EDUCATION SIGNALLING AND UNCERTAINTY

Transcription

1 Universität des Saarlandes University of Saarland Volkswirtschaftliche Reihe Economic Series EDUCATION SIGNALLING AND UNCERTAINTY Jürgen EICHBERGER AND David KELSEY Economic Series No March 1999 University of Saarland Department of Economics (FB 2) P.O. Box D SAARBRUECKEN, GERMANY Phone , Fax-4823 http//

2 Abstract Applying the new concept of a Dempster-Shafer equilibrium to signalling games, we show that a pooling equilibrium is the unique equilibrium outcome. With strategic uncertainty, signalling productivity by education may no longer be feasible.

3 EDUCATION SIGNALLING AND UNCERTAINTY Jürgen Eichberger Department of Economics (FB 2) Universität des Saarlandes David Kelsey Department of Economics The University of Birmingham Abstract. Applying the new concept of a Dempster-Shafer equilibrium to signalling games, we show that a pooling equilibrium is the unique equilibrium outcome. With strategic uncertainty, signalling productivity by education may no longer be feasible. 1. Introduction Twenty-five years ago, Spence (1973) wrote in his now famous article on Job Market Signaling The fact that it takes time to learn an individual s productive capabilities means that hiring is an investment decision. The fact that these capabilities are not known beforehand makes the decision one under uncertainty. To hire someone, then, is frequently to purchase a lottery.... Primary interest attaches to how the employer perceives the lottery, for it is these perceptions that determine the wages he offers to pay. We have stipulated that the employer cannot directly observe the marginal product prior to hiring. What he does observe is a plethora of personal data in the form of observable characteristics and attributes of the individual, and it is these that must ultimately determine his assessment of the lottery he is buying. (pp. 356/7). Personal characteristics which the individual can influence, such as education, determine the employer s assessment of a job applicant s productivity. The willingness of an employer to accept a wage claim of a job applicant in turn depends on the employer s

4 JÜRGEN EICHBERGER and DAVID KELSEY belief about the job applicant s productivity. Knowing this mechanism, a job applicant has good reason to consider what beliefs the choice of education level will entail. But can expectations about the productivity level based on the education level of an applicant be reliable? Spence (1973) shows that it can, if the employer interprets an applicant s education level in a particular way. Tirole (1988) confirms this claim in a fully specified game-theoretic model. With the concept of a Perfect Bayesian Equilibrium (PBE) which was designed for signalling games, Spence s analysis could be made completely rigorous. What became obvious, however, was the importance of the outof-equilibrium beliefs. Depending on the particular out-of-equilibrium beliefs, many education-wage combinations could be obtained in equilibrium. Moreover, in a PBE, signalling would always work. The game-theoretic analysis of Spence-like signalling games sparked off a search for refinements of PBE 1 based on more and more sophisticated out-of-equilibrium reasoning. This analysis neglects the inherent weakness of signalling equilibria depending on specific out-of-equilibrium beliefs. Job market signalling, as a reliable means of assessing a job applicant s productivity, becomes more dubious as the degree of sophistication of the refinement increases. In Spence (1973), uncertainty about the lottery which the employer faces is a crucial issue. Traditional game-theoretic analysis leaves no room for uncertainty about an opponent s behaviour. In Nash equilibrium, players predict the behaviour of the opponent precisely. There is no uncertainty about the lottery that players face. Recent attempts to modify game-theoretic equilibrium concepts 2 in order to allow for uncertainty about the opponent s behaviour offer a new perspective on the signalling question. With uncertainty about the other player s strategy, out-of-equilibrium beliefs may have no role to play. Depending on the updating rule, beliefs can be endogenised. In this paper, we use an adaptation of an equilibrium concept introduced in Eichberger and Kelsey (1997a) and adapted to the signalling game structure in Eichberger and Kelsey (1997b) in order to show that, under uncertainty, equilibria of signalling games can have features which differ substantially from those suggested by traditional analysis. For constant degrees of uncertainty, education may no longer be a feasible signalling device. Moreover, with heterogeneous degrees of uncertainty, new equilibria may arise. The following section introduces the new notions of beliefs and updating. Section 3 considers a special case of beliefs. In section 4, signalling games are formally defined and equilibrium concepts presented. Section 5 applies the new concepts to the education-signalling model. Concluding remarks are gathered in section 6. Proofs of propositions are relegated to an appendix. 2. Beliefs and updating Decision makers beliefs are formed subject to complex information patterns. Ellsberg (1961) observed that decision makers prefer to bet on urns with a known colour distribution of balls. It is ambiguity of beliefs which one tries to model by non-additive

5 EDUCATION SIGNALLING AND UNCERTAINTY probabilities. Let S be a finite set of states. Below, in the context of the signalling model, a player is uncertain about the strategy choice of the opponent. Hence, strategy sets will replace the set of states. Definition 2.1 A capacity (non-additive probability) is a real-valued function º on the set of subsets of S with the following properties (i) A µ B ) º(A) º(B) (ii) º(S) =1; º(;) =0 The capacity is convex if for all A; B µ S; º(A [ B) º(A)+º(B) º(A \ B) Capacities capture the imprecision of a decision maker s information by abandoning the restriction to additivity implied by the property º(A [ B) =º(A)+º(B) º(A \ B) for all A; B µ S Convex capacities break this equality in a particular direction which is often associated with greater ambiguity by overweighting bigger events. In order to define an expected value with respect to a capacity some extra notation is useful. Denote by f k the k-th highest value of f on S; then f 1 >f 2 >>f n where f n =minff(s)j s 2 Sg denotes the smallest element of f For convenience, let f 0 be an arbitrary number larger than f 1 =maxff(s)j s 2 Sg Definition 2.2 The Choquet integral of a real-valued function f on S with respect to the capacity º is Z nx fdº= f k [º(fs 2 Sj f(s) f k g) º(fs 2 Sj f(s) f k 1 g)] k=1 The Choquet integral weights outcomes in ascending order by the additional weight attributed to the level set of an outcome. Since lower level sets contain higher level sets, lower outcomes get a higher weight. The downward bias of the Choquet integral models a cautious or pessimistic attitude of the decision maker. In applications, one often wishes to compare situations where a player is confident about his probabilistic assessment with those where ambiguity is experienced. For this purpose, it proves useful to have a measure of deviation of a capacity from an additive probability distribution. One can use the maximal difference of the weight given an event and its complement to the weight of their union as a measure of ambiguity. Definition 2.3 The degree of ambiguity of capacity º is defined as ½ = 1 min AµS [º(A)+º(SnA)] It is then easy to check that a degree of ambiguity of zero implies additivity, provided the capacity is convex.

