9 A Class of Dynamic Games of Incomplete Information:

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1 A Class of Dynamic Games of Incomplete Information: Signalling Games In general, a dynamic game of incomplete information is any extensive form game in which at least one player is uninformed about some other players type and where there are some sequential moves. An interesting and tractable class of such games are called signalling games..1 What is Signaling? The concept of signaling refers to strategic models where one or more informed agents take some observable actions before one or more uninformed agents make their strategic decisions. This leads to situations where the uninformed agent care about the actions taken by the informed agent not only because the actions a ect payo s directly, but also because the action taken say something about the type of the player. This in turn creates incentives to select actions to send the right signal about type.. The Basic Signaling Game Consider the following setup: There are two players. We refer to player 1 as the sender and to player as the receiver. Player 1 has private information about his type. We denote the type space by and write for a generic element. The set of available actions for the sender is A 1 ; so a (pure) strategy is function s 1 with s 1 () in A 1 for every type : We let p denote the (prior) probability distribution over the senders type space. Player, the receiver, observes the action chosen by the sender and then take some action in A : A pure strategy for player is a function s, where s (a 1 ) A for every a 1 A 1 63

2 Utility functions, u 1 (a 1 ; a ; ) for the sender u (a 1 ; a ; ) for the receiver,.3 Equilibrium Sometimes, it is su cient to look at Nash equilibria. However, there are times where the set of Nash equilibria includes i) way to many possibilities; ii) some quite unreasonable equilibria. The reason is that the Nash equilibrium concept has no restrictions on o the equilibrium path play. We therefore want to do something in the spirit of backwards induction/subgame perfection, which is a bit trickier in games with incomplete information. In essence, we want equilibria to satisfy two sorts of criteria: 1. Sequential Rationality. Whenever an agent is called to play, the agent does something optimal (to re ne away Nash equilibria supported by play that is suboptimal o the path).. Consistency of beliefs. What is optimal often depends on what an agent believes about the opponent(s), and we want to rule out an agent thinking something that is contradicted by the equilibrium strategies (coordination assumption much in the same spirit as the rational expectations part of Nash equilibrium). There are several equilibrium concepts that capture these two ideas to a smaller or larger extent. The de nition below is not my favorite version, but we ll stick to it in this course because it is easier to de ne than the more appealing versions: De nition 1 A (pure) perfect Bayesian Equilibrium in a signaling game (of the form described above) is a strategy pro le s and a system of beliefs such that 1. s 1 () solves max a 1 A 1 u 1 (a 1 ; s (a 1 ) ; ) for all. s (a 1 ) solves max a A 1 P (ja 1) u (a 1 ; a ; ) for all a 1 A 1 6

3 3. (ja 1 ) satis es Bayes rule whenever applicable (i.e., whenever you can avoid dividing by zero). Example (0,0) (1,) (-,-) (,3) e e e e e,, p L1 [1/] [1/] jn R1, (,3) t,,, Q q QQQ (-1,0) (0,0) t 1-p L @ (-1,) Figure 1: A Simple Signaling Game (R 1 L ; at ) ; (p; q) = (0; 1) is Perfect Bayesian Bayes rule p = Pr [t 1 jl] = = = 0 q = Pr [t 1 jr] = Pr [Ljt 1 ] Pr [t 1 ] Pr [Ljt 1 ] Pr [t 1 ] + Pr [Ljt ] Pr [t ] Pr [Rjt 1 ] Pr [t 1 ] Pr [Rjt 1 ] Pr [t 1 ] + Pr [Rjt ] Pr [t ] = = 1 CHECK! (L 1 L ; bb) ; (p; q) = ( 1 ; q) is a PBE if q 5 (free to specify beliefs at unreached information sets, but must be done so that they support equilibrium strategies). 65

4 CHECK! (R 1 R ; ct ) is a Nash equilibrium, but not a PBE. 10 Job Market Signaling 10.1 A Simple Version of the Static Spence Model Consider the following formulation of the job market signalling model due to Spence; Types are given by f1; g 0 is the probability that = 1 A Worker may choose (i.e. commit to) any education of length t 0 The utility is given by u (w; t; ) = w t Game: 1) Worker choose education. ) Firms compete Bertrand for workers and get a pro t = w if it manages to attract the worker. Let (t) denote the probability that rms asses that = 1. In (a perfect Bayesian) equilibrium, the rms must both optimize given beliefs, implying that w (t) = (t) Separating Equilibria Given that we restrict attention to equilibria where rms behave optimally after any t 0 is follows that t 1 = 0 in any separating (perfect Bayesian) equilibrium: the reason is that if t 1 > 0 and the low productivity worker deviates to t = 0; then (0) [0; 1] ) w (0) 0 1 > 1 t 1 = w (t 1 ) t 1 : 66

