EC319 Economic Theory and Its Applications, Part II: Lecture 7

EC319 Economic Theory and Its Applications, Part II: Lecture 7 Leonardo Felli NAB.2.14 27 February 2014

Signalling Games Consider the following Bayesian game: Set of players: N = {N, S, }, Nature N strategy space: (1/2, 1/2) s beliefs: µ(s 1 ) = 1/2. The game is known as a signaling game and it is described in the following extensive form. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 2 / 39

Signalling Games: Example (1, 3) (4, 0) U L [p] D L L S 1 ( 1 2 ) U (2, 1) [q] D (0, 0) N (2, 4) (0, 1) U L. [1 p] D L L S 2 ( 1 2 ) U (1, 0). [1 q] D (1, 2) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 3 / 39

Signalling Games: Equilibria We can classify the Perfect Bayesian equilibria of the game according to the beliefs of the receiver at the information set in which he is asked to play. In particular in this game there exist two categories of equilibria. Pooling equilibria: these are equilibria in which the two types of sender S 1 and S 2 send the same signal (in this game a choice of action a Si ) either L or. In this case only one information set of the receiver is on the equilibrium path. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 4 / 39

Signalling Games: Equilibria (cont d) Bayes rule imposes that the beliefs at this information set are the same as the probabilities with which nature takes its choice ( 1 2 ) and ( 1 2 ). Separating equilibria: these are equilibria in which the two types of sender S 1 and S 2 send the two different signals: L and. In this case both information sets of the receiver are on the equilibrium path. Bayes rule imposes that the beliefs at the two information sets are degenerate: p {0, 1} and q {0, 1}. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 5 / 39

Pooling Equilibria Consider now a candidate pooling equilibrium in which both S 1 and S 2 choose strategy L. In this case Bayes rule implies p = 1 2. The expected payoff to are then: Π (L, L; U L ) = 7 2, Π (L, L; D L ) = 1 2 It is therefore a best reply for to choose U L. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 6 / 39

Pooling Equilibria (cont d) (1, 3) (4, 0) U L [p] D L L S 1 ( 1 2 ) U (2, 1) [q] D (0, 0) N (2, 4) (0, 1) U L. [1 p] D L L S 2 ( 1 2 ) U (1, 0). [1 q] D (1, 2) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 7 / 39

Pooling Equilibria (cont d) We now need to specify the off-the-equilibrium-path beliefs q that sustain this equilibrium. At this purpose notice that if response to is U then it is not a best reply for S 1 to choose L: he can gain by deviating and choosing instead. It is however a best reply for S 2 to choose L. If instead response to is D then it is a best reply for both S 1 and S 2 to choose L. Therefore for the pooling strategy (L, L) to be part of a Perfect Bayesian equilibrium we need to choose D at the information set off the equilibrium path. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 8 / 39

Pooling Equilibria (cont d) The expected payoff to at this information set is: Π (U ) = q, Π (D ) = 2 (1 q) Clearly Π (D ) Π (U ) if and only if q 2 3 A Pooling Perfect Bayesian Equilibrium of this game is: [ (L, L); (UL, D ), p = 1 2, q 2 3] Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 9 / 39

Pooling Equilibria (cont d) In other words given the following Bayesian signaling game: (1, 3) (4, 0) U L [p] D L L S 1 ( 1 2 ) U (2, 1) [q] D (0, 0) N (2, 4) (0, 1) U L. [1 p] D L L S 2 ( 1 2 ) U (1, 0). [1 q] D (1, 2) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 10 / 39

ecipe: We used the following recipe to find a solution: make an hypothesis on the sender equilibrium strategies; given these strategies, identify the receiver s beliefs at every information set that is on the equilibrium path; given these beliefs compute the receiver s best reply at the information set that are on the equilibrium path; verify that the sender s equilibrium strategies you started from are a best reply for the different types of sender; identify the restrictions that this last step imposes on the receiver s beliefs at information sets that are off the equilibrium path (if there is any). Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 11 / 39

Pooling Equilibria (cont d) We already found a Pooling Strong Perfect Bayesian Equilibrium: [ (L, L); (UL, D ), p = 1 2, q 2 3] Consider now the other Pooling equilibrium in which both S 1 and S 2 choose the action. Clearly Bayes rule implies that q = 1 2. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 12 / 39

Pooling Equilibria (cont d) Given this beliefs s best response is D : Π (, ; U ) = 1 2 < Π (, ; D ) = 1 Given this the sender S 1 will always find optimal to deviate and choose L independently of whether following L will choose U L or D L. This implies that there there does not exist a Pooling Perfect Bayesian Equilibrium where S 1 and S 2 both choose. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 13 / 39

