Suppose Players 1 and 2 are participating in a first-price sealed bid auction wi

Question

Suppose Players 1 and 2 are participating in a first-price sealed bid auction with private, independent valuations. Each player's valuation of the object to be sold, which is assumed to be worth 0 to the seller, is drawn from a uniform distribution on [0, 1]. Each player knows her own valuation but only the probability distribution on the other player's valuation. Bids are restricted to be in [0, 1]. Remember bids are chosen simultaneously, the highest bidder wins and pays the amount of his bid. If two bidders bid the same amount, one of them gets the object with probability 0.5. (a) (10%) What is a strategy for a player in this game? (b) (15%) Bidder 2 decides to choose his bid b_2= .8v_2, where v_2 is his value. Consider bidder 1 and suppose her value is 0.8. Which is the best bid for her, knowing Bidder 2's bidding function, among these three bids (i) b_1 = .64, (ii) b_1 = .4, (iii) b_1 = .3. Assume she wants to maximize her expected payoff from the auction where she gets .8 - b_1 if she wins and 0 if she loses.

Explanation / Answer

A probability distribution is a mathematicsl function that, stated in simple terms, can be thought of as providing the probability of occurrence of different possible outcomes in an experiment. For instance, if the random variable x is used to denote the outcome of a coin toss ('the experiment'), then the probability distribution of x would take the value 0.5 for {display style x={ ext {heads}}} {display style x={ ext {heads}}}, and 0.5 for {display style x={ ext {trails}}} {display style x= { ext {trails}}}.

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