6 JÜRGEN EICHBERGER and DAVID KELSEY Lemma 2.1 If a convex capacity has zero degree of ambiguity, then it is additive. Proof. See Eichberger and Kelsey (1997a), Proposition 2.1. Thus, ambiguity vanishes, as ½ converges to zero. Additive probabilities remain as the limiting case of a capacity with a degree of ambiguity of zero. 2.1 The support of a capacity Important for applications of the Choquet expected utility (CEU) approach to games is the notion of support for a capacity. There are many different, but equivalent, ways for defining a support for additive probabilities. For capacities however, each of these concepts has a different interpretation 3. Ryan (1997a) studies support concepts in great detail. In this paper, we apply the notion suggested by Dow and Werlang (1994) and Eichberger and Kelsey (1998). The support of a capacity is the smallest event with a complement of measure zero. Definition 2.4 A support of a capacity º; supp º; is an event A such that º(SnA) = 0 and º(SnB) > 0 for all events B ½ A holds. With this support notion, there always exists a support of a capacity; the support, however, may not be unique. 2.2 Dempster-Shafer updating Signalling private information intends to influence the opponent s beliefs. This raises the question of how beliefs represented by a capacity are modified by new information. If beliefs are additive, Bayesian updating is known to be the only consistent way to integrate new information in the belief. A major problem arises if the information received is inconsistent with the probability distribution representing the beliefs. If an event occurs which the decision maker believed to have zero probability, then no consistent updating is possible. In signalling games, this problem has been recognised as the reason for the multiplicity of equilibria. For non-additive beliefs, several updating methods are known and have been investigated in the literature. Gilboa and Schmeidler (1993) provide an axiomatic foundation for several updating rules. All of these share the property that they converge to a Bayesian update if a sequence of non-additive beliefs converges to an additive belief. The Dempster-Shafer updating rule for capacities which will be adopted in this paper can be interpreted as a maximum likelihood procedure 4. Definition 2.5 Dempster-Shafer updating rule (DS-update ) For all events A µ S; º((A \ E) [ SnE) º(SnE) º(AjE) = 1 º(SnE) Note that the DS-updating rule is well-defined if an event E occurs that had measure zero, º(E) =0, provided the complement of E does not have full measure. In game-

7 EDUCATION SIGNALLING AND UNCERTAINTY theoreticapplications with strategicuncertainty,½ >0, this property makes equilibrium predictions much tighter. The following lemma shows that Bayesian updating is the limit of DS-updating. Lemma 2.2 Let º n be a sequence of capacities converging to an additive probability ¼ Suppose that ¼(A) > 0; then the sequence of DS-updates º n ( ja) converges to the Bayesian update ¼( ja) Proof. See Eichberger and Kelsey (1997a), Proposition E-capacities In signalling games, beliefs of a player concern strategies and types. The capacity representing a player s beliefs is therefore defined on a product space S T of finite sets S and TIn this context, one often wants to maintain the assumption that a player is better informed in regard to possible types, possibly because the proportion of types in a population of players is common knowledge, and that ambiguity affects the opponent s choice of strategy. E(llsberg)-capacities 5 offer a convenient way to combine ambiguity about strategies with knowledge about types. Let F t = S ftg be the set of type-strategy combinations with the same type t 2 T Define the following capacity º t (E) by ½ 1 º t (E) = if Ft µ E 0 otherwise An E-capacity with knowledge of an additive probability distribution p on T is defined as follows. Definition 3.1 An E-capacity on S T compatible with the probability distribution p on T is defined by º(E) = ¼(E)+(1 ) X º t (E) p(t) t2t where ¼ is an additive probability distribution with ¼(F t )=p(t) for all t 2 T and is a confidence parameter. The additive probability distribution ¼ on S T can be chosen arbitrarily as long as it satisfies the condition on its marginal distribution ¼(F t )=p(t) for all t 2 T The probability distribution ¼ will be chosen endogenously in an equilibrium of a game. The confidence parameter is interpreted as an exogenously given degree of confidence in the equilibrium distribution ¼ One checks easily that the degree of ambiguity of an E-capacity equals ½ =1. E-capacities of this type will be used extensively throughout this paper. It is therefore useful to record some properties of these capacities before turning to the analysis of the signalling games. For E-capacities, the Choquet integral and the DS-update take particularly simple forms. Moreover, the support of an E-capacity is unique and equal to the support of the additive

8 part of the capacity. JÜRGEN EICHBERGER and DAVID KELSEY Proposition 3.1 The Choquet integral of an E-capacity is Z X fdº= ¼(s; t) f(s; t)+(1 ) X p(t) minff(s; t)j (s; t) 2 F t g t2t (s;t)2s T Proof. See Proposition 2.1 in Eichberger and Kelsey (1997b). Proposition 3.2 The support of an E-capacity is equal to the support of the additive probability distribution on which the capacity is based, supp º = supp ¼ Proof. See Lemma 2.2 in Eichberger and Kelsey (1997b). Proposition 3.3 The DS-update of an E-capacity º compatible with the prior distribution p on T with respect to es 2 S is º(tjes) = ¼(es; P t)+(1 ) p(t) ¼(es; t 0 )+(1 ) t 0 2T Proof. The proof follows from a direct application of Lemma 4.2 in Eichberger and Kelsey (1997b). In Eichberger and Kelsey (1997b) (Lemma 4.1), we show that the DS-update of an E- capacity is again an E-capacity. Moreover, for the case of a product capacity which is compatible with an additive prior distribution, the updated capacity º( jes) is additive. This is quite intuitive, since beliefs about strategy choice s were ambiguous, while there was no ambiguity in regard to types t In contrast to a Bayesian update, the DS-update is well-defined even if º(f(es; t)j t 2 T g) =0for some strategy-type pair holds. Corollary 3.1 The DS-update of an E-capacity on es 2 S 1 with º(f(es; t)j t 2 T g) = 0 is º(tjes) =p(t) Proof. From º(f(es; t)j t 2 T g) =0; it follows for all t 2 T; º(f(es; t)g) =0and, therefore, ¼(f(es; t)g) =0. Because of the additivity of the DS-update of an E-capacity consistent with a probability distribution on types, the Choquet integral conditional on an observed signal is simply the expected value with respect to the additive DS-updated capacity. Proposition 3.4 Let º( jes) be a DS-update of an E-capacity º compatible with the prior distribution p on T with respect to es 2 S 1 The Choquet integral of the updated capacity º( jes) is Z fdº( jes) = X ¼(es; t)+(1 ) p(t) f(es; t) P ¼(es; t)+(1 ) t2t t2t

