Distinguishing between equilibria
For each player i the action vi weakly dominates all her other actions
That is: player i can do no better than bid vi no matter what the other players bid.
If the highest of the other players’ bids is at least vi, then
if player i bids vi her payoff is 0
if player i bids x = vi her payoff is either zero or negative.
If the highest of the other players’ bids is b < vi, then
if player i bids vi her payoff is vi − b (she obtains the object at the price b)
if player i submits some other bid then she either obtains the good and gets the same payoff, or does not obtain the good and gets the payoff of zero.
Second-price auction has many Nash equilibria, but the equilibrium (b1, . . . , bn) = (v1, . . . , vn) is the only one in which every players’ action weakly dominates all her other actions.