First price auctions: Monotonicity and uniqueness

I study monotonicity and uniqueness of the equilibrium strategies in a two-person first price auction with affiliated signals. Existing results establish uniqueness within the class of non-decreasing bidding strategies. I show that there is an effectively unique Nash equilibrium within the class of piece-wise monotone strategies. The main result is that in equilibrium, the strategies must be strictly increasing within the support of winning bids. This result provides the missing link for the analysis of uniqueness in two-person first price auctions. The analysis applies to asymmetric environments as well and does not require risk neutrality.