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Cournot’s model of oligopoly - page 32 / 38

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Nash equilibria of second-price sealed-bid auction

One Nash equilibrium

(b1, . . . , bn) = (v1, . . . , vn)

Outcome: player 1 obtains the object at price v2; her payoff is v 1 v 2 a n d e v e r y o t h e r p l a y e r s p a y o ff i s z e r o .

Reason:

  • Player 1:

    • if she changes her bid to some x b2 the outcome does not change (remember she pays the second highest bid)

    • if she lowers her bid below b2 she loses and gets a payoff of 0 (instead of v1 b2 > 0).

  • Players 2, . . . , n:

    • if she lowers her bid she still loses

    • if she raises her bid to x b1 she still loses

    • if she raises her bid above b1 she wins, but gets a payoff vi v1 < 0.

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