Adverse Selection Asymmetric information is feature of many markets - some market participants have information that the others do not have 1) The hiring process – a worker might know more about his ability than the firm does - the idea is that there are several types of workers - some are more productive than others are 2) Insurance – insurance companies do not observe individual characteristics such as driving skills 3) Project financing – entrepreneurs might have more information about projects than potential lenders 4) Used cars – sellers know more about the car’s quality than buyers Adverse selection is often a feature in these settings - it arises when an informed individual’s decisions depend on his privately held information in a way that adversely affects uninformed market participants .

Lemon’s problem A classic example: Alerlof’s “lemons” used car market. Nobel 2001: Akerlof, Spence, Stiglitz - sellers of used cars have private information on vehicle quality, which buyer’s lack - suppose we have two types of cars high and low quality; only sellers observe type .

Signaling In the example above, -sellers with high quality cars would want to convince buyers that the car is high quality. -the signaling device has to be one that sellers with bad cars cannot use effectively. A “This is a Good Car” sign is ineffective: every type of seller will use it, and it will provide no new info Rule 1: in order to be effective, signaling must be costly Rule 2: costs must differ for different “types,” with the cost structure favoring high types Signaling devices can generate efficient outcomes, but need not always - Since buyers have positive surplus for both types of cars and sellers prefer to sell at any price above their reservation price, efficiency requires that both good and bad cars be offered for sale - An effective signal might be a guarantee, since people with bad cars would be unwilling to guarantee their cars, or let a mechanic inspect the car - Since employing signaling devices requires incurring costs (otherwise the devices are ineffective) they can actually make agents worse off .

Perfect Bayesian Equilibrium In order to analyze signaling games, we sometimes use a refinement of WPBE Recall that given the equilibrium an information set is on-the-path if it will be reached with positive probability if the game is played according to the equilibrium strategies, and it is off-the-path if the information set is certain not to be reached with positive probability with play of the equilibrium strategies Perfect Bayesian Equilibrium (PBE) -Requires that at info sets off-the-path, all players agree on what their beliefs are .

Signaling Game Dynamic game of incomplete information Two players: - one with private info : Sender - another without private info : Receiver Structure of signaling game: 1) Nature moves first - determines type of sender 2) Sender moves – sender observes his own type and sends a signal - sender’s optimal strategy depends on receiver’s strategy 3) Receiver observes signal -chooses action, optimal strategy depends on a signal Equilibrium definition: WPBE 1) σ is sequentially rational, given beliefs μ 2) system of beliefs μ is consistent with σ (eqm-on-the-path) determined using Bayes’ rule whenever possible PBE adds 3) all players have the same beliefs off-the-path .

Qual Question Fall 2002 (#75) . .

Two types of equilibria μ (1-μ) .

General Procedure Start analyzing a Signaling game by deciding whether to look for a Separating or Pooling Equilibrium. We’ll start with a Separating Equilibrium: 1) Along the equilibrium path, type is revealed. This gives us a place to start. 2) Assess the receiver’s BR, under the assumption that the receiver knows the sender’s type 3) Back-up the game tree; determine the sender’s optimal choice given how the receiver plays in response to the signal 4) Then apply the incentive compatibility condition: high type must prefer sending high signal; low type must prefer sending low signal 5) Finally, check off-the-path beliefs - here the equilibrium concept does not specify the receiver’s beliefs - to obtain the widest range of equilibrium paths, make the receiver pessimistic about the sender’s type off-the-path - then check that the high type prefers the high signal to any off-the-path signal, and that the low type prefers the low signal to any off-the-path signal .

Separating equilibrium (#75) 1) Separating equilibrium path, type is revealed 2) C’s BR (sequentially rational strategy), given that it knows P’s type on seeing th , C believes that P is H with prob. 1; C does not pay on seeing tl , C believes that P is L with prob. 1; C pays 3) Back-up the game tree H’s payoff is Wh (T - th) therefore in eqm. th = 0 L’s payoff is Wl (T - tl) + d Proof of th = 0: Suppose not. Then th > 0. Consider th’ < th. . C cannot have more pessimistic beliefs, so the worse that can happen to P is that C does not pay. Even in this case, P is better off because he wastes less time lobbying. Thus, th cannot be optimal. If C does not pay given th’ , H is better off because Wh (T – th’) > Wh (T - th) If C pays, H is better off because Wh (T – th’) + d > Wh (T - th) QED. .

Incentive compatibility .

Check Off-the-path beliefs 5) Here the equilibrium concept does not specify off-the-path beliefs, so we choose them - usually, the easiest way to specify these beliefs is to assume that the receiver is highly pessimistic about sender’s type off-the-path. - if any t other than th or tl is received, C believes with prob. 1 that the sender is H - then C does not pay .

Efficiency . Typically in a separating equilibrium the worst type does not incur costs to signal that it is the worst type With n types we can order the cost of signaling by type and again the worst type does not incur costs to signal that it is the worst type

Pooling equilibrium (#75) Pooling equilibria exist under general conditions and are often not very interesting 1) Along the equilibrium path, type is not revealed: th = tl = t*. Posteriors equal priors: μ = q 2) C’s best response on observing t*, C’s payoff from paying is (1-q) A-d; C’s payoff from not paying is 0 C’s optimal choice: if (1-q) A-d < 0 then C does not pay if (1-q) A-d > 0 then C pays There are no incentive compatibility conditions because both types choose the same t 3) Check off-the-path beliefs: assume pessimisms for t ≠ t*, On observing any t ≠ t*, C updates beliefs that P is H w/ prob. 1 Given this, μ =1 and C does not pay .

Further Steps Both types must prefer t* to any other t 1st Case: (1-q) A-d < 0 and C does not pay In this case, the relevant condition is Wi (T - t*) ≥ Wi (T- t) for all t ≠ t*, which yields t ≥ t* Given this, t* = 0 This is intuitive: why spend anything to get nothing? 2nd Case: (1-q) A-d > 0 and C pays In this case, the relevant condition is Wi (T - t*) + d ≥ Wi (T- t) for all t ≠ t*, which yields d/Wh ≥ t* t* cannot be too high or P would prefer choosing t= 0 and not receiving d .

The WPBE WPBE: th = tl = t* 1st Case: (1-q) A-d < 0; th = tl = t* = 0 On observing 0, C updates to μ = q and does not pay On observing t ≠ 0 , C updates to μ = 1 and does not pay 2nd Case: (1-q) A-d ≥ 0 and d/Wh ≥ t* On observing t*, C updates to μ= q and pays On observing t ≠ t*, C updates to μ = 1 and does not pay Note that in the second case, the pooling equilibrium makes the senders worse off than in the case where no signal is possible. This is an example of a general result. Pooling equilibria generate no new information for the receiver, so if signaling is costly then the senders would be better off if no signaling is possible .