Question

An all-pay auction is a type of auction which allocates the good to the player with the highest bid, and where all bidders pay their bid (even if they lose). Suppose that there are two bidders, with private valuations uniformly distributed on the interval 0 to 1. A bidder who wins the auction has a payoff of v_i ? b_i. A bidder who loses the auction has a payoff of ?b_i. a. Suppose that player 2 pays the bidding strategy b₂ (v₂) = ? (v₂)² , where ? is a positive constant we will solve for in a later part of the problem. If player 1 has a bid equal to b₁ what is the probability that it will win the auction?

b. If player 1 has a bid equal to b₁ and valuation equal to v₁, what are its expected payoffs?

c. Find player 1’s payoff-maximizing bidding strategy.

d. Using your answer to part c, solve for ?. (Hint: Your answer to part c should be b₁ = ?v₁² for some number ? > 0. Equate ? and ? to solve for the ? that we conjectured in part a.)

e. What are the seller’s expected revenues of this auction?

Answer 1

In an All-Pay Auction every bidder pays regardless of whether he wins or not so In this game In this case there are two bidders with price valuations in the interval [0,1]. No bidder out of the two will bid more than 1 Let us consider the player 1 bid is an amount equal to b1 and valuation equal to v2. Let us consider the player 2 bid is an amount equal to b2 and valuation equal to v2 As winner's Payoff is given by the probablity of wining the bid multiplied by their respective valuations and bidding amount or the winner is the highest bidder with maximum payoff given by v and the other player payoff will be given by -

As there is no pure strategy equilibrium of this game. So In order to find the Probablity distribution for the Winner or Player1 . A mixed strategy of a Player 1 as a probability distribution over the bids between 0 and 1. • A probability distribution function P =P(x) which is the probability that a player bids less than some amount between 0 and 1.

We look for a symmetric equilibrium. Let suppose that both Player1 and Player2 used the mixed strategy P and consider player 1 who bids His expected payoff is given by

In order for Player 1 to be willing to randomize we require

, As P is a probability distribution function It means P(0)=0 – Hence, q=0

Hence,

b.Expected payoff of the Player 1 with bid equal to and valuation will be zero as each player will be playing mixed strategy.

c. It is a type of Bayesian Nash Equilibrium and Player 1's payoff maximizing bidding strategy will be given by choosing the bid of an amount with valuation from interval [0,1]   such that it should be a best response to Player 2's strategy given by and vice versa  Or the pair of strategies ( constitutes a BNE, if for each , solves:

e. Seller's expected revenues of this auction will be equal to the value of the price.