# Another Nash equilibrium

(v_{2}, v_{1}, 0, . . . , 0) is also a Nash equilibrium: Outcome: player 2 gets object at price v_{2}; all payoffs 0. Reason:

•

Player 1:

•

if she raises her bid to x < v

_{1 }she still loses•

if she raises her bid to x ≥ v

_{1 }she wins, and gets a payoff of 0•

Player 2

•

if she raises her bid or lowers it to x > v

_{2}, outcome remains same•

if she lowers her bid to x ≤ v

_{2 }she loses and gets 0•

Players 3, . . . , n:

•

if she raises her bid to x ≤ v

_{1 }she still loses•

if she raises her bid above v

_{1 }she wins, but gets a negative payoff v_{i }− v_{1}.

Player 2’s may seem “risky”—but isn’t if the other players adhere to their equilibrium actions.

N a s h e q u i l i b r i u m r e q u i r e s o n l y t h a t e a c h p l a y e r ’ s a c t i o n b optimal, given the other players’ actions. e

In a dynamic setting, player 2’s bid isn’t credible (why would she keep bidding above v_{2}?) [Will study this issue later.]

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