Question

Consider a second price auction for a single item with two bidders. Suppose the bidders have independent private values, uniformly drawn in the interval [0, 1]. Suppose the seller sets a reserve price p = 0.5; that is, only bids above p = 0.5 can win. If a bidder bids above p and the other bids below p, then the first bidder wins and pays a price p. If both bid above p, then the highest bidder wins and pays the second highest price.

In the Bayesian equilibrium in undominated strategies, what is the probability that the item will not be sold?

Answer 1

In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information (sequential Bayesian games). It is a refinement of Bayesian Nash equilibrium (BNE). A PBE has two components - strategies and beliefs:

• The strategy of a player in given information set determines how this player acts in that information-set. The action may depend on the history. This is similar to a sequential game.
• The belief of a player in a given information set determines what node in that information-set the player believes that they are playing at. The belief may be a probability distribution over the nodes in the information-set.

In a study, Vickrey (in 1961) showed that when a = 1 the auction has a unique equilibrium in dominant strategies which is symmetric: i puts probability 1 on the bid ti . Again, Milgrom (in 1981) has shown that when a = 0 the auction has a symmetric equilibrium (which is in undominated strategies). In the example here when N = 2, it is a symmetric equilibrium where each bidder bids 0.5 when observing Si = 0, bids 1/2 when observing Si = 1/2 , and bids 1 when observing Si = 1. Thus, for the extremes of a = 0 and a = 1, equilibria exist.

For any N ≥ 2 and any 0.5 > a > 0 there exists m > 0 and ε > 0 such that if m < m and ε < ε, then the second price auction does not have either a symmetric equilibrium or an equilibrium in undominated strategies.

First, note that if bidders use strategies in the closure of the set of undominated strategies, then conditional on any Ti and Si ∈ {0, 1}, a bidder must bid aTi + (1 − a)Si , and conditional on any Ti and Si = 1/2 , a bidder must bid in [aTi , aTi + (1 − a)]. Under a symmetric equilibrium these strategies must also be followed. For example, suppose that instead there was some Ti and Si ∈ {0, 1} such that each i bid above aTi + (1 − a) with positive probability (the argument ruling out bids below is similar). In that case some i would win with positive probability when the price was above aTi + (1 − a) and their value was at most aTi + (1 − a). They could improve their payoff by lowering their bid to aTi + (1 − a). Next, consider the strategy followed by a bidder i conditional on observing Ti = 0, Si = 1/2 . From above, we know that his or her bids are confined to the interval [0, 1 − a]. Let ε < (1−a)/ 2a , so that aε < (1−a)/ 2 . Let us argue that if m is small enough, then neither bidder places positive probability in the range [aε, 1 − a]. Conditional on winning with a bid bi ∈ [aε, 1 − a], if m is small enough, then conditional on the price being below the maximal possible value of 1 − a the overwhelming probability is that the other bidder observed a signal regarding the common component of 0 (all bidders having observed signals of 1 bid at least 1−a). Thus, for small enough m, the overall conditional expected value when winning in this interval is below aε, and so the bidder has a negative expected value conditional on winning in this interval and could improve by instead bidding below aε. So, both bidders conditional on observing Ti = 0, Si = 1/2 , bid in the range [0, aε). A bidder i faces two possibilities in this range. First that the other bidder observed Tj = 0, Sj = 0 and bid 0 in which case the price is 0. This is inconsequential to i’s incentives. Second, the other bidder observed Tj = 0, Sj = 1/2 . In this case, i’s conditional valuation is 1−a 2 > aε and i would strictly prefer to win against the other bidder at any price in [0, aε). This contradicts the fact that both bidders place probability one in this interval. We are left with no possible bids conditional on having observed Ti = 0, Si = 1/2 and so equilibrium does not exist.

The three possible changes to the setting that could potentially restore existence of equilibrium. These are to change the setting so that there are:

(1) finite bidding grids,

(2) type-dependent tie-breaking rules in the auction, or

(3) atomless type and information distributions.