Question

Game theory: Consider a sealed-bid auction in which the winning bidder pays the average of the two highest bids. Assume that players have valuations v₁ > v₂ > … > v_n, that ties are won by the tied player with the highest valuation, and that each player’s valuation is common knowledge.

a. Is there any Nash equilibrium in which the two highest bids are different? If there is, give an example. If there is not, prove that no such equilibrium exists.

b. Is there any Nash equilibrium in which a player other than the one with the highest valuation wins? If there is, give an example. If there is not, prove that no such equilibrium exists.

c. Will bidding more than one’s own valuation be weakly dominated in this auction? Will bidding one’s own valuation exactly be weakly dominated? Will bidding less than one’s own valuation be weakly dominated?

d. What is the highest possible winning bid in any Nash equilibrium of this game? What is the lowest possible winning bid in Nash equilibrium?

Answer 1

a. In every Nash equilibrium a player with the highest valuation is the winner.

• If b_i > v_i , then player’s i payoff is negative and it increases to 0 if he submits the bid equal to v_i .
• If max_j≠_i v_j > bi , then player j such that v_j > b_i can win the object by submitting a bid in the open interval (b_i , v_j ), say v_j − . Then his payoff increases from 0 to .
• If b_i > max_j≠_i b_j , then player i can increase his payoff by submitting a bid in the open interval (max_j≠_i b_j , b_i), say b_i − . Then his payoff increases from v_i − b_i to v_i − b_i + .

So if any of the conditions (i) − (iii) is violated, then b is not a Nash equilibrium.

b. Suppose that a vector of bids b satisfies (i)−(iii).

Player i is the winner and by (i) his payoff is non-negative.

His payoff can increase only if he bids less, but then by (iii) another player (the one who initially submitted the same bid as player i) becomes the winner,

while player’s i payoff becomes 0.

The payoff of any other player j is 0 and can increase only if he bids more and becomes the winner.

But then by (ii), max_j≠_iv_j< b_j , so his payoff becomes negative. So b is a Nash equilibrium.

c. Choose an arbitrary sequence of bids that for the clarity of the argument we denote by (v_i , b−_i). Suppose that i = argsmax (v_i , b−_i. We establish the following four claims.

But even after that each strategy b_i ≠ v_i is weakly dominated by v_i , i.e., that v_i is a weakly dominant strategy.

As an aside, recall that each weakly dominant strategy is unique, so we characterized bidding one’s valuation in the second-price auction in game theoretic terms.

d. If 2 players submit the same bid, and this bid is the highest, then the prize goes to the player with lower index,

e.g. if both player₁ and player₂ bid the same amount of money, the prize goes to player₁. Players value the object differently, player_i valuation is v_i, and v1>v2>⋯>vn.