Pooling Equilibria (cont d) (1, 3) (4, 0) U L [p] D L L S 1 ( 1 2 ) U (2, 1) [q] D (0, 0) N (2, 4) (0, 1) U L. [1 p] D L L S 2 ( 1 2 ) U (1, 0). [1 q] D (1, 2) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 14 / 39

Separating Equilibria Consider now the Separating equilibrium in which S 1 chooses L and S 2 chooses. Bayes rule in this case implies p = 1 and q = 0. Given these beliefs it is then a best reply for to choose: U L following the observed choice of L. D following the observed choice of. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 15 / 39

Separating Equilibria (cont d) Notice however that given s best reply S 2 can gain a payoff of 2 by deviating and choosing L instead of the payoff of 1 he gets by choosing. This implies that there does not exist a Separating Perfect Bayesian Equilibrium where S 1 chooses L and S 2 chooses. Finally consider the Separating equilibrium in which S 1 chooses and S 2 chooses L. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 16 / 39

Separating Equilibria (cont d) (1, 3) (4, 0) U L [p] D L L S 1 ( 1 2 ) U (2, 1) [q] D (0, 0) N (2, 4) (0, 1) U L. [1 p] D L L S 2 ( 1 2 ) U (1, 0). [1 q] D (1, 2) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 17 / 39

Separating Equilibria (cont d) Bayes rule implies that p = 0 and q = 1. Given these beliefs it is then a best reply for to choose: U following the observed choice of. UL following the observed choice of L. Notice that in this case player S 1 earns an equilibrium payoff of 2 and can only earn a payoff of 1 by deviating. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 18 / 39

Separating Equilibria (cont d) Similarly in this case player S 2 earns an equilibrium payoff of 2 and can only earn a payoff of 1 by deviating. In other words it is a best reply for players S 1 and S 2 to choose strategies, respectively L. The Separating Perfect Bayesian Equilibrium of this game is: [ (, L); (U, U L ), p = 0, q = 1 ] Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 19 / 39

Spence Signaling Model Consider the following educational choice model: There are two types of workers: high productivity workers θ H and low productivity workers θ L A worker is of type θ i wth probability π i Before entering the labor market workers may signal their type by acquiring education. Different types of workers have different costs of acquiring education. Employers can distinguish workers only by their education and do not observe their ability. Employers compete in wages to hire workers given such information. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 20 / 39

Timing 1. Nature determines the type θ i of each worker with probability π i. 2. Workers decide how much education e i to acquire. 3. Employers simultaneously and independently make wage offers w(e i ) to each worker (they Bertrand compete for each worker). 4. Workers decide which offer to accept. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 21 / 39

Educational Investment: Workers Workers acquire a level e [0, 1] of education The cost of acquiring education e for type θ i is c(e θ i ) Assume the cost of education is convex: c(0 θ i ) = 0 c e (e θ i ) > 0 c ee (e θ i ) > 0 Marginal cost of education is higher for the low productivity worker θ L than for the high productivity worker θ H : c e (e θ H ) < c e (e θ L ) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 22 / 39

Workers Preferences If the worker accept the wage offer w(e) then the payoff of a worker of type θ i with education e is: u(e θ i ) = w(e) c(e θ i ) Assumptions on the marginal costs of education and on preference guarantee that the single crossing condition is satisfied. The indifference curves of the two types of workers cross once: the Spence-Mirrlees Single Crossing Condition are satisfied. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 23 / 39

Employers The employers technology is such that the productivity of a worker of type θ i is f (θ i ) where f (θ H ) > f (θ L ) Initially employers know only the prior probability π i that a worker is of type θ i. Employers do observe the educational investment e of workers After workers have made their educational investments, employers update their beliefs on the basis of this new observed information e, offer a wage w(e) that depends on education. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 24 / 39

Types of Equilibria: Separating Equilibria This model has two different types of equilibria: Separating equilibria these are characterized by the fact that the two types of worker: 1. choose different education levels: e s H > es L 2. are paid different wages: w(e s H ) > w(es L ) 3. prefer not to mimic the other type by choosing the educational level selected by the other type. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 25 / 39

Types of Equilibria: Pooling Equilibria Pooling equilibria these are characterized by the fact that the two types of worker: 1. choose the same education level: e p H = ep L 2. are paid the same wage: w(e p H ) = w(ep L ) 3. prefer not to be separated from the other type by choosing a different educational level than the one selected by the other type. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 26 / 39

