Asymmetric Nash equilibria of second-price sealed-bid common value auctions:-

Show that when α = γ = 1, for any value of λ > 0 the game studied above has an (asymmetric) Nash equilibrium in which each type t1 of player 1 bids (1 + λ)t₁ and each type t₂ of player 2 bids (1 + 1/λ)t₂.

Note that when player 1 calculates her expected value of the object, she finds the expected value of player 2's signal given that her bid wins.

If her bid is low then she is unlikely to be the winner, but if she is the winner, player 2's signal must be low, and so she should impute a low value to the object. She should not base her bid simply on an estimate of the valuation derived from her own signal and the (unconditional) expectation of the other player's signal. If she does so, then over all the cases in which she wins, she more likely than not overvalues the object.

A bidder who incorrectly behaves in this way is said to suffer the winner's curse. (Bidders in real auctions know this problem: when a contractor gives you a quotation to renovate your house, she does not base her price simply on an unbiased estimate out how much it will cost her to do the job, but takes into account that you will select her only if her competitors' estimates are all be higher than hers, in which case her estimate may be suspiciously low.)

Nash equilibrium in a first-price sealed-bid auction I claim that under the assumptions on the players' signals and valuations in the previous section, a first-price sealed-bid auction has a Nash equilibrium in which each type t_i of each player i bids ½ (α + γ)ti. This claim may be verified by arguments like those in the previous section. In the next exercise, you are asked to supply the details.