Introduction to Game Theory Part 3. Dynamic games of incomplete information Chapter 3. Job Market Signaling Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 1 / 57
Job market signaling We restate Spence s (QJE 1973) model as an extensive-form game and describe some of its perfect Bayesian equilibria The timing is as follows 1 Nature determines a worker s productive ability, η, which can be either high H or low L. The probability that η = H is q 2 The worker learns his or her ability and then chooses a level of education, e 0 3 Two firms observe the worker s education but not the worker s ability, and then simultaneously make wage offers to the worker 4 The worker accepts the higher of the two wage offers, flipping a coin in case of a tie V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 2 / 57
Payoffs Let w denote the wage the worker accepts The payoff to worker is w c(η, e) where c(η, e) is the cost to a worker with ability η of obtaining education e The payoff to the firm that employs the worker is y(η, e) w where y(η, e) is the output of a worker with ability η who has obtained education e The payoff to the firm that does not employ the worker is zero V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 3 / 57
Assumption on production We allow for the possibility that output increases not only with ability but also with education We assume that high-ability workers are more productive, i.e., e, y(h, e) > y(l, e) We assume that education does not reduce productivity, i.e., (η, e), y e (η, e) 0 where y e (η, e) is the marginal productivity of education for a worker of ability η at education e y e (η, e) = y (η, e) 0 e V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 4 / 57
Interpretation of education We interpret differences in e as differences in the quality of a student s performance Not as differences in the duration of the student s schooling Thus, the game could apply to a cohort of high school graduates, or to a cohort of college graduates or MBAs Under this interpretation, e measures the number and kind of courses taken and the caliber of grades and distinctions earned during an academic program of fixed length V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 5 / 57
Interpretation of education Tuition costs (if they exist at all) are independent of e, so the cost function c(η, e) measures non-monetary (or psychic) costs Students of lower ability find it more difficult to achieve high grades at a given school, and also more difficult to achieve the same grades at a more competitive school Firm s use of education as a signal thus reflects the fact that firms hire and pay more to the best graduates of a given school and to the graduates of the best schools V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 6 / 57
Assumption on costs The crucial assumption in Spence s model is that low-ability workers find signaling more costly than do high-ability workers More precisely, we assume that the marginal cost of education is higher for low-ability than for high-ability workers e, c e (L, e) > c e (H, e) where c e (η, e) denoted the marginal cost of education for a worker of ability η at education e c e (η, e) = c (η, e) e V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 7 / 57
Assumption on costs: Interpretation Consider a worker believing that with education e 1 he would get paid wage w 1 We investigate the increase in wages that would be necessary to compensate this worker for an increase in education from e 1 to e 2 The answer depends on the worker s ability: Low-ability workers find it more difficult to acquire the extra education and so require a larger increase in wages to compensate them for it w = w 2 w 1 = e2 e 1 c (η, e)de e The graphical statement of this assumption is that low-ability workers have steeper indifference curves than do high-ability workers V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 8 / 57
Assumption on costs: Interpretation I L is an indifference curve of a low-ability worker I H is an indifference curve of a high-ability worker V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 9 / 57
Competition among firms Spence also assumes that competition among firms will drive expected profits to zero One can build this assumption into our model by replacing the two firms in stage 3 with a single player called the market The market makes a single wage offer w and has the payoff [y(η, e) w] 2 Doing so would make the model belong to the class of one-receiver signaling games defined previously V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 10 / 57
