Abstract - uni-saarland.de · 2013-02-13 · means that hiring is an investment decision. The fact that these capabilities are not known beforehand makes the decision one under uncertainty

EDUCATION SIGNALLING

AND UNCERTAINTY

Jürgen EICHBERGER AND David KELSEY

Economic Series No. 9903March 1999

University of SaarlandDepartment of Economics (FB 2)

P.O. Box 151 150D-66041 SAARBRUECKEN, GERMANY

Phone: +49-681-302 -4822, Fax:-4823http://www.uni-sb.de/rewi/fb2/eichberger/

E-mail: [email protected]

Volkswirtschaftliche ReiheEconomic Series

Universität des SaarlandesUniversity of Saarland

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 pro-

ductivity by education may no longer be feasible.

EDUCATION SIGNALLING ANDUNCERTAINTY

Jürgen Eichberger

Department of Economics (FB 2)Universität des Saarlandes

David Kelsey

Department of EconomicsThe University of Birmingham

Abstract.Applying the new concept of a Dempster-Shafer equilibrium to signalling games, weshow that a pooling equilibrium is the unique equilibrium outcome. With strategicuncertainty, 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 MarketSignaling��: ��The fact that it takes time to learn an individual�s productive capabilitiesmeans that hiring is an investment decision. The fact that these capabilities are notknown beforehand makes the decision one under uncertainty.To hire someone, then, is frequently to purchase a lottery. ..... Primary interest attachesto how the employer perceives the lottery, for it is these perceptions that determine thewages he offers to pay. We have stipulated that the employer cannot directly observethe marginal product prior to hiring. What he does observe is a plethora of personaldata in the form of observable characteristics and attributes of the individual, and it isthese 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, deter-mine the employer�s assessment of a job applicant�s productivity. The willingness of anemployer to accept a wage claim of a job applicant in turn depends on the employer�s

JÜRGEN EICHBERGER and DAVID KELSEY

belief about the job applicant�s productivity. Knowing this mechanism, a job appli-cant 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 anapplicant be reliable? Spence (1973) shows that it can, if the employer interprets anapplicant�s education level in a particular way. Tirole (1988) confirms this claim in afully specified game-theoretic model. With the concept of a Perfect Bayesian Equilib-rium (PBE) which was designed for signalling games, Spence�s analysis could be madecompletely rigorous. What became obvious, however, was the importance of the out-of-equilibrium beliefs. Depending on the particular out-of-equilibrium beliefs, manyeducation-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 forrefinements of PBE1 based on more and more sophisticated out-of-equilibrium reason-ing. This analysis neglects the inherent weakness of signalling equilibria depending onspecific out-of-equilibrium beliefs. Job market signalling, as a reliable means of assess-ing a job applicant�s productivity, becomes more dubious as the degree of sophisticationof the refinement increases.In Spence (1973), uncertainty about the lottery which the employer faces is a crucialissue. Traditional game-theoretic analysis leaves no room for uncertainty about an op-ponent�s behaviour. In Nash equilibrium, players predict the behaviour of the opponentprecisely. There is no uncertainty about the lottery that players face. Recent attemptsto modify game-theoretic equilibrium concepts2 in order to allow for uncertainty aboutthe opponent�s behaviour offer a new perspective on the signalling question. With un-certainty about the other player�s strategy, out-of-equilibrium beliefs may have no roleto play. Depending on the updating rule, beliefs can be endogenised.In this paper, we use an adaptation of an equilibrium concept introduced in Eichbergerand Kelsey (1997a) and adapted to the signalling game structure in Eichberger andKelsey (1997b) in order to show that, under uncertainty, equilibria of signalling gamescan have features which differ substantially from those suggested by traditional analy-sis. For constant degrees of uncertainty, education may no longer be a feasible sig-nalling device. Moreover, with heterogeneous degrees of uncertainty, new equilibriamay arise.The following section introduces the new notions of beliefs and updating. Section 3considers a special case of beliefs. In section 4, signalling games are formally de-fined and equilibrium concepts presented. Section 5 applies the new concepts to theeducation-signalling model. Concluding remarks are gathered in section 6. Proofs ofpropositions 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 dis-tribution of balls. It is ambiguity of beliefs which one tries to model by non-additive

EDUCATION SIGNALLING AND UNCERTAINTY

probabilities.Let S be a finite set of states. Below, in the context of the signalling model, a player isuncertain about the strategy choice of the opponent. Hence, strategy sets will replacethe set of states.

Definition 2.1 A capacity (non-additive probability) is a real-valued function º onthe 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 abandoningthe 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 whichis often associated with greater ambiguity by overweighting bigger events.In order to define an expected value with respect to a capacity some extra notation isuseful. Denote by fk the k-th highest value of f on S; then f1 > f2 > ::: > fn wherefn = minff(s)j s 2 Sg denotes the smallest element of f: For convenience, let f0 bean arbitrary number larger than f1 = maxff(s)j s 2 Sg:

Definition 2.2 The Choquet integral of a real-valued function f on S with respectto the capacity º isZ

f dº :=nX

k=1

fk ¢ [º(fs 2 Sj f(s) ¸ fkg) ¡ º(fs 2 Sj f(s) ¸ fk¡1g)]:

The Choquet integral weights outcomes in ascending order by the additional weightattributed to the level set of an outcome. Since lower level sets contain higher levelsets, lower outcomes get a higher weight. The downward bias of the Choquet integralmodels a cautious or pessimistic attitude of the decision maker.In applications, one often wishes to compare situations where a player is confidentabout his probabilistic assessment with those where ambiguity is experienced. For thispurpose, it proves useful to have a measure of deviation of a capacity from an additiveprobability distribution. One can use the maximal difference of the weight given anevent 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¡ minAµS

[º(A) + º(SnA)]:

It is then easy to check that a degree of ambiguity of zero implies additivity, providedthe capacity is convex.


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 thelimiting 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 gamesis the notion of support for a capacity. There are many different, but equivalent, waysfor defining a support for additive probabilities. For capacities however, each of theseconcepts has a different interpretation3. Ryan (1997a) studies support concepts in greatdetail. In this paper, we apply the notion suggested by Dow and Werlang (1994) andEichberger and Kelsey (1998). The support of a capacity is the smallest event with acomplement 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, how-ever, may not be unique.

