Lemma: The symmetric equilibrium is defined as follows:

if the buyer is the current high bidder at any auction, or if the buyerâ€™s valuation is less than or equal to the lowest standing bid, the buyer should pass;

otherwise, if there is a unique lowest standing bid, the buyer should submit a bid with the seller offering the lowest standing bid. The bid should be equal to the smallest valuation that exceeds this lowest standing bid;

otherwise, let be the set of sellers who have the lowest standing bid. Let be the subset of sellers in whose current high bidder submitted his bid while the standing bid was strictly below its current level; let be the set of sellers in who have not received a bid. If is not empty, the buyer should bid with equal probability with every sellers in . If is empty but is nonempty, the buyer should bid with equal probability with every seller in . Otherwise, if both subsets are empty, the buyer should bid with equal probability with every seller in .
Share with your friends: 