9 EDUCATION SIGNALLING AND UNCERTAINTY Proof. See the appendix. 4. Signalling Games Signalling games are a special case of dynamic two-player games where players 6 move sequentially. Player 1, the sender, has a characteristic, a type, which is unknown to the opponent. Player 1 moves first and chooses a usually costly action, the signal. Player 2 observes the action of player 1 and uses this information to update his prior beliefs, based on which he will choose his action. Since player 2 does not know the type of player 1, signalling games are two-player games with incomplete information Players I = f1; 2g Strategy sets S 1 = fs 1 1; ; s 1 M g;s2 = fs 2 1;;s 2 N g Typesetofplayer1 T finite. Payoff functions u 1 (s 1 ;s 2 ;t); u 2 (s 1 ;s 2 ;t) Prior distribution p on T It is assumed that the description of the game is common knowledge. 4.1 Equilibrium concepts with additive beliefs From the sequential structure of the game it is clear that player 1 s choice of strategies will depend on her private information, i.e. her type. Since player 2 observes the action of player 1, his response will depend on the observed action. In traditional game theory, a player s belief, represented by an additive probability, coincides with the opponent s actual mixed strategy. Following Milgrom and Weber (1986), we represent type-contingent strategies by a probability distribution ¼ 1 on the strategy-type space of player 1, S 1 T; with the following constraint on the marginal distribution X ¼(s 1 ;t)=p(t) s 1 2S 1 The most commonly used equilibrium concept is Perfect Bayesian Equilibrium 7. Definition 4.1 A Perfect Bayesian Equilibrium (PBE) for the signalling game consists of probability distributions ¼ 1 on S 1 T; ¼ 2 ( ; s 1 ) on S 2 for all s 1 2 S 1 ; and beliefs ¹( js 1 ) on T for all s 1 2 S 1 such that P (i) (bs 1 ;t) 2 supp ¼ 1 implies bs 1 2arg max ¼ 2 (s 2 ; s 1 ) u 1 (s 1 ;s 2 ;t); s 1 2S 1 s 2 2S 2 P (ii) bs 2 2 supp ¼ 2 ( ; s 1 ) implies bs 2 2arg max ¹(tjs 1 ) u 2 (s 1 ;s 2 ;t); s 2 2S 2 P (iii) ¼ 1 (s 1 ;t) > 0 implies ¹(tjs 1 )=¼ 1 (s 1 ;t)á P ¼ 1 (s 1 ;t) t2t t2t t2t

10 JÜRGEN EICHBERGER and DAVID KELSEY Notice that restrictions on beliefs (iii) obtain only for those strategies s 1 which are played with positive probability by some type t of player 1. When choosing her strategy, player 1 takes into consideration that the mixed strategy of player 2, ¼ 2 ( ; s 1 ) will depend on his signal s 1 Player 2 in turn holds beliefs about player 1 s type-contingent behaviour represented by the probability distribution ¼ 1 (s 1 ;t) He will update these beliefs in the light of the signal that he observes accordingtobayeslaw,¹( js 1 ) In a PBE, both beliefs must be justified by the actual play of the two players, i.e., strategies in the support of a player s beliefs must be best responses given the opponent s beliefs. For additive beliefs, this condition implies that beliefs coincide with the mixed strategies that are actually played. 4.2 Equilibrium concepts with non-additive beliefs If one studies games in which players face strategic uncertainty, one can no longer maintain the equality of actual behaviour and beliefs. Dow and Werlang (1994) suggest an equilibrium concept for two-player games which requires consistency of actual behaviour with beliefs in the sense that the strategies in the support of a player s beliefs are best-responses of the opponent 8. In contrast to additive beliefs, however, the concept of support is no longer obvious, and the equilibrium condition does not imply that equilibrium behaviour coincides with equilibrium beliefs. This concept has to be adapted in order to take into account the dynamic structure of a signalling game. In Eichberger and Kelsey (1997a), an equilibrium concept based on DS-updating has been suggested and studied in detail. Definition 4.2 A Dempster-Shafer Equilibrium (DSE) consists of capacities º 1 on S 1 T and º 2 ( ; s 1 ) on S 2 for all s 1 2 S 1 such that R (i) (bs 1 ;t) 2 supp º 1 implies bs 1 2arg max u 1 (s 1 ;s 2 ;t) dº 2 (s 2 ; s 1 ); s 1 2S 1 (ii) bs 2 2 supp º 2 ( ; s 1 ) implies bs 2 2arg max s 2 2S 2 R u 2 (s 1 ;s 2 ;t) d¹ DS (tjs 1 ); where ¹ DS (tjs 1 ) denotes the DS-update of º 1 conditional on s 1 The capacity º 1 and its update in response to signal ¹ DS ( js 1 ) represent the beliefs of player 2 about the strategy-type pair of player 1, before and after the signal s 1 is observed. The capacity º 2 ( ; s 1 ), on the other hand, is the belief of player 1 about player 2 s behaviour which she expects in response to her strategy choice s 1 A DSE is a straightforward adaptation of the PBE concept to games where players face strategic uncertainty in addition to incomplete information. The following existence result is proved in Eichberger and Kelsey (1997a), Proposition 3.1. Proposition 4.1 For any 2 [0; 1] and any probability distribution p on T; there exists a DSE which is compatible with p and where players have degrees of ambiguity ½ 1 ;½ 2

11 EDUCATION SIGNALLING AND UNCERTAINTY In Eichberger and Kelsey (1997a), we show in fact a slightly more general result. Furthermore; we explore there the relationship between DSE and the traditional equilibrium concepts. One can show that an appropriately defined limit of a sequence of DSE equilibria, where beliefs become additive in the limit, is not necessarily a PBE. Moreover, there are PBE which cannot be obtained as an additive limit of a sequence of DSE. Because DS-updates are well-defined even if an event occurs that was given zero weight in the beginning, DSE is in general a more determinate equilibrium concept. In the definition of a DSE, there was no need to restrict updated beliefs. These updates are generated by DS-updating. If players face strategic uncertainty, DS-updates are defined even if a capacity gives zero weight to an event. Taking away the arbitrariness of beliefs about out-of-equilibrium play is in our opinion a major advantage of DSE over PBE. Whether the behaviour in a DSE appears a sensible description of actual behaviour has to be studied in specific applications. Applying the equilibrium notion of a DSE equilibrium to the education-signalling game in the next section and comparing the results to the traditional analysis may provide such a test 9. Traditionally, PBE of signalling games have been classified as separating equilibria, pooling equilibria, or hybrid equilibria. A separating equilibrium is a PBE in which all types of players choose different actions. Player 2 can therefore identify the type of player 1 by observing her action. In a pooling equilibrium, all types of player 1 choose the same action. Player 2 receives therefore no signal which would allow him to distinguish player 1 s type. Many PBE, however, do not fall in either of these two classes, i.e. some types may be discerned by their choice of action while others remain indistinguishable. For DSE, we adapt these concepts as follows. Denote by ¾ t the set of strategies of player 1 in the support of the capacity º 1 ; ¾ t (º 1 )=fs 1 2 S 1 j (s 1 ;t) 2 supp º 1 g; and consider the following definition. Definition 4.3 ADSE(º 1 ; (º 2 ( ; s 1 )) s 1 2S1) is called (i) separating equilibrium if ¾ t (º 1 ) \ ¾ t 0(º 1 )=; for all t; t 0 2 T; (ii) pooling equilibrium if ¾ t (º 1 )=¾ t 0(º 1 ) for all t; t 0 2 T 5. Education Signalling In this section, we present a model based on the labour market signalling model of Spence (1973). Tirole (1988) has adapted the Spence model to make it conform to the structure of a signalling game. To simplify exposition, we have further modified the model by restricting attention to finite strategy sets.

