Question

Two players compete in an auction to win an object. Players’ valuation for the object are different: player i’s value for winning the object is vi , for i = 1, 2, with v1 > v2 > 0. These values are publicly known. The rules of the auction are the following: (a) Each player submits a bid bi ∈ [0,∞) to the auctioneer. (b) The player who makes the highest bid wins the object and pays the auctioneer bi . The player who does not win the object, does not pay anything. (c) If the two players submit the same bid, then the auctioneer gives the object to player 1. Find all the Nash equilibria in pure strategies of this game

Answer 1

Nash equilibrium is a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice.

In this auction, both players know the valuations of the other player. Further, if the two players submit the same bid, then the auctioneer gives the object to Player 1.

Each player wants to win the auction, and pay as little as possible to do so. We also know that V1>V2, i.e. Player 1 has a greater valuation than Player 2.

If Player 1 bids below V2, Player 2 will bid V2 and win the auction. But Player 1 has the incentive to increase his bidding, since he has not yet reached his valuation of the product. If Player 1 bids at V2, Player 2 will also bid at V2, but according to the rules of the auction, Player 1 will win. Beyond this, Player 1 has no incentive to increase his bidding since it will only result in lowering his utility, and neither will Player 2 since he does not value the product any higher.

Therefore, the Nash equilibrium of the game is where both Player 1 and 2 bid V2, and Player 1 wins the auction.