Nash equilibrium of first-price sealed-bid auction Show that (b 1 , …, b n ) = (v 2 , v 2 , v 3…

Nash equilibrium of first-price sealed-bid auction Show that (b1, …, bn) = (v2, v2, v3,…, vn) is a Nash equilibrium of a first-price sealed-bid auction. A first-price sealed-bid auction has many other equilibria, but in all equilibria the winner is the player who values the object most highly (player 1), by the following argument. In any action profile (b1,…, bn) in which some player i = 1 wins, we have bi > b1. If bi > v2 then i’s payoff is negative, so that she can do better by reducing her bid to 0; if bi ≤ v2 then player 1 can increase her payoff from 0 to v1 − bi by bidding bi, in which case she wins. Thus no such action profile is a Nash equilibrium.