# Distinguishing between equilibria

For each player i the action v_{i }weakly dominates all her other actions

That is: player i can do no better than bid v_{i }no matter what the other players bid.

## Argument:

•

If the highest of the other players’ bids is at least v

_{i}, then•

if player i bids v

_{i }her payoff is 0•

if player i bids x = v

_{i }her payoff is either zero or negative.•

If the highest of the other players’ bids is b < v

_{i}, then•

if player i bids v

_{i }her payoff is v_{i }− 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.

# Summary

Second-price auction has many Nash equilibria, but the equilibrium (b_{1}, . . . , b_{n}) = (v_{1}, . . . , v_{n}) is the only one in which every players’ action weakly dominates all her other actions.

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