Class #4 Highlights

In this class, we studied decentralized solutions to the information problem. Specifically, we considered a situation where sellers were on the short side of the market, buyers had independent private valuations and unit demand. We studied the case where there was a single unit available though this is easily extended to the case where there are k units and n > k potential bidders.

Our solution concept for this game was a Bayes-Nash equilibrium. This is a strategy that is optimal in expectation and where expectations are formed using Bayes rule whenever possible. Specifically, given private information, v, a bidder's strategy is a schedule b(v).

In a first-price auction, we saw that bid shading (bidding below value) is optimal, where the degree of shading depends on the number of rival bidders. Specifically, we showed that a symmetric BNE consists of a bidding function:

b(v) = E[v_1:n-1 | v_1:n-1 <= v]

The expression v_1:n-1 is an order statistic, it denotes the highest value out of n-1 draws from the value distribution F. 

We performed a similar analysis for the second-price (Vickrey) auction. Here, we could use the stronger equilibrium concept of dominance to establish that b(v) = v is an equilibrium in weakly dominant strategies. The reason that truthful bidding is optimal is that an individual's bid only determines circumstances for winning or losing, it does not determine the price paid. 

Given this bidding strategy, the expected payment of the winning bidder is simply the highest rival value conditional on this being below the value of the winning bidder. That is, the expected payment of the winner whose valuation is v is:

p(v) = E[v_1:n-1 | v_1:n-1 <= v]

Notice that, under each auction, the payment of the winning bidder is the same. It therefore follows that the seller earns the same expected revenue under each auction form. Moreover, since the individual with the highest value wins in both cases, the allocation is efficient--no reallocation can produce a Pareto improvement. 

In a sense, the auction "solves" the information problem completely. Note, however, that we have assumed that the auctioneer simply chooses an auction with no reserve price. That is, there is no profit maximization motive imposed on the choice of auction. One can easily show that, given the option to impose a positive reserve price, a profit maximizing auctioneer can produce higher revenue by choosing a reserve. To see this, notice that, evaluated at the lowest possible bid, the efficiency losses from an infinitesimal increase in the reserve price are zero while the gains are strictly positive. Therefore, under profit maximization, we still obtain the familiar inefficiency of the solution to the information problem. As in the pricing case, this stems from market power on the part of the seller rather than the private information per se. Determining the optimal reserve price depends on the assumption that the seller knows the exact distribution of values, F.

Key things should be able to do after this class:

1. Compute a Bayes-Nash equilibrium of an auction.
2. Understand why truthful bidding is a dominant strategy in a Vickrey auction.