Competition among firms To maximize its expected payoff, as required by Signaling Requirement 2R, the market would offer a wage equal to the expected output of a worker with education e, given the market s belief about the worker s ability after observing e w(e) = µ(h e) y(h, e) + [1 µ(h e)] y(l, e) (W) µ(h e) is the market s assessment of the probability that the worker s ability is H The purpose of having two firms bidding against each other in Stage 3 is to achieve the same result without resorting to a fictitious player called the market V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 11 / 57
Firms beliefs To guarantee that firms will always offer a wage equal to the worker s expected output We need to impose that, after observing education choice e, both firms hold the same belief about the worker s ability, again denoted µ(h e) Signaling Requirement 3 determines the belief that both firms must hold after observing a choice of e that is on the equilibrium path The assumption is that the firms also share a common belief after observing a choice of e that is off the equilibrium path Given this assumption, it follows that in any perfect Bayesian equilibrium the firms both offer the wage w(e) given in (W) Equation (W) replaces Signaling Requirement 2R for this two-receiver model V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 12 / 57
The complete information case First, consider temporarily that the worker s ability is common knowledge among all the players, rather than privately known by the worker Competition between the two firms in Stage 3 implies that a worker of ability η with education e earns the wage ŵ(η, e) = y(η, e) A worker with ability η therefore chooses e (η) to solve max y(η, e) c(η, e) e 0 The associated wage (when it exists) is denoted by w (η), i.e., w (η) = y[η, e (η)] V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 13 / 57
The complete information case Assume that e y(η, e) is concave and e c(η, e) is strictly convex Assume that y(η, ) and c(η, ) are such that and y c lim (η, e) (η, e) > 0 e 0 + e e lim e y e c (η, e) (η, e) < 0 e Then the maximization problem has a unique solution e (η) satisfying y e (η, e (η)) = c e (η, e (η)) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 14 / 57
The complete information case V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 15 / 57
The complete information case We propose to strengthen the assumption e 0, y(h, e) > y(l, e) that states that high-ability workers are more productive Assume that inf y e(h, e) max y e(l, e) e 0 e 0 The previous assumption is automatically satisfied if e y(η, e) is linear V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 16 / 57
The complete information case Proposition Under the previous assumption, one must have e (L) < e (H) and w (L) < w (H) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 17 / 57
The private information case We now return to the assumption that the worker s ability is private information A low-ability worker could try to masquerade as a high-ability worker Two cases can arise The additional effort c[l, e (H)] c[l, e (L)] needed to obtain the education level e (H) is not compensated by the additional wage w (H) w (L) The additional effort c[l, e (H)] c[l, e (L)] needed to obtain the education level e (H) is compensated by the additional wage w (H) w (L) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 18 / 57
The private information case The low-ability worker has no incentives to pretend being a high-ability worker by choosing e (H), i.e., w (L) c[l, e (L)] w (H) c[l, e (H)] V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 19 / 57
The private information case The low-ability worker has incentives to pretend being a high-ability worker by choosing e (H), i.e., w (L) c[l, e (L)] w (H) c[l, e (H)] V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 20 / 57
Perfect Bayesian Nash equilibria Each kind of equilibrium pooling separating hybrid can exist in this model In a pooling equilibrium both worker-types choose a single level of education, say e p Requirement 3 then implies that the firm s belief after observing e p must be the prior belief µ(h e p ) = q and µ(l e p ) = 1 q V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 21 / 57
Perfect Bayesian Nash equilibria This in turn implies that the wage offered by the firm after observing e p must be w p = q y(h, e p ) + (1 q) y(l, e p ) To complete the description of a pooling perfect Bayesian equilibrium, it remains 1 to specify the firm s belief µ( e) for out-of-equilibrium education choices e e p (Requirement 1) 2 these beliefs will then determine the firm s strategy e w(e) through w(e) = µ(h e) y(h, e) + [1 µ(h e)] y(l, e) (W) (Requirement 2R) 3 to show that both worker-types best response to the firm s strategy w is to choose e = e p (Requirement 2S) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 22 / 57