5 We conclude that the deviation is pro table. In words the logic is simply that the low type reveals to be the worst possible worker. Clearly, it cannot be worth anything to the low productivity worker to do so, so the worker must take the least costly action. By consistency of beliefs so w (t ) = (t ) = = 0 (t ) = : In order for type 1 to be better o at t 1 = 0 than with t it must be that, w (t 1 ) w(t ) t 1 t, t 1 and for type to be better of with t than with t 1 = 0 it must be that w (t ) t w(0) t 1, t () Hence, if 1 t neither type has an incentive to pretend to be the other, and by considering beliefs; 8 < 0 if t = t (t) = : 1 if t 6= t it is immediate that neither type has a pro table deviation. In fact, it is su cient to use beliefs 8 < 0 if t t or e (t) = : 1 if t < t () ; which you can verify by drawing a graph. e is nicer than because beliefs are monotonic in education. 67

6 10.1. Pooling Equilibria To support as large a set of education levels as pooling equilibria, suppose that t 1 = t = t is a pooling equilibrium and let beliefs be 8 < (t) = : 0 if t = t ; 0 if t 6= t which obviously is consistent with Bayes rule where relevant. The best deviation for both types is then to t = 0 and this is not pro table for the low productivity/high cost of education type if 0 t 1: Clearly, the high productivity type has no incentive to deviate if the low productivity type has no incentive to deviate, so any 0 t 1 0 can be supported as a pooling equilibrium. 11 Limit Pricing Consider the following environment: There are two periods and a market for a homogenous good with inverse demand p(q) = 1 q in each period (i.e., a non-durable good) Two rms, labeled I (incumbent) and E (entrant) incumbent has marginal cost c I fc L ; c H g entrant has marginal cost c E Incumbent (the sender) is a monopolist in the rst period. Sets some q I 68

7 Entrant (the receiver) observes the realized q I and decides whether or not to enter. For simplicity it is assumed that once the receiver has decided whether or not to enter the incomplete information magically gets resolved. Firms compete Cournot in second period if there is entry Incumbent acts a monopolist in second period if no entry Cost of entry given by K: The monopoly and cournot pro ts are readily computed as in the following table (recall that the asymmetric information disappears after entry). Entrant enters and c = c L Entrant enters and c = c H Entrant stays out Pro t for incumbent c L (1 c L + c E ) = (1 c L ) = Pro t for incumbent c H (1 c H + c E ) = (1 c H ) = Pro t for entrant (1 c E + c L ) = K (1 c E + c H ) = K 0 YOU SHOULD CHECK THESE CALCULATIONS! To make things interesting we assume that (1 c E + c H ) = > K > (1 c E + c L ) =; meaning that under symmetric information the entrant would enter if and only if the incumbent would be a high cost rm Separating Equilibria In a separating equilibrium, the entrant will (as would be the case with symmetric information) enter if and only if the incumbent has a high cost. Let q L and q H denote the quantities chosen by the low and the high cost type respectively. Observe that a necessary condition is that (1 q L c L ) q L + (1 c L) (1 q H c H ) q H + (1 c H + c E ) (1 q H c L ) q H + (1 c L + c E ) (1 q L c H ) q L + (1 c H) 6 (IC-L) (IC-H)

8 The rst condition states that a low cost rm should not be tempted to select the quantity of a high cost rm, the second the opposite. Next we observe that; q H = (1 c H) in any Perfect Bayesian equilibrium (not true in a Nash equilibrium). Reason: The worst case scenario for a high cost type is entry. There will be entry after q H : Hence, if q H 6= (1 c H) there would be a pro table deviation for the high cost rm. The reason why this is not true in every Nash equilibrium is that the entrant in A Nash equilibrium may play something di erent than the Cournot duopoly equilibrium o the equilibrium path. For example, let q L ; q H be the equilibrium outputs in the rst period and consider a strategy where the entrant stays out if and only if q = q L ; where the entrant enters and plays the Cournot quantity if q = qh ; and where the entrant enters and plays q E = 1 if q = fql ; q Hg : This strategy would give the incumbent a pro t=0 if q = fql ; q Hg, implying that this construction can support a range of values for q H based on the non-credible threat by the entrant to ood the market in case the incumbent plays the wrong rst period quantity. Symmetrically, (1 c L+c E ) is the worst possible second period payo for the low cost type)if the low cost type would choose the static monopoly quantity in the rst period, it must get a payo of at least (1 c L ) + (1 c L + c E ) (1 q H c L ) q H + (1 c L + c E ) : (1) Taking these considerations together, this leads us to focus on the following necessary incentive compatibility constraints (1 q L c L ) q L + (1 c L) (1 c H ) + (1 c H + c E ) (1 c L) + (1 c L + c E ) (1 q L c H ) q L + (1 c H) ; where we have; 1. replaced the right hand side in the IC constraint for L by using (1) 70