12 JÜRGEN EICHBERGER and DAVID KELSEY Consider firms which intend to hire a worker. There is a large pool of workers with differing productivities. Workers know their productivity, while firms cannot observe the productivity of job applicants directly. What the firm can confirm however is the education level of a worker. If the education level is positively correlated with a worker s productivity, then education may serve as a signal for a worker s productivity. Workers A worker s strategy is a level of education e 2 E and a wage claim w 2 W Assume that E = f0; 1; 2;;Eg and W = f0; 1; 2;;W g arefinitesets. The payoff of a worker depends on her productivity Á t which can be either high, Á H ; or low, Á L ; 0 <Á L <Á H ; and takes the following form u(e; w; Á t )=w e Á t for t = H; L There is a large population of workers with a proportion p of highproductivity workers. Since workers are ex-ante identical, p is also the probability of meeting a high-productivity worker. Firm The firm is the potential employer. Its payoff function does not depend on the level of education that a worker achieves. The productivity of a worker matters however. For simplicity, assume the following payoff function for the firm if it hires a worker of productivity type Á t v(e; w; Á t )=Á t w for t = H;L Workers are assumed to move first. They apply for a job with the firm based on an educationlevel e and a wage claim w The firm responds to the education-wage profile (e; w) by either accepting it, a; or by rejecting it, r The following diagram shows the decision tree for a representative proposal (e; w) 2 E W H q p 0 a 1 p q L (e; w) (e; w) F q. q F a r a r q q q q w e Á H Á H w 0 w e Á L Á L w 0 Figure 1 Decision tree

13 EDUCATION SIGNALLING AND UNCERTAINTY 5.1 Conventional analysis The following two classes of pure-strategy equilibria are usually discussed in conventional analysis. To simplify notation, denote by ¼ H (e; w); ¼ L (e; w) the mixed strategies chosen by a worker of type H and L respectively. In terms of the notation in section 4, ¼ W ((e; w);h) p ¼ H (e; w) and ¼ W ((e; w);l) (1 p) ¼ L (e; w) We present these equilibria in a more formal way than is usually done in textbooks, in order to make similarities and differences to the DSE more transparent. Proposition 5.1 (pooling equilibrium) Let (e ;w ) satisfy the following conditions w e Á Á L and p Á H +(1 p) Á L w Á L L Then the following strategies and beliefs form a PBE (i) ¼ H (e ;w )=¼ L ½(e ;w )=1; 1 for (e; w) =(e (ii) ¼ F (a;(e; w)) = ;w ) or w Á L ½ 0 otherwise p for (e; w) =(e (iii) ¹(Hj(e; w)) = ;w ) 0 otherwise ; Proof. See the appendix. There is a multiplicity of pooling PBE. Figure 2 illustrates the range of education-wage pairs that could be supported as pooling equilibria. All (e; w)-combinations in region P are pooling equilibria. w w e Á L Á H w e Á L S E p Á... P Á L 0 e Figure 2 Perfect Bayesian Equilibria

14 JÜRGEN EICHBERGER and DAVID KELSEY In a pooling equilibrium, education fails as a signal for the firm. Productivity types cannot be distinguished and the employer will accept only wage claims below or equal to the average productivity level p Á H +(1 p) Á L For a low-productivity type such a wage is optimal as long as it does not fall below her productivity level. A highproductivity worker who would benefit from signalling her type cannot do so, because any out of equilibrium education-wage pair will be interpreted as a signal of a lowproductivity type. Notice the importance of out-of-equilibrium beliefs for this set of equilibria. We will demonstrate below with Example 5.2 that the set of equilibrium education-wage pairs would be substantially altered if an out-of-equilibrium (e; w)-pairs were not interpreted as indicating a low-productivity type, ¹(Hj(e; w)) = 0. Pooling equilibria can be ordered in the Pareto sense. From Figure 2, the Paretodominant equilibrium is easily identified as (e ;w )=(0;p Á H +(1 p) Á L ) Separating equilibria form a second class of PBE. Proposition 5.2 (separating equilibrium) Let (e H ;w H ) and (e L ;w L ) satisfy the following conditions w H e H Á H w L = Á L w H e H Á L ; e L =0 and w H Á H The following strategies and beliefs form a PBE (i) ¼ H (e H ;w H )=¼ ½L(0;Á L )=1; 1 for (e; w) 2f(e (ii) ¼ F (a;(e; w)) = H ;w H ); (0;Á L)g or w Á L ½ 0 otherwise 1 for (e; w) =(e (iii) ¹(Hj(e; w)) = H ;w H ) 0 otherwise ; Proof. See the appendix. The multiplicity of separating equilibria is illustrated by the (e; w)-combinations in region S of Figure 2. All these equilibria are supported by out-of-equilibrium beliefs which attribute any non-equilibrium education-wage pair to the low-productivity worker. In many of these equilibria the high-productivity worker is clearly identified by the firm but does not obtain a wage equal to her marginal productivity. A higher wage claim of the high-productivity worker would be rejected by the firm because it would read it as a signal of the low-productivity worker. As in the case of the pooling equilibria, one can Pareto-order these equilibria. Since a low-productivity worker obtains the same wage in every equilibrium, one can Paretorank the separating equilibria in terms of the preferences of the high-productivity worker. The Pareto-dominant separating equilibrium is (e H ;w H )=(Á L (Á H Á L );Á H ) The high-productivity worker receives a wage equal to her marginal product but has to educate herself up to the level Á L (Á H Á L ) There are many more equilibria in mixed strategies as the following example illustrates.

