Download Beer-Quiche Game: Understanding Information Asymmetry and Equilibria and more Lecture notes Political Economy in PDF only on Docsity!
Section 12
Akos Lada∗
Fall 2012
- Adverse Selection
- Signaling with Discrete Signals
- Signaling with Continuous Signals
- Example with Independent Certification
- Today’s topics all revolve around asymmetric information about agent types
- We assume that information is not perfect, some actor does not have full information about what kind of opponent/product they are facing
- First we will see what happens if the actor with perfect information has no way of conveying information to the one without it - this may be mutually detrimental
- Then we will see that a signal or an observable action may reveal information, but need to remember that actors act strategically
- Finally, we will investigate how hiring an outside independent source of verification affects equilibria
1 Adverse Selection
- Adverse selection is probably best illustrated through an example ∗Third-year student in the Political Economy and Government Program at Harvard University. Section notes draw on previous years’ sections. Special thanks to Zhenyu Lai, as well as Chris Avery’s notes on signaling.
- Assume a buyer does not know whether a car is good (with probablity q) or bad (with probability 1 − q)
- A good car is worth $3000, a bad car is worth $2000 to the buyer, while to the seller a good car is worth $2500 and a bad car is worth $
- The seller knows the type of the car
- The buyer does not know the type and offers p to buy the car
- If the offer is p ≥ 2500 then both good and bad cars would be sold so the buyer expects a payoff of 3000q + 2000(1 − q) − p
- If the offer is 2500 > p ≥ 1000 then only bad cars are sold so the buyer expects a payoff of 2000 − p (conditional on getting the car)
- If the offer is 1000 > p then no seller sells the car so the expected payoff is 0
- In a rational expectations equilibrium of this game we look for a price offer p where the buyer knows (rationally expects) which types of cars are sold at any price offered
- For the 1000 > p case, we can have no trade with positive probability
- For the 2500 > p ≥ 1000 case only bad cars are sold, which is worth $2000 to the buyer, so the only equilibrium is when 2000 ≥ p ≥ 1000
- The interesting case really is whether a p ≥ 2500 equilibrium is sustainable since this is the one where both types of cars are sold. Note that this type of equilibrium is only sustainable if 3000q + 2000(1 − q) ≥ p(≥ 2500), or when q ≥ (^12)
- In words what this means is that for a buyer to make an offer p in a rational expecta- tions equilibrium we need the exogenously given probability of having a high type of car on the market to be at least (^12)
- Note that both types of cars are worth more to the buyer than the seller so the economically efficient outcome should involve trade in both types, yet if q < 12 we can find no such equilibrium because of the buyer’s informational disadvantage
- Thus the information asymmetry makes the market break down
- Player 2’s payoffs are such that if he knew player 1’s type perfectly then he would like to back down (concede) facing a strong opponent, but would like to fight a weak opponent: fighting a weak type yields payoff 1 to player 2, backing down with a strong type similarly yields a payoff of 1 to player 2, while all else yields a payoff of 0
- Player 1 gains 2 if player 2 concedes and 0 if player 2 fights, regardless of type
- The dotted lines in the game tree denote information sets for player 2: when choosing an action at some node he does not know whether he is at the true node or some other node in the same information set (e.g. when he observes beer, he does not know whether a strong or a weak type played beer - as long as both types drink beer)
- Assume that having a hearty quiche breakfast is better for the weak type of player 1 but drinking the beer is better for the strong type: choosing the preferred breakfast increases payoff by 1 unit to player 1, regardless of player 2’s actions
- Making payoffs to player 1 dependent on breakfast choice has created a potentially informative signal - his breakfast choice
- If player 2 was not sitting across the hall, unable to observe the breakfast choice both types of player 1 would go for their favorite breakfast
- However now a weak player 1 has a motivation to drink beer so that player 2 does not learn the type of player 1 and decide to fight after observing quiche as breakfast choice
- So player 1 is a ‘sender’ of a message to player 2 by the breakfast choice, while player 2 is the ‘receiver’ of this message
- We will look for a Perfect Bayesian Equilibrium: this equilibrium concept is a subset of Subgame Perfect Nash Equilibria, and includes information asymmetries, roughly each player’s actions need to be compatible with their beliefs
- A Perfect Bayesian equilibrium needs to specify beliefs at decision nodes - this is part of the equilibrium
- In a Perfect Bayesian Equilibrium each player who has multiple types (player 1) has to pick a strategy that fully specifies what each of his types would do (think of this strat-
egy being picked before nature moves), and his actions are characterized by ‘sequential rationality’ which excludes incredible threats like the subgame perfect equilibrium did
- Player 1’s strategy is thus for instance (‘beer’, ‘quiche’): drinking beer when strong and eating quiche when weak
- We restrict attention to pure strategies for simplicity
- In a pooling equilibrium all types choose the same action, i.e. send the same message (in our example the weak and strong both choose to eat quiche, or the weak and strong both drink beer)
- In a separating equilibrium all types choose distinct actions, i.e. all send different messages (in our example the weak chooses quiche while the strong chooses beer; or the weak chooses beer while the strong chooses quiche)
- In an equilibrium the receiver of the signal updates his beliefs about which type of sender he faces after observing the signal using Bayes’ rule
- In a separating equilibrium in the beer-quiche game the receiver perfectly learns the type of the sender
- The problem is that at decision nodes which are off the equilibrium path, Bayes’ rule does not help: here a Perfect Bayesian Equilibrium allows any kind of beliefs
- There are also partially separating/pooling equilibria in which some senders send the same message, others send the same message, in our pure-strategy Bayesian Equilibria such a case is not possible
- To analyze our game define ˆpW = Prob(Player 1 is weak| quiche) or the posterior probability of player 1 being weak after the message sent was quiche
- Notice that the ˆpW depends on both types of player 2’s choices:
pˆqW = Prob(Player 1 is weak| quiche) =
= (^) Prob(Quiche|Player 1 is weak)Prob(QuichepW + Prob(Quiche|Player 1 is weak)|Player 1 is strong)(1pW − pW )
case a separating equilibrium would exist because imitating is too costly for the weak type
- How about pooling equilibria?