9 . substituted q H = (1 c H) into the IC constraint for H: Let L and H be the value of keeping the entrant out for the low and high cost incumbent, that is L = (1 c L) H = (1 c H) (1 c L + c E ) (1 c H + c E ) ; One can check that L may be smaller or larger than H depending on c E. The way to understand this is that as c E increases, eventually one reaches a point where the duopoly pro t equals the monopoly pro t for the low cost rm. At this cost, the entrant will still produce a strictly positive quantity against the high cost rm, so the high cost rm still has a strict gain from the entrant staying out. On the other hand side, it is easy to check that for c E = c L ; then L > H ; so here the larger payo under monopoly dominates. We solve this as real economists: Assumption L H De ne, M L = (1 c L) M H = (1 c H) L (q) = (1 q c L )q H (q) = (1 q c H )q; so that the necessary incentive constraints may be written for short as H M H H (q L ) L M L L (q L ) In Figure the range of separating equilibrium quantities q L is illustrated in terms of these two inequalities. To draw the picture, note that the cost of eliminating entry M J J (q L ) 71

10 has a minimum of zero at the static monopoly quantity, which is a higher quantity for the low cost rm. Using the fact that d dq ( L(q) H (q)) = (1 q c L ) (1 q c H ) = c H c L > 0; one checks that M L L (q) and M H H (q) intersects at most once. It should be clear from the gure that L H is a su cient (but not necessary) condition for the range of separating equilibria to be nonempty. 6 h H (q) M H i h L (q) M L i L H - q H separating equilibria q L q Figure : A Continuum of Separating Equilibria in the -type Limit Pricing Model Finally, to support a quantity q L in the range indicated in the gure as a separating equilibrium (where the high cost rm chooses its static monopoly output in the rst period) in the simplest possible way we let the beliefs o the equilibrium path be as pessimistic as possible. That is, if the entrant thinks that any quantity other than q L must be evidence that the rm is a high cost rm, so that 8 < 1 if q = q L (c = c L jq) = : 0 otherwise then any deviation q for the low cost rm gives a payo (1 q c L ) q H + (1 c L + c E ) 7 (1 c L) ; + (1 c L + c E ) :

11 so deviating to the monopoly output is the best deviation under these beliefs. Moreover, from the point of view of the high cost rm any q = q L gives a pro t (1 q c H ) q + (1 c L + c E ) ; under these beliefs, which obviously is maximized for q = q H : In words, if the only quantity that deters entry is q L ; then the best rst period quantity for the high cost rm must be either q L or the static monopoly quantity Remarks 1. The analysis above shows that it is a possibility that a relatively e cient incumbent rm engages in limit pricing (higher quantity=lower price) to signal to potential entrants that they ll be so strong competitors that they better keep out of the market. Note that in the equilibrium constructed nobody is fooled. However, an e cient rm would be mistaken for a not so e cient rm should they deviate. 3. The example also shows that limit pricing (sometimes considered predatory behavior ) may actually bene t consumers (although one would have to consider more carefully exactly what the alternative is id one would think of this as a serious policy-related exercise) by lowering the price.. There is in general also a continuum of pooling equilibria (as well as semi-pooling). This huge multiplicity of equilibria is typical for (simple) signaling games and has led to a monstrous literature on equilibrium selection. I personally think that re nements beyond standard sequential rationality arguments is a suspect idea and that we have to live with the multiplicity. I may talk a little bit about this re nements literature, mainly to illustrate what I think is wrong with the whole idea. However, to some extent the multiplicity is an artefact of the very simplicity of the model (just a few types) and at least when we look for pure separating equilibria, this is mitigated by a richer (continuous) type space. 73

12 1 Cheap-Talk and Communication A fundamental question that arises in most any context where i) one agent has information that is payo -relevant to the other and ii) there is a divergence in preferences, is whether the player with the better information can be induced to communicate what she knows. One of the cleanest type of models to think about this sort of issues is in cheap talk models. The distinguishing feature of such a model is that signaling is without direct costs. That is, the informed player has a bunch of available messages and the cost of sending any message is zero. However, since the receiver potentially reacts to the message there are potentially endogenous costs arising in equilibrium. 1.1 Cheap-Talk Model A classic in modern economic theory is Crawford and Sobel s cheap talk model. Their model has been applied in many di erent contexts, for example in political economy and economics of expertise. Two players, Sender (S) and Receiver (R) S has private information. Type [0; 1] distributed with density f () : We assume that distribution is uniform on [0; 1] : The inerpretation of is that the sender is an expert who observes the realization of some state of the world. Senders payo U S (y; ) = (y b) where y in [0; 1] is action taken by receiver, is the type=realized random variable. OBSERVE THAT THE ACTION OF SENDER DOESN T ENTER, which is what makes this cheap talk. U R (y; ) = (y ) : We can think of b as the bias of the sender. 1. The Game 1. First S observes type. Then she sends a message m in [0; 1] : 7

