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Second Price sealed bid-auction: Assume n players are bidding in an auction in order to obtain an indivisible object. Denote by vi the value player i attaches to the object; if she obtains the object at the price p her payoff is vi −p. Assume that the players’ valuations of the object are all different and all positive; number the players 1 through n in such a way that v1 > v2 > · · · > vn > 0. Each player i submits a (sealed) bid bi . If player i’s bid is higher than every other bid, she obtains the object at a price equal to the second-highest bid, say bj , and hence receives the payoff vibj . If some other bid is higher than player i’s bid, player i does not obtain the object, and receives the payoff of zero. If player i is in a tie for the highest bid, her payoff depends on the way in which ties are broken. A simple (though arbitrary) assumption is that the winner is the player among those submitting the highest bid whose number is smallest (i.e. whose valuation of the object is highest). (If the highest bid is submitted by players 2, 5, and 7, for example, the winner is player 2.) Under this assumption, player i’s payoff when she bids bi and is in a tie for the highest bid is vi − bi if her number is lower than that of any other player submitting the bid bi , and 0 otherwise.
(a) Formulate this situation as a strategic (normal-from) game.
(b) Find a pure strategy Nash equilibrium in which player 1 obtains the object.
(c) Find a pure strategy Nash equilibrium in which player n obtains the object.
(d) Show that bidding own valuation is a weakly dominant strategy for each player.
(e) Find ALL pure strategy Nash equilibria of this game when n = 2.
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