- Pooling Equilibria:
- First consider the equilibrium in which both types of player 1 choose to drink beer
- In this equilibrium the best response of player 2 depends on his beliefs after seeing beer, knowing both types play beer
- Therefore the posterior is the same as the prior now: seeing beer reveals nothing new about the probability of player 1 being the weak type, remember that this prior is common knowledge
- Player 2 plays fight if ˆpbW = pW > 0 .5 (the weak type is more likely than the strong) and plays concede otherwise
- We can see that in any beer-beer pooling equilibrium player 2 needs to pick concede after seeing beer, otherwise the weak player would be better off deviating to quiche whatever happens in that case: 0 < 1 , 3
- This means that beer-beer involves player 2 conceding after beer and such an equilibrium can only exist if pW ≤ 0. 5
- Do not forget that equilibria need to specify off-equilibrium path behavior and beliefs as well, so we need to answer the question: what would player 2 do if either type picked quiche instead?
- If ˆpbW = pW ≤ 0 .5 then deviating to pick quiche when the type is weak will only not happen if player 2 picks fight after observing quiche, otherwise 1 > 0, so the weak type would have an incentive to eat quiche, knowing that fighting could be thus be avoided, so the equilibrium would break down
- Thus we know that in any beer-beer pooling equilibrium and pW ≤ 0 .5 we need player 2 to play fight when he sees beer and to play concede when he sees quiche
- Note that the quiche node is never reached in this equilibrium, so the beliefs of player 2 when he sees quiche could be anything, to support this equilibrium, his belief needs to be such that ˆpqW ≥ 0 .5 so that fighting is a best response
- So player 1’s weak type is playing a best response, player 2 is playing a best response given his beliefs, all we need is player 1’s strong type playing a best response
- As long as player 2 would pick fight after seeing quiche but concession after beer, the strong type has no incentive to deviate either: 3 > 0, so such a beer-beer pooling equilibrium can exist with pW ≤ 0 .5 and pqW ≥ 0. 5
- How about a quiche-quiche pooling equilibrium? Similar considerations mean that the strong type will only be willing to play this equilibrium if quiche leads to concessions so that ˆpqW = pW ≤ 0 .5, while beer would lead to fighting: ˆpbW > 0 .5. In this equilibrium therefore both types eat quiche and player 2 concedes after quiche and fights after beer
- We have these pooling equilibria when either type is willing to eat a less preferred breakfast as long as he gets concession instead of fighting as a ‘reward’
- When pW ≥ 0 .5 we did not find any pure strategy Bayesian equilibria (when there are too many weak types, a pooling equilibrium does not make the second player pick fight so the incentive to pool would disappear) but there are some in mixed strategies
- In mixed strategies there are a lot of possibilities. In particular, when player 1 mixes between playing beer (probability p) and quiche when weak but playing only beer when strong, while player 2 always fighting quiche but mixing between fighting (probability q) and conceding when seeing beer gives us one equilibrium: for player 2 to be willing to mix after beer he needs: 1 (^) pW p+(1pW^ p−pW ) + 0 (^) pW p^1 +(1−pW−pW ) = 1 (^) pW p^1 +(1−pW−pW ) + 0 (^) pW p+(1pW^ p−pW ) or p = 1 − pWpW , which when pW > 12 is a valid probability, and the weak player 1 is only willing to mix if: 0q + 2(1 − q) = 1 or q = (^12)
- Note that if you perturb the game so that drinking beer is costly for both types, with cWB > cSB > 1 then we can have a separating equilibrium for any probability. In this setup under full information a strong type would choose quiche since 2 > 3 − cSB. However with asymmetric information he is willing to drink beer because that costly message is less costly to the strong type and the weak type would not imitate him since quiche followed by fighting yields 1 while beer followed by concession yields 2 −cWB < 1. In this set-up we also have that drinking beer is not too costly for the strong type so
- To solve, make two observations. First, that in a separating equilibrium all information about types are revealed so competition between firms sets w(eH ) = θH and w(eL) = θL
- Second, that in any separating Perfect Bayesian Equilibrium eL = 0 since the low type is revealed, so there is no reason to waste energy on education for the low type
- This means the two incentive compatibility constraints are such that:
θH − CH (eH ) ≥ θL,
and θL ≥ θH − CL(eH ), which should give us an eH ∈ [C L− 1 (θH − θL), C H− 1 (θH − θL)] and together with eL = 0; this gives an equilibrium (a range of possible equilibria, to be more precise)