2.2 Dempster-Shafer updating

Signalling private information intends to influence the opponent�s beliefs. This raisesthe 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 tointegrate new information in the belief. A major problem arises if the informationreceived is inconsistent with the probability distribution representing the beliefs. Ifan event occurs which the decision maker believed to have zero probability, then noconsistent updating is possible. In signalling games, this problem has been recognisedas the reason for the multiplicity of equilibria.For non-additive beliefs, several updating methods are known and have been investi-gated in the literature. Gilboa and Schmeidler (1993) provide an axiomatic founda-tion for several updating rules. All of these share the property that they converge to aBayesian 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 papercan be interpreted as a maximum likelihood procedure4.

Definition 2.5 Dempster-Shafer updating rule (DS-update )For all events A µ S;

º(AjE) :=º((A \ E) [ SnE) ¡ º(SnE)

1 ¡ º(SnE):

Note that the DS-updating rule is well-defined if an event E occurs that had measurezero, º(E) = 0, provided the complement of E does not have full measure. In game-


theoretic applications with strategic uncertainty, ½ > 0, this property makes equilibriumpredictions much tighter.The following lemma shows that Bayesian updating is the limit of DS-updating.

Lemma 2.2 Let ºn be a sequence of capacities converging to an additive probability¼: Suppose that ¼(A) > 0; then the sequence of DS-updates ºn(¢jA) converges to theBayesian update ¼(¢jA):

Proof. See Eichberger and Kelsey (1997a), Proposition 2.4.

3. E-capacities

In signalling games, beliefs of a player concern strategies and types. The capacityrepresenting a player�s beliefs is therefore defined on a product space S £ T of finitesets S and T: In this context, one often wants to maintain the assumption that a player isbetter informed in regard to possible types, possibly because the proportion of types in apopulation of players is common knowledge, and that ambiguity affects the opponent�schoice of strategy. E(llsberg)-capacities5 offer a convenient way to combine ambiguityabout strategies with knowledge about types.Let Ft = S £ ftg be the set of type-strategy combinations with the same type t 2 T:Define the following capacity ºt(E) by

ºt(E) =

½1 if Ft µ E0 otherwise

:

An E-capacity with knowledge of an additive probability distribution p on T is definedas follows.

Definition 3.1 An E-capacity on S £ T compatible with the probability distributionp on T is defined by

º(E) = ¸ ¢ ¼(E) + (1 ¡ ¸)¢Xt2T

ºt(E) ¢ p(t)

where ¼ is an additive probability distribution with ¼(Ft) = 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 asit satisfies the condition on its marginal distribution ¼(Ft) = p(t) for all t 2 T: Theprobability distribution ¼ will be chosen endogenously in an equilibrium of a game.The confidence parameter ¸ is interpreted as an exogenously given degree of confi-dence in the equilibrium distribution ¼: One checks easily that the degree of ambiguityof an E-capacity equals ½ = 1 ¡ ¸.E-capacities of this type will be used extensively throughout this paper. It is thereforeuseful to record some properties of these capacities before turning to the analysis of thesignalling 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


part of the capacity.

Proposition 3.1 The Choquet integral of an E-capacity isZf dº = ¸¢

X(s;t)2S£T

¼(s; t) ¢f(s; t)+(1¡¸)¢Xt2T

p(t) ¢minff(s; t)j (s; t) 2 Ftg:

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 additiveprobability distribution on which the capacity is based,

supp º = supp¼:

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 distri-bution p on T with respect to es 2 S is

º(tjes) =¸ ¢ ¼(es; t) + (1 ¡ ¸) ¢ p(t)¸¢ P

t02T

¼(es; t0) + (1 ¡ ¸):

Proof. The proof follows from a direct application of Lemma 4.2 in Eichberger andKelsey (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 iscompatible with an additive prior distribution, the updated capacity º(¢jes) is additive.This is quite intuitive, since beliefs about strategy choice s were ambiguous, whilethere 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 2Tg) = 0 for some strategy-type pair holds.

Corollary 3.1 The DS-update of an E-capacity on es 2 S1 with º(f(es; t)j t 2 Tg) =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) = 0 and,therefore, ¼(f(es; t)g) = 0.Because of the additivity of the DS-update of an E-capacity consistent with a proba-bility distribution on types, the Choquet integral conditional on an observed signal issimply 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 theprior distribution p on T with respect to es 2 S1: The Choquet integral of the updatedcapacity º(¢jes) isZ

f dº(¢jes) =Xt2T

f(es; t) ¢ ¸ ¢ ¼(es; t) + (1 ¡ ¸) ¢ p(t)¸¢ P

t2T

¼(es; t) + (1 ¡ ¸):


Proof. See the appendix.

4. Signalling Games

Signalling games are a special case of dynamic two-player games where players6 movesequentially. Player 1, the sender, has a characteristic, a type, which is unknown to theopponent. Player 1 moves first and chooses a usually costly action, the signal. Player2 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 ofplayer 1, signalling games are two-player games with incomplete information:

Players: I = f1;2g:Strategy sets: S1 = fs1

1; :::; s1Mg; S2 = fs2

1; :::; s2Ng:

Type set of player 1: T finite.Payoff functions: u1(s1; s2; t); u2(s1; s2; 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 strategieswill depend on her private information, i.e. her type. Since player 2 observes the actionof 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�sactual mixed strategy.Following Milgrom and Weber (1986), we represent type-contingent strategies by aprobability distribution ¼1 on the strategy-type space of player 1, S1 £ T; with thefollowing constraint on the marginal distribution:X

s12S1

¼(s1; t) = p(t):

The most commonly used equilibrium concept is Perfect Bayesian Equilibrium 7.

