Problem 1

Consider a sealed bid auction for an indivisible item among N bidders with the usual linear payoffs. Types (valuations) are private, players share a common prior that types are drawn independently from the uniform distribution over [0, 1].

1. Consider the second price auction. Ties are broken uniformly: if there are two or more highest bidders each gets the item with equal probability and, if she wins, pays what she bid (since the highest and second-highest bids coincide). Show that, just as in the complete information case, bidding ones value is weakly dominant for every type of every player.

2. Now consider the first price auction. Assume again a uniform tie-breaking rule: each highest-bidder gets the item with equal probability and, if she wins, pays her bid. Just as we did in class for the 2-bidder case, find a linear symmetric Bayesian Nash equilibrium in pure strategies.

3. Whats the probability of winning and expected payment of type vi 2 [0, 1] of player i in the equilibrium in weakly dominant strategies of the second price auction you found in the first point?

4. Whats the winning probability and expected payment of type vi 2 [0, 1] of player i in the linear symmetric Bayesian Nash equilibrium of the first price auction you found in the second point?

5. Find the expected revenue to the auctioneer in each case and comment.

It turns out that the revenue equivalence property you have found holds across a very large set of standard auction formats, the revenue equivalence theorem is a cornerstone and one of the most striking results of auction theory.

Problem 2

We will construct a model of bilateral trade to illustrate how, even if it is common knowl- edge that there are gains from trade, incomplete information may lead to a total breakdown of the market. This example is based on the classic market for lemons paper by Akerlof, which kick-started the study of market failures due to information frictions.

Theres two players, call them buyer and seller. The seller owns an indivisible good and wishes to sell it to the buyer. The goods quality is described by t 2 [0, 1]. Assume that a good of quality t is worth t money units to the seller and 1.5 t money units to the buyer. For example, if the good is a used car, the model says that both agents like cars of higher quality (since payoffs are increasing in quality) and a completely wrecked car (t = 0) is completely worthless to both. However, any car that is not completely wrecked (any car of quality t 2 (0, 1]) is worth more to the buyer than to the seller (since 1.5 t > t for all t 2 (0, 1]). For example, this may be because the buyer needs a car a lot (hes a commuter) while the seller doesnt really need that car (she has many other cars and does not commute, or maybe shes just a salesperson in a shop).

To start with, assume that the quality t 2 [0, 1] of the object is known to the two players. The trade protocol is in the spirit of the Nash demand game: both players simultaneously announce a number in [0,1.5]. If the sellers asked price is equal to or lower than the buyers offered price, trade occurs at a price in the middle of asked and offered prices. If the buyer offers strictly less than what the seller demands, trade does not occur. To summarize: if trade of a good of quality t occurs at price p (given by the midpoint of offered and asked prices), payoffs are p for the seller and 1.5 t p for the buyer. If trade of a good of quality t does not occur, payoffs are t for the seller and 0 for the buyer.

1. Model the problem above as a game in strategic form and find all pure Nash equilibria. 2. Now suppose that that the objects quality is the sellers private information1 and that the buyer holds a uniform prior over the quality of the object (i.e. over the sellers type). Observe that trading is still always Pareto efficient (i.e. no matter what the quality is, it

is efficient if the good is exchanged). The trade protocol is the same as before.Model the problem as a Bayesian game and prove that, in any Bayesian Nash equi- librium, the seller never offers more than 0 so that the probability of trade is 0: trade completely breaks down. Tip: for any candidate equilibrium with the offered price strictly above 0 it is possible to derive a contradiction. Remember that a pure strategy in a Bayesian game associates an action to every type.

3. Recall that for all qualities above 0 the object is worth more to the buyer than to the seller, furthermore this is common knowledge among the players. Do you think that if the two could talk before the game the problem would be resolved?

advanced game theory exercises about bayesian nash equilibrium

advanced game theory exercises