Pooling equilibrium One possibility is that the firms believe that any education level other than e p implies that the worker has low ability e e p, µ(h e) = 0 Nothing in the definition of perfect Bayesian equilibrium rules these beliefs out Requirements 1 through 2 put no restrictions on beliefs off the equilibrium path Requirement 4 is vacuous in a signaling game The refinement we will introduce in a subsequent chapter will rule out the beliefs analyzed here V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 23 / 57
Pooling equilibrium If the firm s beliefs are { 0 for e ep µ(h e) = q for e = e p Then Equation (W) implies that the firms strategy is { y(l, e) for e ep w(e) = w p for e = e p where we recall that w p = q y(h, e p ) + (1 q) y(l, e p ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 24 / 57
Pooling equilibrium A worker of ability η chooses e to solve max e 0 Consider the following example w(e) c(η, e) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 25 / 57
Pooling equilibrium The low-ability worker s indifference curve through the point [e (L), w (L)] lies below that type s indifference curve through (e p, w p ) This implies that the education e p is optimal for the low-ability worker The high-ability worker s indifference curve through the point (e p, w p ) lies above the wage function w = y(l, e) This implies that the education e p is optimal for the high-ability worker This is because the solution e H to the maximization problem max y(l, e) c(h, e) e 0 will lead to a wage w(e H ) = y(l, e H ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 26 / 57
Other pooling equilibria In the previous example, many other pooling perfect Bayesian equilibria exist Some of these equilibria involve a different education choice by the worker Others involve the same education choice but different off the equilibrium path V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 27 / 57
Other pooling equilibria Let ê denote a level of education between e p and e If we substitute e p by ê then the resulting belief and strategy for the firms together with the strategy y(η) = ê for both worker s types form another pooling perfect Bayesian equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 28 / 57
Other pooling equilibria Suppose that the firms belief is defined by 0 for e e except for e = e p µ(h e) = q for e = e p q for e > e The firms strategy is then y(l, e) for e e except for e = e p w(e) = w p for e = e p w p for e > e These belief and strategy for the firms and the strategy (e(l) = e p, e(h) = e p ) for the worker form a third pooling perfect Bayesian equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 29 / 57
Separating equilibrium: the no-envy case We now turn to separating equilibria Consider again the no-envy example V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 30 / 57
Separating equilibrium: the no-envy case The natural separating perfect Bayesian equilibrium involves the strategy e(l) = e (L) and e(h) = e (H) for the worker Signaling Requirement 3 then determines the firms belief after observing either of these two education levels µ[h e (L)] = 0 and µ[h e (H)] = 1 Equation (W) implies that the firms strategy is w(e (L)) = w (L) = y[l, e (L)] and w(e (H)) = w (H) = y[h, e (H)] V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 31 / 57
Separating equilibrium: the no-envy case To complete the description of this separating perfect Bayesian equilibrium, it remains 1 to specify the firms belief µ(h e) for out-of-equilibrium education choices, i.e., values of e other than e (L) and e (H) 2 which then determines the rest of the firms strategy w through Equation (W) 3 to show that the best response for a worker of ability η to the firms strategy w is to choose e (η) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 32 / 57
Separating equilibrium: the no-envy case Consider the belief that the worker has high ability if e is at least e (H) but has low ability otherwise { 0 for e < e µ(h e) = (H) 1 for e e (H) Equation (W) then implies that the firms strategy is { y(l, e) for e < e w(h e) = (H) y(h, e) for e e (H) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 33 / 57
Separating equilibrium: the no-envy case V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 34 / 57
Separating equilibrium: the no-envy case Recall that e (H) is the high-ability worker s best response to the wage function e y(h, e) Since y(l, e) y(h, e) we get that e (H) is still a best response to the wage function w Recall that e (L) is the low-ability worker s best response to the wage function e y(l, e) on the whole real line, this implies that it is also a best response on the interval [0, e (H)) since e (L) < e (H) We should now solve the following maximization problem max y(h, e) c(l, e) e e (H) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 35 / 57