Definition 4.1 A Perfect Bayesian Equilibrium (PBE) for the signalling game con-sists of probability distributions ¼1 on S1 £ T; ¼2(¢; s1) on S2 for all s1 2 S1; andbeliefs ¹(¢js1) on T for all s1 2 S1 such that

(i) (bs1; t) 2 supp¼1 implies bs1 2arg maxs12S1

Ps22S2

¼2(s2; s1) ¢ u1(s1; s2; t);

(ii) bs2 2 supp¼2(¢; s1) implies bs2 2arg maxs22S2

Pt2T

¹(tjs1) ¢ u2(s1; s2; t);

(iii)Pt2T

¼1(s1; t) > 0 implies ¹(tjs1) = ¼1(s1; t)ÁPt2T

¼1(s1; t):


Notice that restrictions on beliefs (iii) obtain only for those strategies s1 which areplayed with positive probability by some type t of player 1.When choosing her strategy, player 1 takes into consideration that the mixed strat-egy of player 2, ¼2(¢; s1) will depend on his signal s1: Player 2 in turn holds beliefsabout player 1�s type-contingent behaviour represented by the probability distribution¼1(s1; t): He will update these beliefs in the light of the signal that he observes accord-ing to Bayes law, ¹(¢js1):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�sbeliefs. For additive beliefs, this condition implies that beliefs coincide with the mixedstrategies 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 longermaintain the equality of actual behaviour and beliefs. Dow and Werlang (1994) sug-gest an equilibrium concept for two-player games which requires consistency of actualbehaviour with beliefs in the sense that the strategies in the support of a player�s beliefsare best-responses of the opponent8. In contrast to additive beliefs, however, the con-cept of support is no longer obvious, and the equilibrium condition does not imply thatequilibrium behaviour coincides with equilibrium beliefs.This concept has to be adapted in order to take into account the dynamic structure of asignalling game. In Eichberger and Kelsey (1997a), an equilibrium concept based onDS-updating has been suggested and studied in detail.

Definition 4.2 A Dempster-Shafer Equilibrium (DSE) consists of capacities º1 onS1 £ T and º2(¢; s1) on S2 for all s1 2 S1 such that

(i) (bs1; t) 2 supp º1 implies bs1 2argmaxs12S1

Ru1(s1; s2; t) dº2(s2; s1);

(ii) bs2 2 supp º2(¢; s1) implies bs2 2argmaxs22S2

Ru2(s1; s2; t) d¹DS(tjs1);

where ¹DS(tjs1) denotes the DS-update of º1 conditional on s1:

The capacity º1 and its update in response to signal ¹DS(¢js1) represent the beliefsof player 2 about the strategy-type pair of player 1, before and after the signal s1 isobserved. The capacity º2(¢; s1), on the other hand, is the belief of player 1 aboutplayer 2�s behaviour which she expects in response to her strategy choice s1:A DSE is a straightforward adaptation of the PBE concept to games where players facestrategic uncertainty in addition to incomplete information. The following existenceresult 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; thereexists a DSE which is compatible with p and where players have degrees of ambiguity½1; ½2 ¸ ®:


In Eichberger and Kelsey (1997a), we show in fact a slightly more general result. Fur-thermore; we explore there the relationship between DSE and the traditional equilib-rium concepts. One can show that an appropriately defined limit of a sequence of DSEequilibria, where beliefs become additive in the limit, is not necessarily a PBE. More-over, there are PBE which cannot be obtained as an additive limit of a sequence ofDSE.Because DS-updates are well-defined even if an event occurs that was given zeroweight 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 updatesare generated by DS-updating. If players face strategic uncertainty, DS-updates are de-fined even if a capacity gives zero weight to an event. Taking away the arbitrariness ofbeliefs about out-of-equilibrium play is in our opinion a major advantage of DSE overPBE. Whether the behaviour in a DSE appears a sensible description of actual behav-iour has to be studied in specific applications. Applying the equilibrium notion of aDSE equilibrium to the education-signalling game in the next section and comparingthe results to the traditional analysis may provide such a test9.Traditionally, PBE of signalling games have been classified as separating equilibria,pooling equilibria, or hybrid equilibria. A separating equilibrium is a PBE in whichall types of players choose different actions. Player 2 can therefore identify the typeof player 1 by observing her action. In a pooling equilibrium, all types of player 1choose the same action. Player 2 receives therefore no signal which would allow himto distinguish player 1�s type. Many PBE, however, do not fall in either of these twoclasses, i.e. some types may be discerned by their choice of action while others remainindistinguishable.For DSE, we adapt these concepts as follows. Denote by ¾t the set of strategies ofplayer 1 in the support of the capacity º1;

¾t(º1) := fs1 2 S1j (s1; t) 2 supp º1g;

and consider the following definition.

Definition 4.3 A DSE (º1; (º2(¢; s1))s12S1) is called(i) separating equilibrium if

¾t(º1) \ ¾t0(º1) = ; for all t; t0 2 T;

(ii) pooling equilibrium if

¾t(º1) = ¾t0(º1) for all t; t0 2 T:

5. Education Signalling

In this section, we present a model based on the labour market signalling model ofSpence (1973). Tirole (1988) has adapted the Spence model to make it conform to thestructure of a signalling game. To simplify exposition, we have further modified themodel by restricting attention to finite strategy sets.


Consider firms which intend to hire a worker. There is a large pool of workers with dif-fering productivities. Workers know their productivity, while firms cannot observe theproductivity of job applicants directly. What the firm can confirm however is the edu-cation level of a worker. If the education level is positively correlated with a worker�sproductivity, 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 2W: Assume that E = f0; 1; 2; :::;Eg and W = f0; 1;2; :::;Wg are finite sets. Thepayoff of a worker depends on her productivity Át which can be either high, ÁH ; orlow, ÁL; 0 < ÁL < ÁH ; and takes the following form:

u(e;w;Át) := w ¡ e

Át

for t = H; L: There is a large population of workers with a proportion p of high-productivity workers. Since workers are ex-ante identical, p is also the probability ofmeeting a high-productivity worker.

Firm: The firm is the potential employer. Its payoff function does not depend onthe level of education that a worker achieves. The productivity of a worker mattershowever. For simplicity, assume the following payoff function for the firm if it hires aworker of productivity type Át :

v(e; w;Át) = Át ¡ w

for t = H;L:

Workers are assumed to move first. They apply for a job with the firm based on aneducation level 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 thedecision tree for a representative proposal (e; w) 2 E £ W:

w ¡ eÁH

w ¡ eÁL

.....................................

q

q

q

q

q

qq

q

a0

H

L

F

F

p

1 ¡ p

(e; w)

(e; w)

a

r

a

r

00

00

ÁH ¡ w

ÁL ¡ w

..

Figure 1: Decision tree


5.1 Conventional analysis

The following two classes of pure-strategy equilibria are usually discussed in conven-tional analysis. To simplify notation, denote by ¼H(e; w); ¼L(e; w) the mixed strate-gies chosen by a worker of type H and L respectively. In terms of the notation insection 4,

¼W ((e; w);H) ´ p ¢ ¼H(e;w) and ¼W ((e; w); L) ´ (1 ¡ p) ¢ ¼L(e; w):

We present these equilibria in a more formal way than is usually done in textbooks, inorder to make similarities and differences to the DSE more transparent.