Separating Equilibria: Characterization Separating Equilibria are such that: Workers of different type choose different education levels e i, i {H, L} Employers upon observing the level of education e i perfectly identify each type θ i. Employers makes an offer w(e i ) to each worker. The worker decides which offer to accept. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 27 / 39

Separating Equilibria: Perfect Bayesian Equilibrium We are constructing a Perfect Bayesian Equilibrium of this imperfect information game. We solve the model backward ad start from theemployers Bertrand competition for each worker. Assume there exists two potential employers h {1, 2} for each worker. Given two offers w 1 (e i ) and w 2 (e i ) the worker will choose the offer (and hence the employer) such that w h (e i ) = max {w 1 (e i ), w 2 (e i )} If two offers are identical w 1 (e i ) = w 2 (e i ) the worker will randomize with equal probability the offer to accept. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 28 / 39

Bertrand Competition Each employer h s payoff is then: f (θ i ) w h (e i ) if w h (e i ) w h (e i ) 1 Π(w h (e i )) = 2 [f (θ i) w h (e i )] if w h (e i ) = w h (e i ) 0 if w h (e i ) < w h (e i ) Each employer h s best reply is then: w h (e i ) = w h (e i ) + ε if w h (e i ) < f (θ i ) w h (e i ) w h (e i ) if w h (e i ) = f (θ i ) w h (e i ) < w h (e i ) if w h (e i ) > f (θ i ) The unique Bayesian Nash equilibrium of the Bertrand Competition subgame is then w h (e i ) = w h (e i ) = f (θ i ), i {H, L} Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 29 / 39

Employers Beliefs We now move back to the employers believes upon observing each worker education investment e i, i {H, L}. Assume that the two types of workers separate at the eduction investment stage: θ H workers choose e H and θ L choose e L where e H e L. Bayes rule implies that the only equilibrium posterior beliefs of employers compatible with a separating equilibrium are such that: Pr {θ = θ i e i } = 1 Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 30 / 39

Educational Investment We now move back to the workers choice of education. It must be a best reply for each worker of type θ i to choose the level of educations e i : w(e H ) c(e H θ H ) w(e L ) c(e L θ H ) w(e L ) c(e L θ L ) w(e H ) c(e H θ L ) Given the equilibrium wage offer w(e i ) these no deviation (incentive compatibility) conditions are: f (θ H ) c(e H θ H ) f (θ L ) c(e L θ H ) f (θ L ) c(e L θ L ) f (θ H ) c(e H θ L ) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 31 / 39

Low Productivity and Education Notice first that in this model education is a pure signal, it has no effect on productivity. Therefore the only Bayesian equilibrium choice of education on the part of θ L workers is e L = 0 Assume not: e L > 0. The incentive compatibility constraints imply e H > e L. Then a θ L worker has a profitable deviation choosing e L < e L, since f (θ L ) > f (θ L ) c(e L θ L ) Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 32 / 39

High Productivity and Education The incentive compatibility constraints can then be written as: c(e H θ H ) f (θ H ) f (θ L ) c(e H θ L ) Hence, incentive compatibility conditions imply: e H [e, e] where c(e θ H ) = f (θ H ) f (θ L ) = c(e θ L ) All Bayesian Nash equilibria of the Spence Education model are such that e H [e, e] e L = 0 with beliefs: { 1 if e = eh Pr {θ = θ H e} = 0 if e = e L Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 33 / 39

Off the Equilibrium Path Beliefs To guarantee that no worker chooses e {e H, e L } impose: { 1 if e eh Pr {θ = θ H e} = 0 if e < e H ecall that Bayes rule does not impose any constraint on the off-the-equilibrium-path beliefs. The beliefs above imply that: { f (θl ) if e < e w(e) = H f (θ H ) if e e H Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 34 / 39

Off the Equilibrium Path Given these beliefs and wage offers no θ H worker wants to deviate: f (θ H ) c(e H θ H ) f (θ H ) c(e θ H ) for all e > e H f (θ H ) c(e H θ H ) f (θ L ) c(e θ H ) for all e < e H Moreover no θ L worker wants to deviate: f (θ L ) f (θ H ) c(e θ L ) for all e > e H f (θ L ) f (θ L ) c(e θ L ) for all e < e H Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 35 / 39

Multiplicity of Equilibria There is a multiplicity of separating Perfect Bayesian equilibria They are characterized by the education levels satisfying: e H [e, e] & e L = 0 Workers receive the efficient wage, namely their productivity But no equilibrium is efficient since θ H workers waist resources to signal their θ H type by investing in education The Pareto dominant equilibrium is the one in which e H = e since the cost of acquiring education is the lowest The multiplicity is due to the off-the-equilibrium-path beliefs. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 27 February 2014 36 / 39