# Property of all equilibria

In all equilibria the object is obtained by the player who values it most highly (player 1)

## Argument:

•

If player i = 1 obtains the object then we must have b

_{i }> b_{1}.•

But there is no equilibrium in which b

_{i }> b_{1}:•

if b

_{i }> v_{2 }then i’s payoff is negative, so she can do better by reducing her bid to 0•

if b

_{i }≤ v_{2 }then player 1 can increase her payoff from 0 to v_{1 }− b_{i }by bidding b_{i}.

# Another equilibrium

(v_{1}, v_{1}, v_{3}, . . . , v_{n}) Outcome: player 1 obtains the object at the price v_{1}.

As before, player 2’s action may seem “risky”: if there is any chance that player 1 submits a bid less than v_{1 }then there is a chance that player 2’s payoff is negative.

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