Proposition 5.1 (pooling equilibrium) Let (e¤;w¤) satisfy the following conditions:

w¤ ¡ e¤

ÁL

¸ ÁL and p ¢ ÁH + (1 ¡ p) ¢ ÁL ¸ w¤ ¸ ÁL:

Then the following strategies and beliefs form a PBE:

(i) ¼H(e¤; w¤) = ¼L(e¤;w¤) = 1;

(ii) ¼F (a; (e;w)) =

½1 for (e;w) = (e¤; w¤) or w · ÁL

0 otherwise;

(iii) ¹(Hj(e; w)) =

½p for (e; w) = (e¤;w¤)0 otherwise

:

Proof. See the appendix.There is a multiplicity of pooling PBE. Figure 2 illustrates the range of education-wagepairs that could be supported as pooling equilibria. All (e;w)-combinations in regionP are pooling equilibria.

e0

w

.....................................................

©©©©

©©©©

©©©©

©©

ÁH

ÁL

EpÁ

w ¡ eÁL

w ¡ eÁL

P

S

Figure 2: Perfect Bayesian Equilibria


In a pooling equilibrium, education fails as a signal for the firm. Productivity typescannot be distinguished and the employer will accept only wage claims below or equalto the average productivity level p ¢ ÁH + (1 ¡ p) ¢ ÁL: For a low-productivity typesuch a wage is optimal as long as it does not fall below her productivity level. A high-productivity worker who would benefit from signalling her type cannot do so, becauseany out of equilibrium education-wage pair will be interpreted as a signal of a low-productivity type.Notice the importance of out-of-equilibrium beliefs for this set of equilibria. We willdemonstrate below with Example 5.2 that the set of equilibrium education-wage pairswould be substantially altered if an out-of-equilibrium (e;w)-pairs were not interpretedas indicating a low-productivity type, ¹(Hj(e; w)) = 0.Pooling equilibria can be ordered in the Pareto sense. From Figure 2, the Pareto-dominant equilibrium is easily identified as (e¤;w¤) = (0; p ¢ ÁH + (1 ¡ p) ¢ ÁL):

Separating equilibria form a second class of PBE.

Proposition 5.2 (separating equilibrium) Let (e¤H ;w¤

H) and (e¤L; w¤

L) satisfy thefollowing conditions:

w¤H ¡ e¤

H

ÁH

¸ w¤L = ÁL ¸ w¤

H ¡ e¤H

ÁL

; e¤L = 0 and w¤

H · ÁH :

The following strategies and beliefs form a PBE

(i) ¼H(e¤H ; w¤

H) = ¼L(0; ÁL) = 1;

(ii) ¼F (a; (e;w)) =

½1 for (e; w) 2 f(e¤

H ; w¤H); (0; ÁL)g or w · ÁL

0 otherwise;

(iii) ¹(Hj(e;w)) =

½1 for (e;w) = (e¤

H ; w¤H)

0 otherwise:

Proof. See the appendix.

The multiplicity of separating equilibria is illustrated by the (e; w)-combinations inregion S of Figure 2. All these equilibria are supported by out-of-equilibrium be-liefs which attribute any non-equilibrium education-wage pair to the low-productivityworker. In many of these equilibria the high-productivity worker is clearly identifiedby the firm but does not obtain a wage equal to her marginal productivity. A higherwage claim of the high-productivity worker would be rejected by the firm because itwould read it as a signal of the low-productivity worker.As in the case of the pooling equilibria, one can Pareto-order these equilibria. Sincea low-productivity worker obtains the same wage in every equilibrium, one can Pareto-rank the separating equilibria in terms of the preferences of the high-productivity worker.The Pareto-dominant separating equilibrium is (eH ;wH) = (ÁL ¢(ÁH ¡ÁL); ÁH): Thehigh-productivity worker receives a wage equal to her marginal product but has to ed-ucate herself up to the level ÁL ¢ (ÁH ¡ ÁL):There are many more equilibria in mixed strategies as the following example illustrates.


Example 5.1 Suppose ÁH = 5 and ÁL = 1, and p = 12 : The following mixed strate-

gies form a PBE (pooling equilibrium):

(i) ¼H(e;w) = ¼L(e;w) =

½12 for (e; w) = (0;4)12 for (e; w) = (0;2)

;

(ii) ¼F (a; (e; w)) =

8>><>>:13 for (e;w) = (0; 4)23 for (e;w) = (0; 2)0 for 1 · w 6= 2; 41 for w < 1

;

(iii) ¹(Hj(e;w)) =

½1 for (e; w) 2 f(0; 4); (0;2)g0 otherwise

:

It is straightforward to check that the worker is indifferent between choosing 2 or 4 andthe 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-productivityworker, provided the (e;w)-pair suggested is less attractive for the low-productivityworker than the marginal-product wage combined with a zero level of education. Basedon such reasoning, many refinements of PBE have been suggested in the literature10.Criteria for eliminating out-of-equilibrium beliefs make usually reference to the equi-librium outcome. Forward induction arguments assume that types of workers whocould not possibly gain from a deviation, compared to what they get in equilibrium, areto be assigned probability zero. The most commonly used criterion in the education-signalling context is the intuitive criterion which selects the Pareto-optimal separatingPBE:

(e¤H ;w¤

H) = (ÁH ¢ (ÁH ¡ ÁL); ÁH); (e¤L;w¤

L) = (0; ÁL):

Many refinements are driven by an effort to justify the Pareto optimal equilibrium in thesignalling game. Unfortunately, no refinement known today guarantees selection of thePareto-optimal PBE in every signalling game. The intuitive criterion selects the Pareto-optimal PBE in the signalling game if workers may have two types of productivity, butfails if there are three possible productivity levels.From the employer�s viewpoint, signals appear ambiguous. Arguments about whata firm should conclude from an out-of-equilibrium education-wage offer are highlyspeculative and require an extreme degree of coordination in beliefs between workerand firm. The only firm knowledge of an employer is the prior distribution of typesand the repeated observation of the equilibrium education-wage pairs. If an unknownworker offers new (e;w)-combination, a reversion to his prior beliefs appears to be areasonable reaction of the employer.The following example illustrates how this assumption about out-of-equilibrium beliefsrestricts the set of PBE.

