context

99-58 Roman Inderst Bargaining with Sequential Buyers under 99-46 Manfred Hassebrauck Der Einfluß von Prototypen auf die Incomplete Information Cornelia Vogt Informationssuche bei Entscheidungen Michael Diehl 99-57 Roman Inderst Bargaining with a Possibly Committed Seller

SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

99-45 Manfred Hassebrauck Das ”prototype matching”-Modell des 99-33 Ulrich Hoffrage Hindsight Bias: A By-product of Knowledge Cornelia Vogt Entscheidungsverhaltens: Der Einfluß kognitiver Ralph Hertwig Updating Michael Diehl Belastung, Zeitdruck und Stimmung Gerd Gigerenzer

99-44 Axel B¨orsch-Supan Pension systems in the Middle East and North 99-32 Ralph Hertwig Begrenzte Rationalit¨at: Die Alternative zu Patrizia Tumbarello Africa: A window of opportunity Ulrich Hoffrage Laplace’schen und schlechter Software Robert Palacios 99-31 Raimond Maurer An Expected Utility Approach to Probabilistic 99-43 Reinhold Schnabel Verm¨ogen und Ersparnis im Lebenszyklus in Ulrich Hoffrage Insurance: A Comment on Wakker, Thaler and Westdeutschland Tversky (1997)

99-42 Reinhold Schnabel The Declining Participation in the German 99-30 Henning Plessner The effects of affect-based attitudes on judgment PAYG-Pension System Susanne Haberstroh and decision making Tilmann Betsch 99-41 Reinhold Schnabel Social Security Reform and Intergenerational Redistribution in Germany 99-29 Tilmann Betsch A Micro-World Simulation to Study Routine Andreas Gl¨ockner Maintenance and Deviation in Repeated Decision 99-40 Reinhold Schnabel The Golden Years of Social Security – Life-cycle Susanne Haberstroh Making Income, Pensions and Savings in Germany 99-28 Jan Walliser Tax incentives, bequest motives and the demand for 99-39 Stefan Schwarz Der Hindsight Bias bei gustatorischen Joachim Winter life insurance: evidence from Germany Sabine Sczesny Entscheidungen Dagmar Stahlberg 99-27 Joachim Winter Okonometrische¨ Analyse diskreter dynamischer Entscheidungsprozesse 99-38 Axel B¨orsch-Supan Family Resources in Retirement. Germany Annette Reil-Held 99-26 Gerd Bohner Einstellungen Dagmar Stahlberg 99-37 Axel B¨orsch-Supan Portfolio Choice with Behavioral Decision Dieter Frey Rob Euwals Mechanisms Angelika Eymann 99-25 Ulrich Hoffrage How to keep children save in traffic: Find the Martin Weber daredevils while they are young. 99-36 Axel B¨orsch-Supan Template for International Savings Comparisons Ralph Hertwig Project Valerie Chase

99-35 Stefan Schwarz Hindsight Bias: The Role of Perfect Memory and 99-24 Elke Kurz Representations of uncertainty and change: Three Dagmar Stahlberg Meta-Cognitions Gerd Gigerenzer case studies with experts Ulrich Hoffrage 99-34 Dagmar Stahlberg Would I Have Known It All Along if I Would Hate Stefan Schwarz to Know It? The Hindsight Bias in Situations of 99-23 Stefan Krauss Simplifying Bayesian Inference: The General Case High and Low Self Esteem Relevance Laura Martignon Ulrich Hoffrage SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

99-22 Ulrich Hoffrage Hindsight Bias: A Price Worth Paying for Fast and 99-10 Dirk Dauenheimer Self-Enhancement, Self-Verification, or Ralph Hertwig Frugal Memory Dagmar Stahlberg Self-Assessment? The Intricate Role of Trait Sandra Spreemann Modifiability in the Self-Evaluation Process 99-21 Ulrich Hoffrage Irren ist wahrscheinlich. Medizinische Experten Constantine Sedikides und Laien bewerten Risiken oft falsch. 99-09 Cornelia Hegele Was kann der Apfel von der Birne lernen, oder 99-20 Claudia Keser Union Bargaining Strength as a Public Good: Peter Walgenbach wozu brauchen Unternehmen Benchmarking? Jean-Louis Rulli´ere Experimental Evidence Marie-Claire Villeval 99-08 Michaela W¨anke Assimilation and Contrast as a Function of the direction of Comparison 99-19 R¨udiger F. Pohl I’m not trying to impress you, but I surely knew it Dagmar Stahlberg all along! Self-presentation and hindsight bias 99-07 Michael Woywode Ein lerntheoretisches Modell zur Erkl¨arung der Dieter Frey Unternehmensent-wicklung