Separating equilibrium: the no-envy case Denote by f the function from [e (H), ) to R defined by f(e) y(h, e) c(l, e) Observe that f (e) = y e (H, e) c e (L, e) y e (H, e) c e (H, e) 0 This implies that w (H) c[l, e (H)] is the highest payoff the low-ability worker can achieve among all choices of e e (H) Since we are in the no-envy case, we have w (L) c[l, e (L)] > w (H) c[l, e (H)] Implying that e (L) is the worker s best response to the strategy w V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 36 / 57
Separating equilibrium: the envy case We consider the envy case: more interesting Now the high-ability worker cannot earn the high wage y(h, ) simply by choosing the education e (H) that he should choose under complete information V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 37 / 57
Separating equilibrium: the envy case To signal his ability, the high-ability worker must choose e s where e s > e (H) is defined by y(h, e s ) c(l, e s ) = y(l, e (L)) c(l, e (L)) This is because the low-ability worker will mimic any value of e between e (H) and e s And will trick the firm into believing that the worker has high ability Formally, the natural separating perfect Bayesian equilibrium involves the strategy for the worker e(l) = e (L) and e(h) = e s V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 38 / 57
Separating equilibrium: the envy case The equilibrium beliefs for the firm must satisfy µ[h e (L)] = 0 and µ[h e s ] = 1 The equilibrium wage strategy for the firms must satisfy w(e (L)) = w (L) = y(l, e (L)) and w(e s ) = y(h, e s ) Actually this is the only equilibrium that survives the refinement we will introduce in a subsequent chapter V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 39 / 57
Separating equilibrium: the envy case We propose the following specification of the firms out-of-equilibrium beliefs that supports this equilibrium behavior { 0 for e < es µ(h e) = 1 for e e s The firms strategy is then { y(l, e) for e < es w(e) = y(h, e) for e e s V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 40 / 57
Separating equilibrium: the envy case V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 41 / 57
Separating equilibrium: the envy case Let us compute the best response of the low-ability worker We already know that e (L) is a best response among all choices of e < e s One should find the worker s best response to the firms strategy among all choices of e e s, i.e., max y(h, e) c(l, e) e e s Denote by g the function defined by g(e) = y(h, e) c(l, e) for all e e s Observe that g (e) = y e (H, e) c e (L, e) y e (H, e) c e (H, e) Recall that the function e y(h, e) c(h, e) is concave and y e (H, e (H)) c e (H, e (H)) = 0 implying that g (e) 0 for all e e s V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 42 / 57
Separating equilibrium: the envy case Therefore, the worker s best response to the firms strategy among all choices of e e s is e s Since w (L) c(l, e (L)) = y(h, e s ) c(l, e s ) The worker has two best responses: e (L) and e s We will assume that this indifference is resolved in favor of e (L) Alternatively, we could increase e s by an arbitrary small amount so that the low-ability worker would strictly prefer e (L) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 43 / 57
Separating equilibrium: the envy case Let us now analyze the best response of the high-ability worker Denote by g the function defined by g(e) = y(h, e) c(h, e) for all e 0 Since g is concave, we have e e s, g (e) = y e (H, e) c e (H, e) y e (H, e (H)) c e (e (H)) This implies that the worker s best response to the firms strategy among all choices of e e s is e s What about the worker s best response among all choices of e < e s? V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 44 / 57
Separating equilibrium: the envy case Let π (L) be the payoff of the low-ability worker at point (e (L), w (L)) Denote by W (L, ) the function defined by W (L, e) = π (L) + c(l, e) This is the equation of the indifference curve I L of the low-ability worker passing through (e (L), w (L)) Denote by W (H, ) the function defined by W (H, e) = [y(h, e s ) c(h, e s )] + c(h, e) This is the equation of the indifference curve I H of the high-ability worker passing through (e s, w(e s )) By definition of e s we have W (L, e s ) = W (H, e s ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 45 / 57
Separating equilibrium: the envy case V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 46 / 57
