Question

Private Value First Price Auction
There are 4 bidders in a first price auction: player 1, 2, 3 and 4.
You are player 1 and your private value of the object is 0.8. You believe that the values of player 2, 3, and 4 are uniformly distributed between [0,1]. If you win by bidding b, your payoff is 0.8-b.
Your opponents’ bidding strategy can be represented by av_i, i=2,3, with v_i the corresponding private value of each of your opponents, respectively, and 0<a<1.
What is your NE bidding strategy? (20 pts)

Answer 1

• I will never bid more than 0.8 because bidding more than 0.8 can only make me lose net value.
• If I bid exactly 0.8, then I will not lose but also not gain any positive value.
• If I bid less than 0.8, then I may have some positive gain, but the exact gain depends on the bids of the others.

My NE bidding strategy depends on:

Probability of winning = Probability[my bid > max {bid (pl. 2), bid(pl.3), bid(pl. 4)}]

Since the value of other players are uniformly distributed,

my bid = my value - Integral from 0 to 1 {F^(n-1) (x)/ F^(n-1) (v)}

Substitutingn the values: v = 0.8, n=4 , F = (x-a)/(b-a)

My NE equilibrium bid = 0.7