Question

Second price auction is quite similar to a first price auction(each player bids secrectly choosing a nonnegative real number) and each player i value the object v_iwhere v₁ > v₂ > ....>v_n > 0, except that the winner pays the amount of the second highest bid.

Prove that for player i, bidding v_i is a weakly dominant strategy. That is, prove that regardless of the actions of the other players, player i payoff when bidding v_i is at least as high as her payoff when making any other bid. Also, prove that there exist Nash equilibria in which player 1 does not win the auction.

Answer 1

In Nash equillibrium game theory bidding, there are "n" players.{1,2,.......n} participating in sealed bid auction. They all simultaneously submit their bids, and the player with the highest bid receives the prize, all other players receive nothing. Player who receive the Prize, pays the bid, other players don't pay anything.

If two players submit the same bid, and this bid is the highest, then the prize goes to the player with the lowest index. Let us understand the same with an example: There are two players- Player 1 and Player 2. Both of them bid the same amount of money, then the prize goes to the player 1. Because, Players value the objective differently. Player 1 values the object as v1 and v1 > v2>........vn.

In case of finite number of players, every player has an object and the player with the lower index values object higher than the player with higher index.i.e., Payment for the object < Valuation.If the player pays the same amount of money they value the object, they are indifferent between this amount of money and object. i.e., Payoff = 0 if the person doesn't receive the prize, or equals v1-b1, where b1 is the bid of player. Suppose, if the bid of the Player 1 is the highest then Payoff is v1-b1 [all other bids are <], or if out of all players with equal highest bids, "i" is the lowest. Otherwise Payoff is "0".

In a Nash equillibrium, no player has incentive to change their action, holding fixed the actions of the others. Here, actions are bids.

Well, suppose someone else wins. If it is an equilibrium, they must have bid no more than their valuation. Otherwise, they would be better off losing the auction and so they would have incentive to bid zero. But, if the winning bidder isn't player 1 and is bidding a value no higher than their own valuation, player 1 has incentive to raise his bid to just above the currently winning bid and win the auction at a price that gives him a positive surplus. Thus, in equilibrium, player 1 has to win.