Consider a firstprice auction with three bidders whose valua

Consider a first-price auction with three bidders, whose valuations are inde- pendently drawn from a uniform distribution on the interval [0, 301. Thus, for each player i and any fixed number y E 10, 30], y 30 is the probability that player i \'s valuation vi is below y. (a) Suppose that player 2 is using the bidding function v) 3/4 and player is using the bidding function by v) (4/5) va. Determine player l\'s optimal bidding function in response. Start by writing player s expected payoff as a function of player l\'s valuation vi and her bid isregard the assumptions made in part (a). Calculate the Bayes- (b) ian Nash equilibrium of this auction game and report the equilibrium bidding functions.

Solution

In this case highest bid in a first place auction is considered as winner, so player 1 is the winner, he/she is having probability of y/30

SEcond player is 30*3/4=22.5

Third Player is 30 * 4/5=24

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b) There is an equilibrium in which all types bid half their valuation b(v) =1/2 v

(0,30),

Y/30 is probability

B(v) = 30

Half of that 0,30, and probability =y/30

a)bidding function b2(v2)= (3/ 4) v2

player 2 bids < ½ v,

30 * ¾= 22.5

y/22.5

Player 3 is using the bidding function b3(v3) =(4/5)V3

=30* 4 /5

=24

y/24

as per rules if player 2 bids more than ½ he /she certainly win because it is lower than all other players

It is payoff function maximum ==3b (v-b)

If v > ½ pay off function will be, b=1/2v while get maximize.

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