15 EDUCATION SIGNALLING AND UNCERTAINTY Example 5.1 Suppose Á H =5and Á L =1,andp = 1 2 The following mixed strategies form a PBE (pooling equilibrium) ½ 1 (i) ¼ H (e; w) =¼ L (e; w) = 2 for (e; w) =(0; 4) 1 ; 8 2 for (e; w) =(0; 2) 1 >< 3 for (e; w) =(0; 4) (ii) ¼ F 2 (a;(e; w)) = 3 for (e; w) =(0; 2) ; 0 for 1 w 6=2; 4 > ½ 1 for w<1 1 for (e; w) 2f(0; 4); (0; 2)g (iii) ¹(Hj(e; w)) = 0 otherwise It is straightforward to check that the worker is indifferent between choosing 2 or 4 and the firm is indifferent about a and r Equilibrium behaviour in a PBE depends crucially on the out-of-equilibrium beliefs. Intuition suggests that the main beneficiary of a signal will be the high-productivity worker, provided the (e; w)-pair suggested is less attractive for the low-productivity worker than the marginal-product wage combined with a zero level of education. Based on such reasoning, many refinements of PBE have been suggested in the literature 10. Criteria for eliminating out-of-equilibrium beliefs make usually reference to the equilibrium outcome. Forward induction arguments assume that types of workers who could not possibly gain from a deviation, compared to what they get in equilibrium, are to be assigned probability zero. The most commonly used criterion in the educationsignalling context isthe intuitive criterion which selects the Pareto-optimal separating PBE (e H ;w H )=(Á H (Á H Á L );Á H ); (e L ;w L )=(0;Á L ) Many refinements are driven by an effort to justify the Pareto optimal equilibrium in the signalling game. Unfortunately, no refinement known today guarantees selection of the Pareto-optimal PBE in every signalling game. The intuitive criterion selects the Paretooptimal PBE in the signalling game if workers may have two types of productivity, but fails if there are three possible productivity levels. From the employer s viewpoint, signals appear ambiguous. Arguments about what a firm should conclude from an out-of-equilibrium education-wage offer are highly speculative and require an extreme degree of coordination in beliefs between worker and firm. The only firm knowledge of an employer is the prior distribution of types and the repeated observation of the equilibrium education-wage pairs. If an unknown worker offers new (e; w)-combination, a reversion to his prior beliefs appears to be a reasonable reaction of the employer. The following example illustrates how this assumption about out-of-equilibrium beliefs restricts the set of PBE. Example 5.2 Let Á H =5and Á L = 1, and p = 1 2 Suppose that firms take an observed deviation from equilibrium play as evidence that their reasoning about the

16 JÜRGEN EICHBERGER and DAVID KELSEY workers behaviour has failed. In this case, all they know is the fact that the proportion of the high-productivity types in the population is p = 1 Based on this reasoning, 2 there is a unique pooling PBE of the education signalling game with the following equilibrium strategies (i) ¼ H (0; 3) = ¼ L (0; ½3) = 1; 1 for w 3 (ii) ¼ F (a;(e; w)) = 0 for w>3 ; (iii) ¹(Hj(e; w)) = p From (i) and (ii), the only signal that the firm can receive if players follow their equilibrium strategies is (0; 3) Bayesian updating yields ¹(Hj(e; w)) = p in this case. Note however that, by assumption, the firm also expects to meet a high-productivity worker with probability p if some out-of-equilibrium signal (e; w) 6= (0; 3) is observed. Given theses beliefs, it is clearly optimal for the firm to accept any offer with a wage rate less than or equal to the average productivity of 3 Hence, behaviour described in (ii) is optimal. Finally, a worker with low productivity, cannot gain by making a wage claim above 3 (with or without extra educational qualifications) because the firm will not accept such a claim. Nor would a high-productivity worker be able to extract a higher wage by obtaining higher education because the firm would take such a deviation as an indication that the equilibrium reasoning has failed and reject any wage except the averagewagerateof3 The out-of-equilibrium beliefs which are not endogenously determined in a PBE largely determine equilibrium behaviour. If a firm takes deviation from a separating equilibrium as a reason for doubts about the equilibrium prediction, then no separating equilibrium can exist. In any separating PBE, the low-productivity worker will obtain at best (e L ;w L )=(0; 1). By deviating to any non-equilibrium education-wage pair with a higher wage, say (e 0 ;w 0 )=(0; 2); the worker could secure this higher wage. Since the firm would conclude that the separating equilibrium hypothesis is false and revert to the belief ¹(Hj(e 0 ;w 0 )) = p; the expected payoff of the firm from hiring the worker would be 3 which makes it optimal to accept the offer. Thus, a low-productivity worker predicting this acceptance, has an incentive to deviate from the separating equilibrium strategy. 5.2 Dempster-Shafer equilibria Conventional analysis of the education signalling model reveals that PBE predictions are driven by assumptions about the interpretation of out-of-equilibrium beliefs. Pure rationality, i.e., optimisation of agents plus rational expectations, hardly restricts the equilibrium outcomes. With strategic uncertainty, i.e., some ambiguity about the equilibrium behaviour of the opponent, modelled by CEU this changes dramatically. Choquet expected utility theory has well-researched decision-theoretic foundations 11. If ambiguity is modelled by E-capacities, out-of-equilibrium beliefs are defined provided there is some positive degree of ambiguity, ½>0 Hence, DSE makes a clear prediction about the equilibrium outcome in the education-signalling game.

17 EDUCATION SIGNALLING AND UNCERTAINTY Proposition 5.3 Suppose that (i) a worker s beliefs are characterised by a simple capacity with constant con dence parameter W 2 (0; 1) and that (ii) the employer s beliefs are represented by an E-capacity compatible with a prior additive probability distribution p on T with p(t) > 0 for all t 2 T and a confidence parameter F 2 (0; 1); then the unique DSE is the Pareto-efficient pooling equilibrium which satisfies supp º W = f(0;e p Á; H); (0;E p Á; L)g with E p Á = p Á H +(1 p) Á L Proof. See the appendix. The logic of this result is easy to explain. Given strategic uncertainty represented by E-capacities with a strictly positive degree of ambiguity, the additive part of the capacities will be determined in equilibrium. Workers will always claim the highest wage which they expect the employer to accept. Knowing that their education-wage pair will signal productivity to the employer, workers will use their private information strategically. The employer updates his beliefs according to the DS-rule. Being uncertain about the equilibrium strategy of the worker, by Corollary 3.1, any out-of-equilibrium education-wage pair will lead the employer to fall back on his prior beliefs. A lowproductivity worker can therefore scramble any signal which the high-productivity worker could send. Since the average wage is higher than the low-productivity wage, low-productivity workers have an incentive to propose this average wage. The employer will accept such a proposal whether it is an equilibrium strategy or an out-ofequilibrium move. The result of Proposition 5.3 depends on the fact that the degree of confidence of the worker W is independent of the worker s signal (e; w). E-capacities have a constant, exogenously chosen degree of ambiguity, ½ =1. A worker s belief about whether the firm will accept or reject her wage claim is contingent on her education-wage signal. In Proposition 5.3, only the endogenously determined additive part of the capacity ¼( ;(e; w)) depends on the wage claim. The degree of confidence in these predictions, W ; is assumed constant. One could argue that the degree of confidence itself be dependent on the signal e.g., that the degree of confidence about the likelihood of acceptance of a proposal increases with a falling wage claim. The following example shows that other pooling equilibria may occur if the degree of confidence varies with the signal (e; w). Example 5.3 Let Á H =5and Á L =1,andp = 1 2 We claim that the following beliefs form a DSE º W is an E-capacity with prior distribution p and confidence parameter F defined by the probability distribution 8 1 < ¼ W 4 for (0; 1;H) and (0; 1;L) 1 (e; w; t) = 4 for (0; 3;H) and (0; 3;L) 0 otherwise

