Confused about what exactly constitutes an optimal auction

Question

When it comes to Bayesian single-item auctions the condition that is usually given for optimality is that the auction maximizes the seller's expected revenue in a Bayes-Nash equilibrium of the auction, given prior distributions. However, isn't it possible for some auction to have several equilibria, with different expected revenues? Why is it ok to assume that bidders will play according to any given Bayes-Nash equilibrium?

Answer 1

• Isn't possible for some auction to have several equilibria?

Yes. When analyzing auction mechanisms, most of times we focus on independent private valuation distribution and monotonically increasing, symmetric strategies. We could also focus on truthfull biding, that is, bidders bid their true valuation for the Single item.

• Isn't possible to have different expected revenues?

It depends. Let us restrict our analysis to sealed-bid auctions with independent private values and symmetric, monotonically increasing strategies. Let us restrict further to standard auctions, that is, (i) auctions where the one who bids the highest value gets the item and (ii) the expected payment made by a bidder with the lowest possible private valuation is zero. Then we can be sure that every possible auction mechanism and every possible strategies satisfying the criteria we set will generate the same expected revvenue for the seller. This is called the Revenue Equivalence Theorem.

This does not say that any possible auction will always generate the same expected revenue. For instance, if the seller could charge an entry fee for the bidders in a sealed-bid second-price auction, then it could be the case that his expected revenue would increase.

• Why is it ok to assume that bidders will play according to any given Bayes-Nash equilibrium?

Well, the rules of the auction, the prior distribution of private values, the bidding set of every player... is assumed to be common knowledge. We also assume they are payoff maximizers. If we accept these assumptions, the Bayes-Nash equilibrium seems a good prediction of how the game will evolve. Note that sometimes we do not even need to assume bidders will bid according a Nash equilibrium rightaway. For instance, in a second-price-sealed-bid auction, we find that truhful binding is the equilibrium strategy simply by elimination what rational players would NOT play, that is, dominated strategies, which is a weaker solution method than the Nash solution.