99-18 Dagmar Stahlberg Preferences for and Evaluation on Self-Relevant 99-06 Tilmann Betsch The Pros and Cons of Expertise: Routine Strength Lars-Eric Petersen Information Depending on the Elaboration of the Susanne Haberstroh and Adaptation in Recurrent Acquisition and Dirk Dauenheimer Self-Schemata Involved Andreas Gl¨ockner Disposal Decisions Klaus Fiedler 99-17 Rob Euwals Do mandatory pensions decrease household savings? Evidence for the Netherlands. 99-05 Ulrich Koch Regel¨anderungsprozesse und organisatorisches Lernen: Zum Ubergang¨ individueller Erfahrungen 99-16 Roman Inderst A Note on the Strategic Foundation of Competitive in eine organisationale Wissensbasis Equilibrium in Buyer Markets 99-04 Alfred Kieser Wie man B¨urokratien das Lernen beibringt 99-15 Michael Adam An Empirical Test of Risk-Adjusted Performance of Ulrich Koch Raimond Maurer Call Option Writing and Put Option Buying Michael Woywode Hedge-Strategies 99-03 Joachim Winter Strukturelleokonometrische ¨ Verfahren zur Analyse 99-14 Annette Reil-Held Vom Arbeitsmarkt in den Ruhestand: Die von Renteneintrittsentscheidungen Reinhold Schnabel Einkommen deutscher Rentner und Rentnerinnen 99-02 Axel B¨orsch-Supan Ersparnisbildung in Deutschland: Meßkonzepte und 99-13 Peter Walgenbach Das Konzept der Vertrauensorganisation - Eine Annette Reil-Held Ergebnisse auf Basis der EVS theoriegeleitete Betrachtung Ralf Rodepeter Reinhold Schnabel 99-12 Herbert Bless Can the same information be typical and atypical? Joachim Winter Michaela W¨anke How perceived typicality moderates assimilation and contrast in evaluative judgements 99-01 Office SFB504 Jahresbericht 1998

99-11 Eric Igou St¨arkere Framing Effekte durch mehr Nachdenken? Herbert Bless Einfl¨usse der Bearbeitungszeit auf L¨osungen des 98-61 Siegfried K. Berninghaus Long-run Evolution of Local Interaction Structures Wolfram Schenck ”Asian-disease”-Problems Karl-Martin Ehrhart in Games

SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

98-60 Isabel G¨odde Does Family Background Matter? - Returns to 98-48 Joachim Winter Does Firms‘ Financial Status Affect Plant-Level Reinhold Schnabel Education and Family Characteristics in Germany Investment and Exit Decision

98-59 Holger M. M¨uller Why Tender Offers Should be Financed with Debt 98-47 Michele Bernasconi Why monetary policy matters — An experimental Oliver Kirchkamp study of saving, inflation and monetary policies in 98-58 Ralf Rodepeter Savings decisions under life-time and earnings an overlapping generations model Joachim Winter uncertainty: 98-46 Oliver Kirchkamp Simultaneous Evolution of Learning Rules and 98-57 Thomas Langer Entscheidungsanalyse Strategies Martin Weber 98-45 Martin Weber Risikomessung im Kreditgesch¨aft: Eine empirische 98-56 Reinhold Schnabel Rates of Return of the German Pay-As-You-Go Jan Pieter Krahnen Analyse bankinterner Ratingverfahren Pension System Frank Voßmann

98-55 Raimond Maurer Immobilienfonds und 98-44 Axel B¨orsch-Supan Anreizprobleme in der Renten- und Steffen Sebastian Immobilienaktiengesellschaften als Krankenversicherung finanzwirtschaftliche Substitute f¨ur Immobiliendirektanlagen 98-43 Martin Hellwig On the Economics and Politics of Corporate Finance and Corporate Control 98-54 Michaela W¨anke Next to a star: Paling, shining or both? Turning Herbert Bless inter-exemplar contrast into inter-exemplar 98-42 Axel B¨orsch-Supan Demographie, Entwicklung und Stabilit¨at der Eric Igou assimilation Sozialversicherung in Deutschland

98-53 Gerd Gigerenzer One-reason decision making. 98-41 Axel B¨orsch-Supan Zur deutschen Diskussion eines Ubergangs¨ vom Laura Martignon Umlage- zum Kapitaldeckungsverfahren in der Ulrich Hoffrage Gesetzlichen Rentenversicherung Joerg Rieskamp Jean Czerlinski 98-40 Axel B¨orsch-Supan A Model under Siege: A Case Study of the Dan G. Goldstein Germany Retirement Insurance System