18 JÜRGEN EICHBERGER and DAVID KELSEY º F ( ;(e; w) is a capacity defined by º F ( ;(e; w) = W (e; w) ¼ F (a;(e; w)); with ½ 1 for w 3 ¼ F (a;(e; w)) = 0 for w 3 and By Definition 3.2, supp º F 8 < W (e; w) = 1 4 for (e; w) =(0; 3) 3 4 for (e; w) =(0; 1) 1 5 otherwise = f(0; 1;H); (0; 3;L); (0; 1;H); (0; 3;L)g; 8 < fag for w<3 supp º W ( ;(e; w)) = fa; rg for w =3 frg for w>3 By Proposition 3.4, for t = H;L; one obtains R [w e Á ] 1 a (s F ) dº F (s F ;(e; w)) 8 t 3 for (e; w) =(0; 3) 4 >< = > 3 4 for (e; w) =(0; 1) 1 5 (w e Á t )+ 4 5 minf0;w e Á t g for (e; w) with ½ w 3; (e; w) 6= (0; 1); (0; 3) 0 for (e; w) with w>3 R Clearly, arg max [w e ] 1 Á a (s F ) dº F (s F ;(e; w)) = f(0; 1); (0; 3)g for each t (e;w)2e W type of worker. Now, consider the DS-updates. By Proposition 3.3, one easily checks that ¹ DS (Hj(e; w)) = 1 2 ; for all (e; w) 2 E W Hence, by Proposition 3.4, if the firm accepts the offer, a; its payoff will be Z [Á t w] d¹ DS (tj(e; w)) = [3 w]; and, for a rejection r; it obtains a payoff of 0 Hence, the firm s best responses are Z arg max [Á t w] d¹ DS (tj(e; w)) = (e;w)2e W This shows that the proposed beliefs form a DSE. 8 < fag for w<3 fa; rg for w =3 frg for w>3

19 EDUCATION SIGNALLING AND UNCERTAINTY 6. Concluding Remarks In signalling games, optimality of a receiver s behaviour depends more on how the player interprets a signal which is not supposed to have been sent according to the equilibrium play. No equilibrium consistency requirement will restrict these beliefs. Multiplicity of equilibria and uncertainty about the behaviour of players is the consequence. Strategic uncertainty has the potential to reduce the indeterminateness of strategic equilibria substantially. Modelled by CEU, the decision maker reserves some weight for outcomes other than those predicted in equilibrium. Thus, Dempster-Shafer equilibria have the potential to restrict beliefs off the equilibrium path. CEU preferences and Dempster-Shafer updating are based on axiomatic foundations. The implied behavioural assumptions are therefore transparent. Applying these concepts to game-theoretic analysis raises issues of the appropriate degree of consistency in equilibrium. Our concept of a DSE provides a possible answer to this question. Education-signalling games are well-known for the plethora of PBE behaviour. DSE with beliefs modelled by E-capacities lead to unique equilibrium behaviour and outcomes if the degree of ambiguity is positive. Sophisticated arguments about out-ofequilibrium beliefs based on forward induction principles can be replaced by assumptions about the degree of ambiguity. The new approach offers an opportunity for a better descriptive analysis of signalling games. APPENDIX Proof of Proposition 3.4 In Eichberger and Kelsey (1997b) (Lemma 4.1), it is shown that, for a subset E µ T; º(Ejes) = b X b¼(es; t)+(1 b ) X bp(t) (i) t2e t2e for appropriately chosen b ; b¼; and bp Furthermore, in Proposition 3.3, º(tjes) = ¼(es; P t)+(1 ) p(t) ¼(es; t 0 )+(1 ) (ii) t 0 2T Hence, applying first 3.1 to equation (i) and, after some simple manipulations, substituting equation (ii), one obtains Z fdº( jes) = b X t2t b¼(es; t) f(es; t)+(1 b ) X t2t bp(t) f(es; t) = X f(es; t) [ b b¼(es; t)+(1 b ) bp(t)] t2t

20 JÜRGEN EICHBERGER and DAVID KELSEY = X f(es; t) º(tjes) t2t = X ¼(es; t)+(1 ) p(t) f(es; t) P ¼(es; t 0 )+(1 ) t2t t 0 2T Proof of Proposition 5.1 First note that ¹ is a probability distribution on the type space T = fh;lg Furthermore, (i) and (ii) imply that only (e ;w ) will be observed in equilibrium. By Bayesian updating, ¹(Hj(e ;w p ¼ H (e ;w ) )) = p ¼ H (e ;w )+(1 p) ¼ L (e ;w ) = p (i) supp ¼ H = supp ¼ L = f(e ;w )g The following estimation shows that (e ;w ) is a maximiser for both types of the worker ¼ F (a;(e ;w )) [w e ]+¼ F (r;(e ;w )) 0 Á H = [w e ] Á H [w e ] Á L Á L ¼ F (a;(e; w)) [w e = ½ 0 for w>ál w e Á t for w Á L Á H ]+¼ F (r;(e; w)) 0 ½ fag for (e (ii) supp ¼ F ( ;(e; w)) = ;w ) frg otherwise or w Á L To show that this behaviour is optimal we note a) For (e; w) =(e ;w ); we have ¹(Hj(e ;w )) [Á H w ]+¹(Lj(e ;w )) [Á L w ] = p Á H +(1 p) Á L w 0 Hence, a is optimal. b) For (e; w) with w Á L ; we obtain ¹(Hj(e; w)) [Á H w]+¹(lj(e; w)) [Á L w] = Á L w 0 Again, a is optimal. Finally, for any other (e; w), one has ¹(Hj(e; w)) [Á H w]+¹(lj(e; w)) [Á L w] = Á L w<0 This implies r is optimal.

