Estimating Value Distribution from Equilibrium Bids

We will look at the case of a single-item First Price Auction. Our goal will be to estimate the distribution of valuations of the players, by observing samples of bid vectors of players from multiple first price auctions. In particular, we consider the case where we have a data set of m First Price Auctions, each among n bidders. The value v i of each bidder i in each first price auction t, is drawn i.i.d. from the same distribution with CDF F (·) and density f(·) and support [0, v̄]. We will not observe these values v i , but we will instead be observing the bid of each player b t i. Moreover, we will be assuming that player’s play according to a symmetric Bayes-Nash equilibrium that corresponds to the value distribution F . In other words, the players know the distribution F and they have computed a symmetric Bayes-Nash equilibrium of a single item first price auction for that distribution. Such a Bayes-Nash equilibrium (BNE) is mapping from a value v ∈ [0, v̄] to a bid b(v), such that for all bi ∈ [0, v̄]: Ev−i [ui(b(v1), . . . , b(vn); vi)] ≥ Ev−i [ui(bi, b−i(v−i); vi)] (1)