Separating equilibrium: the envy case Observe that W e (H, e) W e (L, e) = c e(h, e) c e (L, e) < 0 Implying that the function e W (H, e) W (L, e) is strictly decreasing We then get that e < e s, W (H, e) > W (L, e) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 47 / 57
Separating equilibrium: the envy case By definition of e (L), convexity of e c(l, e) and concavity of e y(l, e) we have e 0, W (L, e) y(l, e) This implies that W (H, e) > y(l, e) It follows that the indifference curve of the high-ability worker passing through (e s, w(e s )) is always above the production function y(l, e), implying that any payoff among e < e s is inferior to the one obtained at e s V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 48 / 57
Separating equilibrium: the envy case There are other separating equilibria that involve a different education choice by the high-ability worker the low-ability worker always separate at e (L) There are other separating equilibria that involve the education choices e (L) and e s but differ off the equilibrium path V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 49 / 57
Hybrid equilibrium We analyze the case of an hybrid equilibrium where the low-ability worker randomizes The high-ability worker chooses the education level e h (h for hybrid) The low-ability worker randomizes between choosing e h with probability π and choosing e L with probability 1 π Signaling Requirement 3 then determines the firms belief after observing e h and e L V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 50 / 57
Hybrid equilibrium Bayes rule yields µ(h e L ) = 0 and µ(h e h ) = q q + (1 q)π Since the high-ability worker always choose e h but the low-ability worker does so only with probability π, observing e h makes it more likely that the worker has high ability so µ(h e h ) > q Second, as π approaches zero, the low-ability worker almost never pools with the high-ability worker so µ(h e h ) approaches 1 Third, as π approaches one, the low-ability worker almost always pools with the high-ability worker so µ(h e h ) approaches the prior belief q V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 51 / 57
Hybrid equilibrium When the low-ability worker separates from the high-ability worker by choosing e L The belief µ(h e L ) = 0 implies the wage w(e L ) = y(l, e L ) We claim that e L = e (L) Suppose the low-ability worker separates by choosing some e L e (L) Such separation yields the payoff y(l, e L ) c(l, e L ) But choosing e (L) would yield the payoff of at least y[l, e (L)] c[l, e (L)] or more if the firms belief µ[h e (L)] is greater than 0 The definition of e (L) implies y[l, e (L)] c[l, e (L)] > y(l, e) c(l, e), e e (L) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 52 / 57
Hybrid equilibrium For the low-ability worker to be willing to randomize between separating at e (L) and pooling at e h The wage w h w(e h ) must make that worker indifferent between the two w (L) c[l, e (L)] = w h c(l, e h ) (P) Recall that Equation (W) and the definition of the belief µ( e h ) imply w h = q q + (1 q)π y(h, e (1 q)π h) + q + (1 q)π y(l, e h) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 53 / 57
Hybrid equilibrium For a given value of e h, if Equation (P) yields w h < y(h, e h ) then there is a unique possible value for π consistent with a hybrid equilibrium in which the low-ability worker randomizes between e (L) and e h If w h > y(h, e h ), then there does not exist a hybrid equilibrium involving e h Observe that Equation (P) yields w h < y(h, e h ) if and only if e h < e s where e s is the education chosen by the high-ability worker in the separating equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 54 / 57
Hybrid equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 55 / 57
Hybrid equilibrium Given w h < y(h, e h ), the probability r solves r y(h, e h ) + (1 r) y(l, e h ) = w h This probability is the firms equilibrium belief µ(h e h ), so π = q(1 r) r(1 q) As e h approaches e s, the probability r approaches 1 so π approaches 0 The separating equilibrium described previously is the limit of the hybrid equilibria considered here V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 56 / 57
Hybrid equilibrium To complete the description of the hybrid perfect Bayesian equilibrium, we should define the firms belief out-of-equilibrium path and check the workers best response Let µ( e) be defined as follows { 0 for e < eh µ(h e) = r for e e h The firms strategy is then { y(l, e) for e < eh w(e) = r y(l, e) + (1 r) y(h, e) for e e h It remains to check that the workers strategy e(l) = eh with probability π and e(l) = e (L) with probability 1 π e(h) = eh is a best response to the firms strategy V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, 2011 57 / 57