21 EDUCATION SIGNALLING AND UNCERTAINTY Proof of Proposition 5.2 First note that ¹ is an additive probability on the type space T = fh;lg Furthermore, (i) and (ii) imply that only (e H ;w H ) or (0;Á L) will be observed in equilibrium. By Bayesian updating, ¹(Hj(e; w)) = p ¼ H (e; w) p ¼ H (e; w)+(1 p) ¼ L (e; w) = ½ 1 for (e; w) =(e H ;w H ) 0 for (e; w) =(0;Á L ) (i) supp ¼ H = f(e H ;w H )g The following estimation shows that (e H ;w H ) is a maximiser ¼ F (a;(e H;w H)) [w H e H Á H ]+¼ F (r;(e H;w H)) 0 = [w H e H ] Á H Á L ¼ F (a;(e; w)) [w e ]+¼ F (r;(e; w)) 0 Á H ½ 0 for w>ál = w e Á for w Á H L Similarly, supp ¼ L = f(0;á L )g The following estimation shows that (0;Á L ) is a maximiser ¼ F (a;(0;á L )) Á L + ¼ F (r;(0;á L )) 0 = Á L = ¼ F (a;(e; w)) [w e ]+¼ F (r;(e; w)) 0 Á 8 H >< w H e H ÁL for (e; w) =(e H ;w H ) 0 for (e; w) 6= (e > H ;w H ) and w>á L w e Á for (e; w) 6= (e L H ;w H ) and w Á L ½ fag for (e; w) 2f(e (ii) supp ¼ F ( ;(e; w)) = H ;w H ); (0;Á L)g or w Á L frg otherwise To show that this behaviour is optimal we note a) For (e; w) =(e H ;w H ); we have ¹(Hj(e H ;w H )) [Á H w H ]+¹(Lj(e H ;w H )) [Á L w H ] = Á H w H 0 Hence, a is optimal. b) For (e; w) =(0;Á L ); we have ¹(Hj(0;Á L )) [Á H Á L ]+¹(Lj(0;Á L )) [Á L Á L ] = Á L Á L 0 Hence, a is optimal. c) For (e; w) with w Á L ; we obtain ¹(Hj(e; w)) [Á H w]+¹(lj(e; w)) [Á L w]

22 JÜRGEN EICHBERGER and DAVID KELSEY = Á L w 0 Again, a is optimal. Finally, for any other (e; w), one has ¹(Hj(e; w)) [Á H w]+¹(lj(e; w)) [Á L w] = Á L w<0 This implies r is optimal. Proof of Proposition 5.3 It is straightforward to compute the payoff functions of the players. Worker Z V W (e; w; t) = [w e ] 1 a (s F ) dº F (s F ;(e; w)) Á t = W [w e Á t ] ¼ F (a;(e; w)) + (1 W ) minf0;w e Á t g Firm Z V F (aj(e; w)) = [Á t w] d¹ DS (tj(e; w)) = [Á H w] ¹ DS (Hj(e; w)) + [Á L w] ¹ DS (Lj(e; w)) = Á L +(Á H Á L ) ¹ DS (Hj(e; w)) w F ¼ W (e; w; H)+(1 F ) p = Á L +(Á H Á L ) F [¼ W (e; w; H)+¼ W (e; w; L)] + (1 F ) w Recall that, by Proposition 3.3 the DS-update of an E-capacity is additive. The proof of this proposition follows now from a sequence of lemmata. First, we show that, in a DSE, there is a strategy (e; w) for each type with positive probability in the prior distribution. Lemma For all t 2 T; there exists (e; w) 2 E W such that ((e; w);t) 2 supp º W Proof. Suppose there exists t 2 T such that ((e; w);t) =2 supp º W for all (e; w) 2 E W Then º W (E W ftg) =0; since E W ftg µ(e W T)n supp º W and monotonicity imply 0 º W (E W ftg) º W ((E W T )n supp º W )=0 Hence, 0=º 1 (E W ftg) =p(t) > 0 a contradiction.. It follows that ¾ H (º W ) 6= ; and ¾ L (º W ) 6= ; Lemma There is at most one (e; w) which both types of worker play in equilibrium, f(e; w)g = ¾ H (º W ) \ ¾ L (º W ) Proof. Suppose there are (e; w); (e 0 ;w 0 ) 2 ¾ H (º W ) \ ¾ L (º W ) By condition (i) of a DSE (Definition 4.2), w e Á H = w 0 e0 Á H and

23 EDUCATION SIGNALLING AND UNCERTAINTY w e = w 0 e0 Á L Á L must be true. This contradicts the assumption Á H >Á L Lemma For all F ; W 2 (0; 1]; a DSE is a pooling equilibrium, f(e; w)g = ¾ H (º W )=¾ L (º W ) Proof. Let Á(e; w) =Á L +(Á H Á L ) ¹ DS (Hj(e; w)) be the expected productivity if the education-wage pair (e; w) is observed. The firm will accept any wage w Á(e; w) Therefore, w = Á(e; w) must hold for any (e; w) 2 ¾ H (º W ) [ ¾ L (º W ) Suppose now the lemma is false. Then there exists either (e L ;w L ) 2 ¾ L (º W )=¾ H (º W ) or (e H ;w H ) 2 ¾ H (º W )=¾ L (º W ). Case (i) (e L ;w L ) 2 ¾ L (º W )=¾ H (º W )Hence, ¼ W (e L ;w L ;H)=0 and ¼ W (e L ;w L ;L) > 0 This implies ¹ DS (Hj(e L ;w L )) <pand w L = Á(e L ;w L ) < Á(0;E p Á)=E p Á For a low-productivity worker, V W (e L ;w L ;L)= W [w L el ] < W E p Á = V W (0;E p Á; L); Á L the proposal (0;E p Á); which will be accepted, yields a strictly higher payoff than (e L ;w L ) This contradicts the equilibrium requirement (e L ;w L ) 2 arg max V W (e; w; L) Note that case (i) implies that there cannot be a separating DSE, i.e., ¾ H (º W )\¾ L (º W ) 6= ; Case (ii) (e H ;w H ) 2 ¾ H (º W )=¾ L (º W ) and f(e; w)g = ¾ H \ ¾ L It follows from ¼ W (e H ;w H ;L)=0 and ¼ W (e H ;w H ;H) > 0; ¼ W (e; w; L) =1 p and ¼ W (e; w; H) > 0; and p = ¼ W (E W fhg) = ¼ W (e; w; H)+¼ W (e H ;w H ;H)+¼ W ((E W )nf(e; w); (e H ;w H )g fhg) that p>¼ W (e; w; H) Hence, ¹ DS (Hj(e; w)) <pand Á(e; w) < Á(0;E p Á)=E p Á Thus, the firm will only accept wages w Á(e; w) By deviating to the proposal (0;E p Á); which will be accepted by the firm, a low-productivity worker can obtain a payoff of W E p Á> W [w e ]. This proves that beliefs with ¼ W (e; w; H) <pcannot be optimal. Á L The following two lemmata establish that f(0;e p Á)g = ¾ H (º W )=¾ L (º W ) Lemma In a pooling DSE, e =0 Proof. Suppose there is a pooling DSE with e>0 By condition (i) of a DSE (Definition 4.2), w e w e Á t Á t for all (e; w) which will be accepted by the firm. Since the DS-update for an out-of-