98-52 Gerd Gigerenzer Cognitive illusions reconsidered 98-39 Martin Hellwig Financial Intermediation with Risk Aversion Ralph Hertwig Ulrich Hoffrage 98-38 Martin Hellwig Risk Aversion and with Ex Peter Sedlmeier Post Information Asymmetry

98-51 Gerd Gigerenzer Overcoming Difficulties in Bayesian Reasoning: A 98-37 Roman Inderst Duopolistic Competition in Search Markets Ulrich Hoffrage Reply to Lewis & Keren and Mellers & McGraw Christian Pfeil

98-50 Roman Inderst Signaling in a Search Market 98-36 Roman Inderst Incentives Schemes as a Signaling Device

98-49 Paul Povel Liquidity Constraints, Production Costs and Output 98-35 Roman Inderst Multi-Issue Bargaining with Endogenous Agenda Michael Raith Decisions 98-34 Roman Inderst Competition Drives Up Prices SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

98-33 Roman Inderst A Note on the Limited Value of Time for Screening 98-12 Siegfried K. Berninghaus Conventions and Local Interaction Structures: Karl-Martin Ehrhart Experimental Evidence 98-32 Roman Inderst Screening With Endogenous Reservation Values Claudia Keser

98-31 Paul Povel optimal bankruptcy laws 98-11 Michael Kilka What Determines the Shape of the Probability Martin Weber Weighting Function under Uncertainty? 98-30 Martin Hellwig Systemische Risiken im Finanzsektor 98-10 Tilmann Betsch The moderating role of category salience and 98-29 Axel B¨orsch-Supan Incentive Effects of Social Security on Labor Force Frank Siebler category focus in judgments of set size and Participation: Evidence in Germany and Across Peter Marz frequency of occurence. Europe Stefan Hormuth Dorothee Dickenberger 98-22 Phillipe Jehiel Efficient Design with Interdependent Valuations Benny Moldovanu 98-08 Peter Albrecht Alterssicherung und Vorsorgebedarf im Spannungsfeld von Versicherungs- und 98-21 Benny Moldovanu Patent Licensing to Bertrand Competitors Investmentprodukten Aner Sela 98-07 Axel B¨orsch-Supan Pension Provision in Germany 98-20 Alfred Kieser How Management Science, Consultancies and Annette Reil-Held Business Companies (Do not) Learn from Each Reinhold Schnabel Other. Applying Concepts of Learning to Different Types of Organizations and to Interorganizational 98-06 Martin Hellwig Discrete-Time Approximations of the Learning Klaus M. Schmidt Holmstr¨om-Milgrom Brownian-Motion, Model of Intertemporal Incentive Provision 98-16 Tilmann Betsch When prior knowledge overrules new evidence: Babette Brinkmann Adaptive use of decision strategies and role 98-05 Tilmann Betsch Natural sampling and base-rate neglect Klaus Fiedler behavioral routines G. - M. Biel Katja Breining C. Eddelbuettel A. Mock 98-15 Klaus Fiedler Illusory Correlations: Explicating and Stimulating Their Apparent and Less Apparent Origins 98-04 Martin Hellwig Allowing for Risk Choices in Diamond’s ”Financial Intermediation as Delegated Monitoring” 98-14 Klaus Fiedler A Sampling Approach to Biases in Conditional Babette Brinkmann Probability Judgments: Beyond Baserate-Neglect 98-03 Martin Weber Hindsight-Bias im Prinzipal-Agent-Kontext: Die Tilmann Betsch and Statistical Format Lukas Mangelsdorff Aktennotiz als Antwort? Beate Wild 98-02 Alfred Kieser Limited Rationality, Formal Organizational Rules, 98-13 Tilmann Betsch Eine Kritik an der klassischen Framing - Studie, Nikolaus Beck and Organizational Learning (OL) Stefan Krauss eine konzeptuelle Replikation und eine Bewertung Risto Tainio der Prospect Theory. 98-01 Office SFB504 Sonderforschungsbereich 504 Jahresbericht 1998

SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

97-44 Raimond Maurer Analytische Evaluation des Risiko-Chance-Profils 97-31 Axel B¨orsch-Supan Das deutsche Rentenversicherungssystem: Michael Adam kombinierter Aktien- und Optionsstrategien Probleme und Perspektiven

97-43 Holger M. M¨uller The Mirrlees-Problem Revisited 97-30 Claudia Keser Principals Marc Willinger 97-42 Annette Reil-Held Bequests and Aggregate Wealth Accumulation in Germany 97-29 Siegfried K. Berninghaus Coordination Games: Recent Experimental Results Karl-Martin Ehrhart 97-41 Axel B¨orsch-Supan Ubergangsmodelle¨ vom Umlage - zum Claudia Keser Kapitaldeckungsverfahren in der deutschen Rentenversicherung 97-28 Peter Albrecht A Stochastic Approach for the Quantification of Default Risk of OTC-Financial Derivatives 97-40 Siegfried K. Berninghaus The invisible hand: Experiments on strategy Karl-Martin Ehrhart selection in population games 97-27 Dagmar Stahlberg Hindsight bias: Impaired memory or biased Claudia Keser A. Maass reconstruction?

