Question

Consider a single-item auction with n bidders where n ≥ 3. The highest bidder wins the auction and pays the amount of the third highest bid. Show that this auction is not dominant strategy incentive compatible

Answer 1

In game theory for any game to be dominant strategy compatible it needs to have a strategy profile such that every player is playing his/her best strategy and have no incentive to deviate from his/her strategy.

In this single item auction game, there are n bidders and each one of them bids an amount Xi where i=1,2,3,.......,n. Let's suppose every bidders bids X1, X2, X3...... Xn such that X1>X2>X3>.......... Xn-1> Xn.

In this Case player 1 has the highest bid and wins the auction but is this dominant strategy compatible. Let's see, player 2 has an incentive to bid more than player 1 and win the auction. So this can't be nash equilibrium with dominant strategy. So this strategy profile is not dominant compatible.

Now consider second case in which every player bids equal amount. Such that X1=X2=X3=....... =Xn. Is this strategy profile dominant strategy compatible in this auction game?

Let's see, any player can make himself better off by bidding extra amount and by winning auction. So every player has an incentive to deviate from his/her strategy so this can't be compatible dominant strategy.

The thing is, in this each player's payoff depends on how much other players are bidding. Every player will like outbid other players since we don't know their valuation for the good hence they will to any extent to win auction. So there cannot be a compatible dominant strategy profile such that no player has an incentive to deviate from that strategy profile.