Question

N players are bidding on an object in a first price auction. The object has a value of v_i for each player i, where v₁ > v₂> ... >v_n> 0. Each player bids secretly choosing nonnegative real number. The winner is the player who bids the largest number, and that player must pay the amount they bid. If it tie, then the player with the lowest index wins. Formulate this situation as a strategic game( describe the players, actions, and payoff functions) and show that in all the Nash equilibrium, player 1 wins the auction.

Answer 1

Solution:

• 1. Player is bid is the highest and there are such bids he/she gets (vi – p(b))/m, then highest bidder gets 0

  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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