Example 5.2 Let ÁH = 5 and ÁL = 1, and p = 12: Suppose that firms take an

observed deviation from equilibrium play as evidence that their reasoning about the


workers� behaviour has failed. In this case, all they know is the fact that the proportionof the high-productivity types in the population is p = 1

2: Based on this reasoning,

there is a unique pooling PBE of the education signalling game with the followingequilibrium strategies:

(i) ¼H(0; 3) = ¼L(0;3) = 1;

(ii) ¼F (a; (e;w)) =

½1 for w · 30 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 equilib-rium strategies is (0;3): Bayesian updating yields ¹(Hj(e;w)) = p in this case. Notehowever that, by assumption, the firm also expects to meet a high-productivity workerwith probability p if some out-of-equilibrium signal (e; w) 6= (0; 3) is observed. Giventheses beliefs, it is clearly optimal for the firm to accept any offer with a wage rate lessthan or equal to the average productivity of 3: Hence, behaviour described in (ii) isoptimal. Finally, a worker with low productivity, cannot gain by making a wage claimabove 3 (with or without extra educational qualifications) because the firm will notaccept such a claim. Nor would a high-productivity worker be able to extract a higherwage by obtaining higher education because the firm would take such a deviation asan indication that the equilibrium reasoning has failed and reject any wage except theaverage wage rate of 3:The out-of-equilibrium beliefs which are not endogenously determined in a PBE largelydetermine equilibrium behaviour. If a firm takes deviation from a separating equilib-rium as a reason for doubts about the equilibrium prediction, then no separating equi-librium can exist. In any separating PBE, the low-productivity worker will obtain atbest (e¤

L; w¤L) = (0;1). By deviating to any non-equilibrium education-wage pair with

a higher wage, say (e0; w0) = (0;2); the worker could secure this higher wage. Sincethe firm would conclude that the separating equilibrium hypothesis is false and revertto the belief ¹(Hj(e0;w0)) = p; the expected payoff of the firm from hiring the workerwould be 3 which makes it optimal to accept the offer. Thus, a low-productivity workerpredicting this acceptance, has an incentive to deviate from the separating equilibriumstrategy.

5.2 Dempster-Shafer equilibria

Conventional analysis of the education signalling model reveals that PBE predictionsare driven by assumptions about the interpretation of out-of-equilibrium beliefs. Purerationality, i.e., optimisation of agents plus rational expectations, hardly restricts theequilibrium outcomes. With strategic uncertainty, i.e., some ambiguity about the equi-librium behaviour of the opponent, modelled by CEU this changes dramatically.Choquet expected utility theory has well-researched decision-theoretic foundations11.If ambiguity is modelled by E-capacities, out-of-equilibrium beliefs are defined pro-vided there is some positive degree of ambiguity, ½ > 0: Hence, DSE makes a clearprediction about the equilibrium outcome in the education-signalling game.


Proposition 5.3 Suppose that(i) a worker�s beliefs are characterised by a simple capacity with constant con�denceparameter ¸W 2 (0;1) and that(ii) the employer�s beliefs are represented by an E-capacity compatible with a prioradditive probability distribution p on T with p(t) > 0 for all t 2 T and a confidenceparameter ¸F 2 (0;1);then the unique DSE is the Pareto-efficient pooling equilibrium which satisfies

supp ºW = f(0;EpÁ;H); (0;EpÁ;L)gwith EpÁ := p ¢ ÁH + (1 ¡ p) ¢ ÁL:

Proof. See the appendix.The logic of this result is easy to explain. Given strategic uncertainty represented byE-capacities with a strictly positive degree of ambiguity, the additive part of the capac-ities will be determined in equilibrium. Workers will always claim the highest wagewhich they expect the employer to accept. Knowing that their education-wage pair willsignal productivity to the employer, workers will use their private information strate-gically. The employer updates his beliefs according to the DS-rule. Being uncertainabout the equilibrium strategy of the worker, by Corollary 3.1, any out-of-equilibriumeducation-wage pair will lead the employer to fall back on his prior beliefs. A low-productivity worker can therefore scramble any signal which the high-productivityworker 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 em-ployer will accept such a proposal whether it is an equilibrium strategy or an out-of-equilibrium move.The result of Proposition 5.3 depends on the fact that the degree of confidence of theworker ¸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 whetherthe firm will accept or reject her wage claim is contingent on her education-wage sig-nal. In Proposition 5.3, only the endogenously determined additive part of the capacity¼(¢; (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 increaseswith a falling wage claim. The following example shows that other pooling equilibriamay occur if the degree of confidence varies with the signal (e;w).

Example 5.3 Let ÁH = 5 and ÁL = 1, and p = 12 : We claim that the following beliefs

form a DSE:ºW is an E-capacity with prior distribution p and confidence parameter ¸F defined bythe probability distribution

¼W (e; w; t) =

8<:14 for (0; 1; H) and (0;1; L)14 for (0; 3; H) and (0;3; L)0 otherwise

:


ºF (¢; (e; w) is a capacity defined by

ºF (¢; (e; w) := ¸W (e;w) ¢ ¼F (a; (e; w));

with

¼F (a; (e;w)) =

½1 for w · 30 for w ¸ 3

and

¸W (e;w) =

8<:14 for (e; w) = (0;3)34 for (e; w) = (0;1)15 otherwise

:

By Definition 3.2,

supp ºF = f(0;1;H); (0; 3; L); (0; 1; H); (0;3; L)g;

supp ºW (¢; (e;w)) =

8<: fag for w < 3fa; rg for w = 3frg for w > 3

:

By Proposition 3.4, for t = H;L; one obtainsR[w ¡ e

Át] ¢ 1a(sF ) dºF (sF ; (e; w))

=

8>>>><>>>>:34

for (e;w) = (0;3)34

for (e;w) = (0;1)

15

¢ (w ¡ eÁt

) + 45

¢ minf0;w ¡ eÁt

g for (e;w) with½

w · 3;(e; w) 6= (0;1); (0;3)

0 for (e;w) with w > 3

:

Clearly, argmax(e;w)2E£W

R[w ¡ e

Át] ¢ 1a(s

F ) dºF (sF ; (e; w)) = f(0;1); (0; 3)g for each

type of worker.Now, consider the DS-updates. By Proposition 3.3, one easily checks that

¹DS(Hj(e;w)) =1

2;

for all (e;w) 2 E £ W: Hence, by Proposition 3.4, if the firm accepts the offer, a; itspayoff will be Z

[Át ¡ w] d¹DS(tj(e;w)) = [3 ¡ w];

and, for a rejection r; it obtains a payoff of 0: Hence, the firm�s best responses are

argmax(e;w)2E£W

Z[Át ¡ w] d¹DS(tj(e; w)) =

8<: fag for w < 3fa; rg for w = 3frg for w > 3

:

This shows that the proposed beliefs form a DSE.