13 . Reciever observes m (and updates beliefs about ) and then picks some y in [0; 1] : Since the receiver is allowed to pick an action rule this is strategically simultaneous with choice of q even though in real time it must obviously happen after. Hence it will be su cient to look for (Bayesian) Nash rather than anything with sequential rationality involved. 1.3 Equilibrium Conditions We will only consider pure strategy equilibria. 1. Let q () denote the strategy for the sender and y (m) denote the strategy of the receiver. Then, q () must solve max U S (y (m) ; ) = max m m (y (m) b) ; for every :. Moreover, y(m) arg max Z 1 0 U R (y; ) p (jm) d = max y Z 1 0 (y ) p (jm) d where p (jm) is the consitional density over types if observing message m: It should be clear that there is no separating equilibrium. If there was such an equilibrium y (q ()) = max y (y ) = ; implying that type sender gets payo (y (q ()) b) = b : Any message between and + b would make the sender better o, so there is a pro table deviation. A natural guess is an equilibrium where the type space is partitioned by limits [a 0 ; a 1 ; ::::; a N ] and where only messages [m 1 ; ::::; m N ] are active and 75

14 1. All types in [a i 1 ; a i ) sends message m i. y (m i ) solves max y Z ai (y ) d a i 1 a i a i 1 3. a 0 = 0; a N = 1 and [a 1 ; ::::; a N 1 ] solves 0 = U S (y (m i+i ) ; a i ) U S (y (m i ) ; a i ) = (y (m i+i ) a i b) (y (m i ) a i b). For simplicity, we assume that o the equilibrium path messages are interpreted as = 1: The objective for the receiver is a quadratic, so the rst order condition is necessary and su cient. We can thus compute y (m i ) as Z ai a i 1 (y )d = 0 ) y (a i a i ) = ai a i 1 = (a i) (a i 1 ) = (a i + a i 1 ) (a i a i 1 ) y (m i ) = (a i+1 + a i ) : That is, the receiver behaves as if he would if he would know that the type would be the expected type over the interval The di erence equation that makes the critical types indi erence between the higher and lower message is then ) (ai+1 + a i ) 0 = U S (y (m i+1 ) ; a i ) U S (y (m i ) ; a i ) (ai+1 + a i ) (ai + a i 1 ) = a i b (ai + a i 1 ) a i b = a i b a i b, 76

15 For xed a i > a i 1 ; the right hand side is just a (positive) real number and it looks like we get a quadratic in a i+1 : However, suppose that (a i+1 + a i ) (ai + a i 1 ) a i b = a i b ) (a i+1 + a i ) = (a i + a i 1 ) ) a i+1 = a i 1 ; which is inconsistent with the partition structure. It follows that (a i+1 + a i ) a i b =, (a i + a i 1 ) + a i + b a i+1 a i = (a i a i 1 ) + b, a i+1 = a i a i 1 + b Summing up, we have that we seek solutions to a 0 = 0 a N = 1 a i+1 = a i a i 1 + b; where we note that a 1 is arbitrary. However, given any a 1 > 0 indi erence requires that a = a b or a a 1 = a 1 a 0 + b: Hence, the second step is larger than the rst step. Indeed, this is generally true which is immediate from We then have that: a i+1 a i {z } step i+1 = a i a {z i } 1 + b step i 77

16 1. First step is of length a 1. Second step is of length a 1 + b 3. Third step is of length a 1 + 8b. ith step is of length a 1 + (i 1) b Since ::: + N 1 = we have that for the nal step we must have that N (N 1) a 1 + NX a 1 + (i 1) b = Na 1 + i= N (N 1) b Hence, given b the nest partition that is consistent with equilibrium is the largest integer such that N (N 1) b 1 Now: 1. Easy to check that N is decreasing in b: Closer preferences leads to ner partitions.. By suitable choice of a 1 one can support equilibria with any number of partitions between the one (babbling) and the maximal number. 3. One can also grind out from these expressions that equilibria are better (ex ante) for both agents the ner the partition is (holding b constant).. Moreover (this is less non-trivial) for equilibria of a given courseness, the smaller is b, the happier are the agents ex ante. The rough intuition for this is that when b is smaller the intervals are of more equal size, which reduces the variance. 78