- Note that firms know the type of the worker in this equilibrium if they see eH and eL, but their beliefs off the equilibrium education levels can be anything that supports this equilibrium
- There are different equilibria refinement criteria that specify certain off-equilibrium beliefs, for instance that beliefs at an off-equilibrium node only put positive weight on types for whom that action is not a strictly dominated action
- The only equilibrium satisfying the dominance refinement^1 is the one with eH = C L− 1 (θH − θL) because in any other equilibrium the low type deviating to eH is a dominated action,^2 so firms would know a deviator must be a high type and so they would need to offer the high wage, making it possible for the high type to earn the same wage with lower education levels (^1) A Perfect Bayesian Equilibrium satisfies the dominance requirement if and only if beliefs are such that they put positive probability on any type only if it is not a strictly dominated action for that type to play that action (if the given action is dominated for all types, there are no restrictions). 2 To see this notice that at this point cL(eH ) = θH − θL the low type is exactly indifferent between choosing 0 education and θL wage or high education eH and θH wage. Which also means that if eH was at any higher level θH − cL(eH ) would be strictly less than θL(−cL(0)) so the low type strictly would prefer his own wage schedule to picking education eH > c− L^1 (θH − θL). The dominance refinement then means that in any equilibrium where eH > c− L^1 (θH − θL) when firms see a deviation to e′, where eH^ > e′^ > c− L 1 (θH − θL), they must be thinking it is coming from a high type, so a high wage should be offered. But in that case high types can get the same θH wage with a lower education so they would deviate to e′. Therefore only when eH^ = c− L 1 (θH − θL) would the dominance refinement not rule out the separating equilibrium.
- There are also pooling equilibria, which we do not consider now
- If you write up pooling equilibria, you may want to consider a stronger equilibrium refinement: the intuitive criterion, which specifies that beliefs should not put positive weight on actors for whom an action leading to the node was either strictly dominated or would lead to a strictly lower payoff than his equilibrium one regardless of other’s behavior - this would actually rule out all pooling equilibria
- In the beer-quiche game the intuitive criterion rules out the pooling equilibrium ‘quiche-quiche’ (recall that player 2 must respond by concession to quiche and fighting to beer in any such equilibrium) because in that case a weak type is getting a payoff of 3, which is his strictly maximal payoff, so if the second player saw a deviation to ‘beer’, he should know it cannot be the weak type, so would realize it is the strong type, in which case he would concede rather than fight; the ‘beer-beer’ equilibrium survives the restriction of the intuitive criterion since any deviation to ‘quiche’ now should be interpreted as coming from a weak player, which player 2 would indeed want to fight
4 Information Asymmetry with Certification: Ex- ample
Roses get a lot of love, but Dotty Gael is in the business of rubies. Ruby certification, that is. Her company will report a ruby to be either “Authentic” or “Fake” at a cost of c per ruby. Assume that the actual value of a ruby to consumers v is distributed uniformly from 0 to 100, and that Dotty’s firm certifies a ruby as “Authentic” if it has a value of 50 or more, and “Fake” otherwise. She provides no further information with certification, but her rubric is public information.
The price a retailer pays to buy rubies is d, which does not depend on the ruby’s quality. The retailer knows the true value of its rubies, and chooses whether or not to buy a ruby and then whether or not to certify it based on that true value. The retailer also functions as a local monopolist, and sets the purchase price for each ruby offered to consumers. Assume consumers are effectively risk-neutral for this kind of
or equivalently,
c ≤ 50 c + d ≤ 75.
We can now identify the possible equilibria that result.
(a) If the above conditions hold and d < 25, then Authentic rubies will be certified and sold at a price of 75, while Fake rubies will be sold at a price of 25. (b) If the above conditions hold and d > 25, then Authentic rubies will be certified and sold at a price of 75, while Fake rubies will not be sold at all. (c) If the above conditions do not both hold and d < 50, all rubies will be sold (uncertified) at a price of 50. (d) If the above conditions do not both hold and d > 50, then no rubies will be sold at all.
