Chapter Signaling, Screening, and Sequential Equilibrium 2-Player Signaling Games Informed and uninformed players Let the buyer beware Three varieties of market failure, and one variety of market success 2
Two-Player Signaling Games A good item is worth V to the buyer A bad item is worth W to the buyer Any item, either good or bad, is offered at price p V > p > W The seller of a bad item must pay a cleanup cost, c 3 Caveat Emptor, extensive form p(good) p(bad) 2 g b p, V - p p - c, W - p -c, 4
Caveat Emptor, playing the game Price of the car: $4 Value: good car (red card) $6 bad car (black card) $3 Clean-up cost: $2 5 Caveat Emptor, action sheet Name: 2 3 4 5 good/bad action taken payoff 6
Total Market Failure All sellers, even the good-type sellers, fearing rejection by the buyers, withhold their goods from the market. The market ceases to function, even though gains from trade are available. An equilibrium where all informed players do the same thing is called a pooling equilibrium. Total market failure is an especially sinister pooling equilibrium. 7 Total Market Success Only sellers with good items offer them for sale. Since all items offered for sale are good, buyers buy everything offered for sale and the market works perfectly. An equilibrium where the types of informed players do different thing is called a separating equilibrium. Here, the very act of offering the item for sale signals to the buyer that it is a good type. 8
Partial Market Success All sellers offer their items for sale, good or bad. All buyers buy whatever is offered for sale. This is only a partial success: the market functions, but there are a lot of bad deals, which reduces market efficiency. This is another example of pooling equilibrium. Unlike total market failure, however, this pooling equilibrium does generate some gains from trade. 9 Near Market Failure Some, but not all, bad-type sellers offer their items for sale. Buyers buy what is offered for sale with a certain probability, and reject what is offered with some probability. Thus, both buyers and bad-type sellers adopt a mixed strategy response to the imperfect information. In this market, total gains from trade are smaller than in complete or partial market success.
Sequential Equilibrium: Pure Strategies Sequential equilibrium as an extension of subgame perfect equilibrium Sequential equilibria that pool informed players Sequential equilibria that separate informed players Partial market success, reaching player 2 s information set g 2 p(good) b p(bad) 2
Bayes Rule Suppose, in a sequential equilibrium: offers an item for sale no matter the item is good or bad By conditional probability, p(good offer) = p(good) p(offer good)/p(offer) p(offer) = p(good) p(offer good) + p(bad) p(offer bad) = Also, p(offer good) = p(offer bad) = p(good offer) = p(good) / = p(good) Similarly, p(bad offer) = p(bad) This is known as Bayes Rule 3 Partial market success, backward induction, player s move p, V - p p(good) p - c, W - p p(bad) 4
Sequential equilibrium when p > c The following is a sequential equilibrium: offers the item for sale if it is good offers the item for sale if it is bad 2 buys whatever that is offered for sale p(node g offer) = p(good) and p(node b offer) = p(bad) p(node g offer) = p(good offer) / (p(good offer) + p(bad offer)) = p(good) / (p(good) + p(bad)) = p(good) Similarly, p(node b offer) = p(bad) 5 Sequential equilibrium when p > c Eu 2 = p(node g offer) (V - p) + p(node b offer) (W - p) = p(good offer) (V - p) + p(bad offer) (W - p) = p(good) (V - p) + p(bad) (W - p) Require p(good) (V - p) + p(bad) (W - p) > p(good) must be sufficiently large. This is partial market success 6
Complete market success, reaching player 2 s information set 2 g p(good) b p(bad) 7 Complete market success, backward induction, player s move p, V - p p(good) p - c, W - p p(bad) 8
Sequential equilibrium when c > p The following is a sequential equilibrium: offers the item for sale if it is good stops if the item is bad 2 buys whatever that is offered for sale p(node g offer) = and p(node b offer) = p(node g offer) = p(good offer) = p(good) p(offer good) / p(offer)= p(good) / p(offer) p(offer) = p(good) p(offer good) + p(bad) p(offer bad) = p(good) p(node g offer) = p(good) / p(good) = Similarly, p(node b offer) = p(bad offer) = 9 Sequential equilibrium when c > p Eu 2 = p(node g offer) (V - p) + p(node b offer) (W - p) = p(good offer) (V - p) + p(bad offer) (W - p) = p(good) (V - p) + p(bad) (W - p) = (V - p) + (W - p) > This is complete market success 2
Complete market failure, backward induction, player s move p(good) p(bad) -c, 2 Sequential equilibrium when we have complete market failure The following sequential equilibrium is not based on information about the items and leads to complete market failure: stops whatever the item is 2 says no to any item offered for sale p(node g offer) = and p(node b offer) = Eu 2 = p(node g offer) (V - p) + p(node b offer) (W - p) = (V - p) + (W - p) < This is complete market failure 22
Sequential Equilibrium: Mixed Strategies Sequential equilibria for markets on the verge of total failure A sequential equilibrium that partially separates and partially pools informed players A regime diagram for sequential equilibria 23 Caveat Emptor, near market failure.5.5 2 g b 2,, -2-24