24 JÜRGEN EICHBERGER and DAVID KELSEY equilibrium event equals p; ¹ DS (Hj(e; w)) = p = ¹ DS (Hj(0;w)) and V F (aj(e; w)) = V F (aj(0;w) follow. Yet, for e =0;w>w e contradicting Á t the optimality of (e; w) with e>0 Lemma In a pooling DSE, w = E p Á Proof. By the same argument as in the previous lemma, one has ¹ DS (Hj(0;w)) = p = ¹ DS (Hj(0;E p Á)) and V F (aj(0;e p Á) 0=V F (rj(0;e p Á) Hence, w<e p Á cannot be optimal for a worker of either type. On the other hand, for w > E p Á; the firm will reject the proposal, V F (aj(0;w) < 0=V F (rj(0;w) This completes the proof of the proposition Notes 1. Mailath (1992) contains a survey and discussion of the most commonly used refinements. 2. Compare Dow and Werlang (1994), Eichberger and Kelsey (1998), and Eichberger and Kelsey (1997a). 3. A special feature of the support notions for capacities which distinguishes them from the support notion of an additive probability distribution is the fact that the outcome on an event which is not contained in the support may still alter the Choquet integral and, thus, influence behaviour. 4. Compare Gilboa and Schmeidler (1993). 5. E-capacities have been studied in Eichberger and Kelsey (1997b) in great detail. 6. Throughout the paper, we will refer to player 1 as she and player 2 as he. 7. In order to distinguish updates of a measure on an observed signal s 1 from beliefs conditional on the signal s 1 ; we write ¹( js 1 ) for the update, and ¼( ; s 1 ) for the conditional beliefs. 8. This equilibrium concept is discussed and compared with alternative approaches in Eichberger and Kelsey (1998). 9. Eichberger and Kelsey (1997a) and Ryan (1997b) contain further applications of the DSE concept. 10. Mailath (1992) provides a survey and discussion of many refinements suggested in the literature for signalling games. 11. Schmeidler (1989), Gilboa (1987), and Sarin and Wakker (1992) provide axiomatic foundations for decision making with ambiguity.

25 EDUCATION SIGNALLING AND UNCERTAINTY References Dow, J., Werlang, S.R.d.C. (1994). Nash Equilibrium under Knightian Uncertainty Breaking Down Backward Induction. Journal of Economic Theory 64, Eichberger, J., Kelsey, D. (1998). Non-Additive Beliefs and Strategic Equilibria. Mimeo. University of Saarland, Saarbrücken. Eichberger, J., Kelsey, D. (1997a). Signalling Games with Uncertainty. Discussion Paper No Department of Economics. University of Birmingham. Eichberger, J., Kelsey, D. (1997b). E-Capacities and the Ellsberg Paradox. Theory and Decision, forthcoming. Ellsberg, D. (1961). Risk, Ambiguity and the Savage Axioms. Quarterly Journal of Economics 75, Gilboa, I. (1987). Expected Utility Theory with Purely Subjective Probabilities. Journal of Mathematical Economics 16, Gilboa, I., Schmeidler, D. (1993). Updating Ambiguous Beliefs. Journal of Economic Theory 59, Mailath, G. (1992). Signalling Games. In Creedy, J., Borland, J. and Eichberger, J. (1992). Recent Developments in Game Theory. Aldershot Edward Elgar, Milgrom, P., Weber, R. (1986). Distributional Strategies for Games with Incomplete Information. Mathematics of Operations Research 10, Ryan, M. (1997a). CEU Preferences and Game-Theoretic Equilibria. Working paper No. 167, Auckland Business School, Auckland, NZ. Ryan, M. (1997b). A Refinement of Dempster-Shafer Equilibrium. Mimeo. University of Auckland, NZ. Sarin, R., Wakker, P. (1992). A Simple Axiomatization of Non-Additive Expected Utility. Econometrica 60, Schmeidler, D. (1989). Subjective Probability and Expected Utility without Additivity. Econometrica 57, Spence, M. (1973). Job Market Signalling. Quaterly Journal of Economics 87, Tirole, J. (1988). The Theory of Industrial Organisation. Cambridge, Mass. MIT Press.

26 Volkswirtschaftliche Reihe/Economic Series Prof. Dr. Hermann ALBECK Prof. Dr. Jürgen EICHBERGER Prof. Dr. Ralph FRIEDMANN Prof. Dr. Robert HOLZMANN PD Dr. Udo BROLL Prof. Dr. Christian KEUSCHNIGG Prof. Dr. Dieter SCHMIDTCHEN Prof. Dr. Volker STEINMETZ Nationalökonomie, insbesondere Wirtschaftsund Sozialpolitik http// Nationalökonomie, insbesondere Wirtschaftstheorie http// Statistik und Ökonometrie http// Nationalökonomie, insbesondere Internationale Wirtschaftsbeziehungen http// http// Nationalökonomie, insbesondere Finanzwissenschaft http// Nationalökonomie, insbesondere Wirtschaftspolitik http// Statistik und Ökonometrie http// Christian KEUSCHNIGG Rationalization and Specialization in Start-up Finanzwissenschaft Investment 9902 Jürgen EICHBERGER Non-Additive Beliefs and Strategic Equilibria David KELSEY Economic Theory 9903 Jürgen EICHBERGER Education Signalling and Uncertainty David KELSEY Economic Theory 9904 Christian KEUSCHNIGG Eastern Enlargement of the EU Wilhelm KOHLER How Much is it Worth for Austria? Finanzwissenschaft 9905 Aymo BRUNETTI More Open Economies Have Better Governments Beatrice WEDER Internationale Wirtschaftsbeziehungen 9906 Ralph FRIEDMANN Effects Of The Order Of Entry On Market Share, Markus GLASER Trial Penetration, And Repeat Purchases Empirical Statistik und Ökonometrie Evidence Or Statistical Artefact?