6. Concluding Remarks

In signalling games, optimality of a receiver�s behaviour depends more on how theplayer interprets a signal which is not supposed to have been sent according to theequilibrium play. No equilibrium consistency requirement will restrict these beliefs.Multiplicity of equilibria and uncertainty about the behaviour of players is the conse-quence.Strategic uncertainty has the potential to reduce the indeterminateness of strategic equi-libria substantially. Modelled by CEU, the decision maker reserves some weight foroutcomes other than those predicted in equilibrium. Thus, Dempster-Shafer equilibriahave 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 con-cepts to game-theoretic analysis raises issues of the appropriate degree of consistencyin 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. DSEwith beliefs modelled by E-capacities lead to unique equilibrium behaviour and out-comes if the degree of ambiguity is positive. Sophisticated arguments about out-of-equilibrium beliefs based on forward induction principles can be replaced by assump-tions about the degree of ambiguity. The new approach offers an opportunity for abetter descriptive analysis of signalling games.

APPENDIX

Proof of Proposition 3.4: In Eichberger and Kelsey (1997b) (Lemma 4.1), it is shownthat, for a subset E µ T;

º(Ejes) = b̧¢Xt2E

b¼(es; t) + (1 ¡ b̧)¢Xt2E

bp(t) (i)

for appropriately chosen b̧; b¼; and bp: Furthermore, in Proposition 3.3,

º(tjes) =¸ ¢ ¼(es; t) + (1 ¡ ¸) ¢ p(t)¸¢ P

t02T

¼(es; t0) + (1 ¡ ¸): (ii)

Hence, applying first 3.1 to equation (i) and, after some simple manipulations, substi-tuting equation (ii), one obtains:Z

f dº(¢jes)= b̧¢

Xt2T

b¼(es; t) ¢ f(es; t) + (1 ¡ b̧)¢Xt2T

bp(t) ¢ f(es; t)=

Xt2T

f(es; t) ¢ [b̧ ¢ b¼(es; t) + (1 ¡ b̧) ¢ bp(t)]


=Xt2T

f(es; t) ¢ º(tjes)=

Xt2T

f(es; t) ¢ ¸ ¢ ¼(es; t) + (1 ¡ ¸) ¢ p(t)¸¢ P

t02T

¼(es; t0) + (1 ¡ ¸):

Proof of Proposition 5.1: First note that ¹ is a probability distribution on the typespace T = fH;Lg: Furthermore, (i) and (ii) imply that only (e¤; w¤) will be observedin equilibrium. By Bayesian updating,

¹(Hj(e¤;w¤)) =p ¢ ¼H(e¤;w¤)

p ¢ ¼H(e¤;w¤) + (1 ¡ p) ¢ ¼L(e¤; w¤)= p:

(i) supp¼H = supp¼L = f(e¤;w¤)g: The following estimation shows that (e¤; w¤)is a maximiser for both types of the worker:

¼F (a; (e¤;w¤)) ¢ [w¤ ¡ e¤

ÁH

] + ¼F (r; (e¤;w¤)) ¢ 0

= [w¤ ¡ e¤

ÁH

]

¸ [w¤ ¡ e¤

ÁL

]

¸ ÁL

¸ ¼F (a; (e;w)) ¢ [w ¡ e

ÁH

] + ¼F (r; (e;w)) ¢ 0

=

½0 for w > ÁL

w ¡ eÁt

for w · ÁL:

(ii) supp¼F (¢; (e;w)) =

½ fag for (e¤; w¤) or w · ÁL

frg otherwise: To show that this be-

haviour is optimal we note:a) For (e; w) = (e¤; w¤); we have

¹(Hj(e¤; w¤)) ¢ [ÁH ¡ w¤] + ¹(Lj(e¤; w¤)) ¢ [ÁL ¡ w¤]= p ¢ ÁH + (1 ¡ p) ¢ ÁL ¡ w¤ ¸ 0:

Hence, a is optimal.b) For (e; w) with w · ÁL; we obtain

¹(Hj(e;w)) ¢ [ÁH ¡ w] + ¹(Lj(e; w)) ¢ [ÁL ¡ w]

= ÁL ¡ w ¸ 0:

Again, a is optimal.Finally, for any other (e; w), one has

¹(Hj(e;w)) ¢ [ÁH ¡ w] + ¹(Lj(e; w)) ¢ [ÁL ¡ w]

= ÁL ¡ w < 0:

This implies r is optimal.


Proof of Proposition 5.2: First note that ¹ is an additive probability on the type spaceT = fH;Lg: Furthermore, (i) and (ii) imply that only (e¤

H ;w¤H) or (0; ÁL) will be

observed in equilibrium. By Bayesian updating,

¹(Hj(e;w)) =p ¢ ¼H(e;w)

p ¢ ¼H(e;w) + (1 ¡ p) ¢ ¼L(e; w)=

½1 for (e;w) = (e¤

H ;w¤H)

0 for (e;w) = (0; ÁL):

(i) supp¼H = f(e¤H ;w¤

H)g: The following estimation shows that (e¤H ; w¤

H) is a max-imiser:

¼F (a; (e¤H ;w¤

H)) ¢ [w¤H ¡ e¤

H

ÁH

] + ¼F (r; (e¤H ;w¤

H)) ¢ 0

= [w¤H ¡ e¤

H

ÁH

]

¸ ÁL

¸ ¼F (a; (e; w)) ¢ [w ¡ e

ÁH

] + ¼F (r; (e; w)) ¢ 0

=

½0 for w > ÁL

w ¡ eÁH

for w · ÁL:

Similarly, supp¼L = f(0; ÁL)g: The following estimation shows that (0; ÁL) is amaximiser:

¼F (a; (0; ÁL)) ¢ ÁL + ¼F (r; (0; ÁL)) ¢ 0= ÁL

¸ ¼F (a; (e; w)) ¢ [w ¡ e

ÁH

] + ¼F (r; (e; w)) ¢ 0

=

8><>:w¤

H ¡ e¤H

ÁLfor (e; w) = (e¤

H ;w¤H)

0 for (e; w) 6= (e¤H ;w¤

H) and w > ÁL

w ¡ eÁL

for (e; w) 6= (e¤H ;w¤

H) and w · ÁL

:

(ii) supp¼F (¢; (e;w)) =

½ fag for (e; w) 2 f(e¤H ; w¤

H); (0; ÁL)g or w · ÁL

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)) ¢ [ÁH ¡ w¤H ] + ¹(Lj(e¤

H ;w¤H)) ¢ [ÁL ¡ w¤

H ]

= ÁH ¡ w¤H ¸ 0:

Hence, a is optimal.b) For (e; w) = (0; ÁL); we have

¹(Hj(0; ÁL)) ¢ [ÁH ¡ ÁL] + ¹(Lj(0; ÁL)) ¢ [ÁL ¡ ÁL]

= ÁL ¡ ÁL ¸ 0:

Hence, a is optimal.c) For (e; w) with w · ÁL; we obtain

¹(Hj(e;w)) ¢ [ÁH ¡ w] + ¹(Lj(e; w)) ¢ [ÁL ¡ w]


= ÁL ¡ w ¸ 0:

Again, a is optimal.Finally, for any other (e; w), one has

¹(Hj(e;w)) ¢ [ÁH ¡ w] + ¹(Lj(e; w)) ¢ [ÁL ¡ w]

= ÁL ¡ w < 0:

This implies r is optimal.

Proof of Proposition 5.3: It is straightforward to compute the payoff functions of theplayers.Worker:

V W (e; w; t) :=

Z[w ¡ e

Át

] ¢ 1a(sF ) dºF (sF ; (e;w))

= ¸W ¢ [w ¡ e

Át

] ¢ ¼F (a; (e; w)) + (1 ¡ ¸W ) ¢ minf0;w ¡ e

Át

g:Firm:

V F (aj(e;w)) :=

Z[Át ¡ w] d¹DS(tj(e; w))

= [ÁH ¡ w] ¢ ¹DS(Hj(e;w)) + [ÁL ¡ w] ¢ ¹DS(Lj(e;w))

= ÁL + (ÁH ¡ ÁL) ¢ ¹DS(Hj(e;w)) ¡ w

= ÁL + (ÁH ¡ ÁL) ¢ ¸F ¢ ¼W (e;w; H) + (1 ¡ ¸F ) ¢ p¸F ¢ [¼W (e;w;H) + ¼W (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 showthat, in a DSE, there is a strategy (e;w) for each type with positive probability in theprior distribution.Lemma 5.3.1: For all t 2 T; there exists (e;w) 2 E £ W such that

((e;w); t) 2 supp ºW :

Proof. Suppose there exists t 2 T such that ((e;w); t) =2 supp ºW for all (e;w) 2E £W: Then ºW (E £W £ftg) = 0; since E £W £ftg µ (E £W £T )n supp ºW

and monotonicity imply

0 · ºW (E £ W £ ftg) · ºW ((E £ W £ T )n supp ºW ) = 0:

Hence, 0 = º1(E £ W £ ftg) = p(t) > 0 a contradiction..It follows that ¾H(ºW ) 6= ; and ¾L(ºW ) 6= ;:Lemma 5.3.2: There is at most one (e;w) which both types of worker play in equilib-rium,

f(e; w)g = ¾H(ºW ) \ ¾L(ºW ):Proof. Suppose there are (e;w); (e0;w0) 2 ¾H(ºW ) \ ¾L(ºW ): By condition (i) of aDSE (Definition 4.2),

w ¡ e

ÁH

= w0 ¡ e0

ÁH

and


w ¡ e

ÁL

= w0 ¡ e0

ÁL

must be true. This contradicts the assumption ÁH > ÁL:Lemma 5.3.3: For all ¸F ; ¸W 2 (0;1]; a DSE is a pooling equilibrium,

f(e;w)g = ¾H(ºW ) = ¾L(ºW ):

Proof. LetÁ(e;w) = ÁL + (ÁH ¡ ÁL) ¢ ¹DS(Hj(e;w))

be the expected productivity if the education-wage pair (e;w) is observed. The firmwill accept any wage w · Á(e;w): Therefore, w = Á(e;w) must hold for any (e; w) 2¾H(ºW ) [ ¾L(ºW ):Suppose now the lemma is false. Then there exists either (eL; wL) 2 ¾L(ºW )=¾H(ºW )or (eH ;wH) 2 ¾H(ºW )=¾L(ºW ).Case (i): (eL;wL) 2 ¾L(ºW )=¾H(ºW ) : Hence,

¼W (eL;wL; H) = 0 and ¼W (eL;wL; L) > 0:

This implies ¹DS(Hj(eL; wL)) < p and wL = Á(eL; wL) < Á(0;EpÁ) = EpÁ: Fora low-productivity worker,

V W (eL;wL; L) := ¸W ¢ [wL ¡ eL

ÁL

] < ¸W ¢ EpÁ =: V W (0; EpÁ; L);

the proposal (0;EpÁ); which will be accepted, yields a strictly higher payoff than(eL; wL): This contradicts the equilibrium requirement (eL; wL) 2 arg maxV W (e;w;L):Note that case (i) implies that there cannot be a separating DSE, i.e., ¾H(ºW )\¾L(ºW ) 6=;:Case (ii): (eH ;wH) 2 ¾H(ºW )=¾L(ºW ) and f(e; w)g = ¾H \ ¾L: It follows from

¼W (eH ; wH ; L) = 0 and ¼W (eH ;wH ;H) > 0;¼W (e;w;L) = 1 ¡ p and ¼W (e; w;H) > 0;

and

p = ¼W (E £ W £ fHg)= ¼W (e;w;H) + ¼W (eH ;wH ;H) + ¼W ((E £ W )nf(e;w); (eH ;wH)g £ fHg)

thatp > ¼W (e; w; H):

Hence, ¹DS(Hj(e;w)) < p and Á(e;w) < Á(0; EpÁ) = EpÁ: Thus, the firm willonly accept wages w · Á(e;w): By deviating to the proposal (0;EpÁ); which will beaccepted by the firm, a low-productivity worker can obtain a payoff of ¸W ¢ EpÁ >

¸W ¢ [w ¡ eÁL

]. This proves that beliefs with ¼W (e;w;H) < p cannot be optimal.

