Two ways to auction off an uncertain good

Abstract

An auction framework is examined where each seller is uncertain about whether or not he will have a good available to sell. A timely example includes the auctioning off of radio spectrum by licensed primary users to unlicensed secondary users. A licensed primary user may not use the spectrum all of the time, but may not know in advance whether or not he will need the spectrum himself and thus whether or not he will have spectrum bandwidth available to rent out. We consider two types of auctions: an ex-post auction which takes place after each seller learns about the availability of the good and an ex-ante auction which takes place before each seller learns about availability. The expected payoffs to buyers, sellers, and to society are then compared. The ex-ante auction can cause buyers to bid on multiple expected units to ensure that they end up with at least one actual unit. Such over purchasing can create a loss to society. Sellers will prefer the ex-post auction since if fewer units are actually available to sell, then they can be sold at a higher price. Whereas in the ex-ante auction all sellers have one expected unit available to sell which results in a lower price. However, if high value buyers are common, then a buyer may prefer the ex-ante auction as it can be easier to win an expected unit than an ex-post actual unit. In contrast, if a buyer has a high valuation and if high value buyers are uncommon, then he may prefer the ex-post auction since he has a strong chance to win an actual unit.

Appendix

In the ex-ante auction, we say that a buyer bids sincerely if for all $\kappa \in \{1,2,\ldots ,m\}$ he drops out of the auction for the $\kappa$th expected unit when the bidding equals his $\kappa$th expected value.

Proposition 7 shows that in the ex-ante auction buyers have incentive to bid sincerely. Note that Ausubel (2004)’s Theorem 2 shows that as long as long as a full support assumption is met sincere bidding is the unique outcome of the iterated elimination of weakly dominated strategies and Ausubel’s Theorem 1 proves that sincere bidding by all bidders is an ex post perfect equilibrium. Ausubel introduces the full support assumption in a model where the buyers’ marginal valuations are integer valued while these valuations are continuous random variables in the current paper. Although Ausubel states that Theorems 1 and 2 hold in the continuous case, the proofs are omitted for brevity and so we provide a proof. Ausubel’s full support assumption guarantees that every bid matters or that given any bid b there is a positive probability that the next bid will be directly above or below bid b. Our valuations also satisfy this assumption (with no additional condition necessary) as the expected value of the $\kappa$th expected unit to buyer j is $(1-q)^{\kappa -1}q\cdot v_{j}.$ Since every $v_{j}$ is a continuous random variable over $[\underline{v},\bar{v}]$, j’s marginal valuation for the $\kappa$th unit is a continuous random variable over $(1-q)^{\kappa -1}q\cdot [\underline{v},\bar{v}]$ and thus there is always a positive probability that the next sincere bid is directly above or below any $b\in (1-q)^{\kappa -1}q\cdot (\underline{v},\bar{v})$.

Proposition 7

Assume the auction is ex-ante. For each buyer j the strategy, to remain in the bidding for $\kappa$ units up to j’s valuation for the $\kappa$th expected unit, is a perfect Bayesian equilibrium and follows the iterated elimination of weakly dominated strategies.

Proof

First, we show that the proposed strategy is a perfect Bayesian equilibrium or that bidder j will bid sincerely if everyone else bids sincerely. Clearly bidder j will not bid above his expected value for an expected unit. We just need to show that he also will not bid below his expected value for a unit. Assume to the contrary that bidder j sets a maximum bid of $b<(1-q)^{\kappa -1}q\cdot v_{j}$ for the $\kappa$th unit; this is only advantageous to j if he can get a lower price on one of his other units which will only occur if j clinches a unit at price b. Let j drop out at price b and let an expected unit be clinched at this price. Thus, at price $b-\varepsilon$ demand was greater than supply and aggregate demand of all bidders excluding k was also greater than aggregate supply for all k. At price b, j decreases his demand by one as this is his maximum bid for the $\kappa$th expected unit; additionally, we assume that a unit is clinched. However, as j has decreased his demand he cannot clinch the good unless another bidder also decreases his demand at price b. Since we assume that the expected valuations are continuous the probability that two bidders drop out at the same time is zero; thus, it must be that the good is not clinched by j. So, j does not gain by dropping out of the auction for the $\kappa$th good before $(1-q)^{\kappa -1}q\cdot v_{j}$. Therefore, no j has incentive to deviate from our strategy.

Second, we show that the proposed strategies are an equilibrium following the iterated elimination of weakly dominated strategies. Consider any history such that the aggregate demand of all bidders equals $(m+1)$ at the current price. We show that bidder j does best to bid sincerely by demonstrating that j will not drop out of the auction for the $\kappa$th expected unit at $b<(1-q)^{\kappa -1}q\cdot v_{j}$. Assume that j is only bidding on one unit or that $\kappa =1$. If j drops out at price b, then he may not win the expected unit and since b is below his marginal valuation for the unit he is better off waiting until the price equals his marginal valuation. Now assume that j is bidding on at least two units or that $\kappa \ge 2$. If j decreases his demand by one unit at price b, then the aggregate demand will decrease to m, the market will clear, and j will win $\kappa -1$ expected units at price b. However, if j does not change his demand at price b and if another bidder changes his demand at price $b+\varepsilon$ then j will win $\kappa$ units at price $b+\varepsilon$ which he prefers to winning $\kappa -1$ units at price b. Since there is a positive probability that another bidder will change his demand at price $b+\varepsilon$, bidder j prefers to not drop out at price b. Additionally, bidder j will never choose to drop out of the auction for the $\kappa$th good at price $b'>(1-q)^{\kappa -1}q\cdot v_{j}$. If j follows this strategy and if another bidder lowers his demand at price p such that $(1-q)^{\kappa -1}q\cdot v_{j}<p<b'$, then j would win $\kappa$ units at price p; this price is above his marginal valuation for the $\kappa$th good. Since there is a positive probability that another bidder will drop out at p, bidder j will not drop out at price $b'$. Thus, bidder j does best to bid sincerely and the insincere strategies can be eliminated.

Next consider any history such that the aggregate demand of all bidders equals $(m+2)$ at the current price. Let bidder j set his maximum bid for the $\kappa$th item at $b<(1-q)^{\kappa -1}q\cdot v_{j}$. If j decreases his demand by one at price b, then either the aggregate demand will equal $(m+1)$ and we will go to the case described above or a buyer $k\ne j$ will clinch a good at price b and aggregate demand will decrease to m while available supply will equal $(m-1)$. Either way we return to the first situation described above where aggregate demand minus aggregate supply equals one. However, if j stays in the auction until the bidding reaches his marginal valuation, then there is a positive probability that someone else will decrease his bid and that j will clinch a unit. Thus, it is not beneficial to j to drop out at price b which is below his marginal valuation. Additionally, j will not want to bid above his marginal valuation as there is a positive probability that he will end up purchasing the unit at a price above his marginal valuation. So j’s best strategy is to bid sincerely. We can continue this process to show that for any aggregate demand greater than m, j’s best strategy is to bid sincerely and the iterated elimination of weakly dominated strategies will yield a strategy of bidding sincerely. $\square$