97-39 Axel B¨orsch-Supan Retirement Income: Level, Risk, and Substitution 97-26 Manfred Hassebrauck Das ”prototype matching”-Modell des Annette Reil-Held Among Income Components Cornelia Vogt Entscheidungsverhaltens: Darstellung des Modells Michael Diehl und erste Ergebnisse 97-38 Holger M. M¨uller The First-Best Sharing Rule in the Continuous-Time Principal-Agent Problem with 97-24 Claudia Keser SUPER: Strategies used in public goods Exponential Utility experimentation rounds

97-37 Holger M. M¨uller Randomization in Dynamic Principal-Agent 97-23 Axel B¨orsch-Supan Germany: A social security system on the verge of Problems collaps

97-36 Gy¨ongyi Bug`ar International Portfolio Diversification for European 97-22 Axel B¨orsch-Supan Privatisierungsm¨oglichkeiten der Raimond Maurer countries: The viewpoint of Hungarian and German Sozialversicherung in Europa investors 97-21 Axel B¨orsch-Supan Capital productivity and the nature of competition 97-35 Martin Hellwig Banks, Markets, and the Allocation of Risks in an Economy 97-20 Axel B¨orsch-Supan Social security and retirement in germany Reinhold Schnabel 97-34 Nikolaus Beck Standard Operating Procedures and Organizational Alfred Kieser Learning 97-19 Raimond Maurer Ertrag und Shortfall Risiko von Wertsicherungsstrategien mit Optionen unter 97-33 Thomas Langer Implementing Behavioral Concepts into Banking alternativen Zielrenditen: Empirische Evidenzen f¨ur Peter Waller Theory: The Impact of on den deutschen Aktienmarkt Collateralization 97-18 Peter Albrecht Risk based capital allocation and risk adjusted 97-32 Guenther Franke Risk-Value Efficient Portfolios and Asset Pricing performance management in Martin Weber property/liability-insurance: A risk theoretical framework SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES SONDERFORSCHUNGSBereich 504 WORKING PAPER SERIES Nr. Author Title Nr. Author Title

97-17 Peter Albrecht Shortfall-Risiko/Excess-Chance- 97-04 Phillipe Jehiel Multidimensional for Auctions Raimond Maurer Entscheidungskalk¨ule: Grundlagen und Benny Moldovanu with Externalities Matthias M¨oller Beziehungen zum Bernoulli-Prinzip E. Stacchetti

97-16 Claudia Keser Coordination and local interaction: Experimental 97-03 Karsten Fieseler Bidding for unit-price contracts - How craftsmen Karl-Martin Ehrhart evidence should bid Siegfried K. Berninghaus 97-02 Martin Hellwig Unternehmensfinanzierung, Unternehmenskontrolle 97-15 Herbert Bless Framing the framing effect: The impact of context und Ressourcenallokation: Was leistet das Tilmann Betsch cues on solutions to the ”asian disease” problem Finanzsystem? Axel Franzen 97-01 Ralf Rodepeter Identifikation von Sparprofilen im Lebenszyklus 97-14 Michael Kilka Home Bias in International Stock Return Martin Weber Expectation Daniel Schunk The Pennsylvania Reemployment Bonus Experiments: How a survival model helps in the 97-13 Jan Vleugels Bidding against an unknown number of analysis of the data competitiors sharing affiliated information Vo l ker St o cke ´ Measuring Information Accessibility and Predicting 97-12 Dov Monderer and- no-cycling conditions Response-Effects: The Validity of Aner Sela Response-Certainties and Response-Latencies

97-11 S. Hon-Suir A learning approach to auctions Vo l ker St o cke ´ Umfrageeinstellung und Umfrageerfahrung. Die Dov Monderer Bettina Langfeldt relative Bedeutung unterschiedlicher Aspekte der Aner Sela Interviewerfahrung f¨ur die generalisierte Umfrageeinstellung 97-10 Karl H. Schlag You play (an auction) only once Aner Sela

97-09 Aner Sela One against all in the fictitious play process

97-08 Benny Moldovanu und die Auktionstheorie - Anmerkungen zum Nobelpreis 1996

97-07 M. Tietzel Goethe Benny Moldovanu

97-06 Phillipe Jehiel Auctions with Downstream Interaction among Benny Moldovanu Buyers

97-05 Phillipe Jehiel Resale Markets and the Assignment of Property Benny Moldovanu Rights