The following two lemmata establish that f(0; EpÁ)g = ¾H(ºW ) = ¾L(ºW ):Lemma 5.3.4: In a pooling DSE, e = 0:Proof. Suppose there is a pooling DSE with e > 0: By condition (i) of a DSE (Defini-tion 4.2),

w ¡ e

Át

¸ w ¡ eeÁt

for all (ee; w) which will be accepted by the firm. Since the DS-update for an out-of-


equilibrium event equals p;

¹DS(Hj(e; w)) = p = ¹DS(Hj(0; w))

and V F (aj(e;w)) = V F (aj(0;w) follow. Yet, for ee = 0; w > w ¡ eÁt

contradictingthe optimality of (e; w) with e > 0:Lemma 5.3.5: In a pooling DSE, w = EpÁ:Proof. By the same argument as in the previous lemma, one has

¹DS(Hj(0; w)) = p = ¹DS(Hj(0;EpÁ))

and V F (aj(0;EpÁ) ¸ 0 = V F (rj(0; EpÁ): Hence, w < EpÁ cannot be optimal fora worker of either type. On the other hand, for w > EpÁ; the firm will reject theproposal, 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 thesupport notion of an additive probability distribution is the fact that the outcome on an eventwhich is not contained in the support may still alter the Choquet integral and, thus, influencebehaviour.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 s1 from beliefs conditionalon the signal s1; we write ¹(¢js1) for the update, and ¼(¢; s1) for the conditional beliefs.8. This equilibrium concept is discussed and compared with alternative approaches in Eichbergerand Kelsey (1998).9. Eichberger and Kelsey (1997a) and Ryan (1997b) contain further applications of the DSEconcept.10. Mailath (1992) provides a survey and discussion of many refinements suggested in the lit-erature for signalling games.11. Schmeidler (1989), Gilboa (1987), and Sarin and Wakker (1992) provide axiomatic founda-tions for decision making with ambiguity.


References

Dow, J., Werlang, S.R.d.C. (1994). ��Nash Equilibrium under Knightian Uncertainty: BreakingDown Backward Induction��. Journal of Economic Theory 64, 305-324.

Eichberger, J., Kelsey, D. (1998). ��Non-Additive Beliefs and Strategic Equilibria��. Mimeo. Uni-versity of Saarland, Saarbrücken.

Eichberger, J., Kelsey, D. (1997a). ��Signalling Games with Uncertainty��. Discussion Paper No.95-13. Department of Economics. University of Birmingham.

Eichberger, J., Kelsey, D. (1997b). ��E-Capacities and the Ellsberg Paradox��. Theory and Deci-sion, forthcoming.

Ellsberg, D. (1961). ��Risk, Ambiguity and the Savage Axioms��. Quarterly Journal of Economics75, 643-669.

Gilboa, I. (1987). ��Expected Utility Theory with Purely Subjective Probabilities��. Journal ofMathematical Economics 16, 65-88.

Gilboa, I., Schmeidler, D. (1993). ��Updating Ambiguous Beliefs��. Journal of Economic Theory59, 33-49.

Mailath, G. (1992). ��Signalling Games��. In Creedy, J., Borland, J. and Eichberger, J. (1992).Recent Developments in Game Theory. Aldershot: Edward Elgar, 65-93.

Milgrom, P., Weber, R. (1986). ��Distributional Strategies for Games with Incomplete Informa-tion��. Mathematics of Operations Research 10, 619-631.

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 ofAuckland, NZ.

Sarin, R., Wakker, P. (1992). ��A Simple Axiomatization of Non-Additive Expected Utility��.Econometrica 60, 1255-1272.

Schmeidler, D. (1989). ��Subjective Probability and Expected Utility without Additivity��. Econo-metrica 57, 571-587.

Spence, M. (1973). ��Job Market Signalling��. Quaterly Journal of Economics 87, 355-374.

Tirole, J. (1988). The Theory of Industrial Organisation. Cambridge, Mass.: MIT Press.

Volkswirtschaftliche Reihe/Economic Series

Prof. Dr. Hermann ALBECK Nationalökonomie, insbesondere Wirtschafts-und Sozialpolitikhttp://www.uni-sb.de/rewi/fb2/albeck/

Prof. Dr. Jürgen EICHBERGER Nationalökonomie, insbesondere Wirtschaftstheoriehttp://www.uni-sb.de/rewi/fb2/eichberger/

Prof. Dr. Ralph FRIEDMANN Statistik und Ökonometriehttp://www.wiwi.uni-sb.de/friedmann/

Prof. Dr. Robert HOLZMANN Nationalökonomie, insbesondere Internationale Wirtschaftsbeziehungenhttp://www.wiwi.uni-sb.de/lst/iwb/

PD Dr. Udo BROLL http://www.wiwi.uni-sb.de/lst/iwb/

Prof. Dr. Christian KEUSCHNIGG Nationalökonomie, insbesondere Finanzwissenschafthttp://www.wiwi.uni-sb.de/fiwi/

Prof. Dr. Dieter SCHMIDTCHEN Nationalökonomie, insbesondere Wirtschaftspolitikhttp://www.uni-sb.de/rewi/fb2/schmidtchen/

Prof. Dr. Volker STEINMETZ Statistik und Ökonometriehttp://www.uni-sb.de/rewi/fb2/steinmetz/

9901 Christian KEUSCHNIGG Rationalization and Specialization in Start-upFinanzwissenschaft Investment

9902 Jürgen EICHBERGER Non-Additive Beliefs and Strategic EquilibriaDavid KELSEYEconomic Theory

9903 Jürgen EICHBERGER Education Signalling and UncertaintyDavid KELSEYEconomic 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 GovernmentsBeatrice WEDERInternationale Wirtschaftsbeziehungen

9906 Ralph FRIEDMANN Effects Of The Order Of Entry On Market Share,Markus GLASER Trial Penetration, And Repeat Purchases: EmpiricalStatistik und Ökonometrie Evidence Or Statistical Artefact ?

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Abstract - uni-saarland.de · 2013-02-13 · means that hiring is an investment decision. The fact that these capabilities are not known beforehand makes the decision one under uncertainty