Median-Price Auction

Consider the following auction game of complete information (all bidders know the value of other bidders). There are three bidders for a single object. Each bidder i has a value vi for the object. Values are v1 = 10, v2 = 5, v3 = 1. Each bidder simultaneously and independently submits bid bi {0, 1, 2, ...} (bids have to be integers) for the object.

(a) Suppose that the winner of the auction has to pay its own bid. For each bidder, identify all weakly dominated strategies.

(b) Consider a standard first-price auction. Ties are broken randomly using a fair randomizing device. Determine all of the pure-strategy Nash equilibria of this game in which bidders do not use weakly dominated strategies. For each Nash equilibrium you find, indicate whether the auction is efficient.

(c) Consider a median-price auction (the winner is the bidder with the median bid who pays its own bid). Ties are broken randomly using a fair randomizing device. Determine all of the pure-strategy Nash equilibria of this game in which bidders do not use weakly dominated strategies. For each Nash equilibrium you find, indicate whether the auction is efficient.