Sequential equilibrium for near market failure In this case, p > c and Eu 2 = p(good) (V - p) + p(bad) (W - p) < Suppose, p(good) = p(bad) =.5, V = $3, p = $2, W = and c = $ p(good) (V - p) + p(bad) (W - p) =.5 +.5-2 < The following is a sequential equilibrium: offers for sale if the item is good offers for sale with probability.5 if the item is bad 2 buys any item with probability.5 p(node g offer) = 2/3 and p(node b offer) = /3 25 Verifying the sequential equilibrium for near market failure p(node g offer) =.5 / (.5 +.25) = 2/3 p(node b offer) =.25 / (.5 +.25) = /3 Eu 2 (buy offer) = p(node g offer) + p(node b offer) (-2) = 2/3 + /3 (-2) = Eu (offer good) =.5 2 +.5 = > Eu (offer bad) =.5 +.5 - = 26
Caveat Emptor, regime diagram Pr(good)(V - p) + Pr(bad)(W - p) Partial market success Near market failure Complete market success c c = p 27 The market for Lemons When price signals quality, and when it fails to signal quality A market fails when only lemons are offered for sale Examples from health insurance and used cars 28
Lemons p(good) p(bad) High price High price Low price Low price 2 2 g b g b p, V - p p - c, W - p -c, q, V - q q, W - q 29 The Market for Lemons In a Market for Lemons, A good item is worth V to the buyer A bad item is worth W to the buyer An item is offered at either a high price p or a low price q Where, V - q > V - p > W - q > > W - p The seller of a bad item must pay a cleanup cost, c 3
Sequential Equilibrium in a market for Lemons when c > p The following is a sequential equilibrium: charges a high price with a good item charges a low price with a bad item 2 buys any item offered for sale p(node g high price) = p(node b low price) = In this case, thanks to the separating sequential equilibrium, Lemons has a complete market success solution. 3 Sequential Equilibrium in a market for Lemons when c = In this case, a bad item can mimic a good item for free and price no longer indicates quality A buyer s expected value at either price is negative: Eu 2 (buy high price) = p(good)(v - p) + p(bad)(w - p) < Eu 2 (buy low price) = p(good)(v - q) + p(bad)(w - q) < In the very worst case the breaks down completely: charges a low price with a good item 2 does not buy at either price p(node g low price) = p(good) p(node g high price) = p(good) 32
The lemons principle The bad drives everything out of the market 33 Costly Commitment as a Signaling Device The principle of costly commitment Money-back-guarantees as costly commitments to solve the lemons problem Costly commitments in all walks of life 34
Money-Back Guarantee p(good) p(bad) High price High price Low price Low price 2 2 g b g b p, V - p P + W - V, V - p q, V - q q, W - q 35 Sequential Equilibrium when the money-back guarantee is costly enough The bad-type seller has to reimburse the unlucky buyer V - W If p - W - V <, the following sequential equilibrium leads to complete market success: charges a high price with a good item charges a low price with a bad item 2 says yes to a high price 2 says no to a low price p(node g high price) = p(node g low price) = 36
Screening Games Uninformed player moves first Offer of contracts by uninformed player Subgame perfection Menu of contracts to achieve full market success Existence of separating equilibrium 37 Screening game yes v, offer contract do not offer contract p(good) p(bad) no yes no -v, 38
Backwards induction in the screening game Both buyers accept insurance contract since > for each. Insurance company has expected value: E(V ) = p(good)v + p(bad) (-v) = [p(good) p(bad)]v Hence insurance company offers contract if expected value is positive, or p(good) > p(bad) and does not offer contract if the expected value is negative, p(good) < p(bad) 39 Backwards induction in the screening game Full market success needs more complexity. Insurance company has two types of contract: contract I: low premium, large deductible contract II: high premium, low deductible Then there could a subgame perfect equilibrium: insurance offers both contract good risk player: accept contract I, reject contract II bad risk player: reject contract I, accept contract II 4
Barbarians at the Gate The power of inside information in corporate takeovers Buying a corporation and buying a used car Bidding behavior that reveals an underlying signal The potential lemons problem in corporate takeovers 4 Repeated Signaling and Track Records isy signals and unknowable player types Bayesian updating during the probationary period Detailed calculations for one-period probation The trade-off between longer probation and the opportunity cost of ordinary performance 42
One-Period Probation Star.5 Ordinary.5 Win.9 Win.5 Lose. Lose.5 Star Ordinary win Star Ordinary Lose 8 3 3 8 3 3 43 One-Period Probation p(star) = p(ordinary) =.5 and discount factor =.95 When the prospective partner is a star performer, EV = Σ(.95) t + [.9 +. ] =.9 / (-.95) = 8 When the prospective partner is a ordinary performer, EV = Σ(.95) t + [.5 +.5 ] =.5 / (-.95) = When you let the prospective partner go, EV = [.5 (8 - ) +.5 ( - )] = 3 44
Deciding on performance in one trial Strategy : no matter what EV = [.5 8 +.5 ] = 4 Strategy 2: if a win, if a loss p(star win) = p(star) p(win star) / p(win) = (.9.5) /.7 = 9/4 p(ordinary win) = - p(star win) = 5/4 p(star loss) = p(star) p(loss star) / p(loss) = (.5.) /.3 = /6 p(ordinary loss) = - p(star loss) = 5/6 EV =.7 [9/4 8 + 5/4 ] +.3 3 = 4.5 45 Deciding on performance in one trial Strategy 3: if a loss, if a win EV =.7 3 +.3 [/6 8 + 5/6 ] = 2.5 Strategy 4:, no matter what EV = [.5 (8 - ) +.5 ( - )] = 3 46
Probability Ten-Period Probation: ordinary performer.25.2.2.2.2.4.4 <. <. 2 3 4 5 6 7 8 9 Number of wins 47 Ten-Period Probation: star performer Probability.39.35.9.6. 2 3 4 5 6 7 8 9 Number of wins 48
Hit and Run: Track Records in Hollywood Sony purchase of Columbia movie studio Asymmetric information in hiring directors Jon Peters and Peter Gruber Use of past performance as signal 49