Auction Theory: Post-Auction Resale

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I. Introduction
This blog entry extends the model of auctions introduced in my prior blog entry by introducing the possibility of post-auction resale. In this model, the winning bidder can sell the item to the losing bidders at a fixed price. Initially, this mechanism may seem to favor the highest bidder, giving him or her the power to extort additional profit from the losing bidders. In the case of the first-price auction, the post-auction resale mechanism results in the same equilibrium bids and outcome as in the standard symmetric first-price auction. Similarly, the post-auction resale mechanism makes little sense to employ in the case of the second-price auction.

II. Auction Resale Problem
We consider the symmetric, sealed bid, first-price auction where each bidder’s valuation is drawn from the probability distribution given by F([0, \omega]), for some \omega \in \mathbb{R}_{++}. Let \beta : [0, \omega] \to \mathbb{R}_{+} be the symmetric equilibrium bidding function in the case of the first-price auction. Recall from my prior blog entry that:

\beta(v) = \displaystyle \dfrac{1}{G(v)} \int_{0}^{v} xG^{\prime}(x)dx

The auctioneer begins by holding the auction. Then, after the auction, each player’s bid is made public. The winning bidder then decides whether or not to sell the item. If the winning bidder chooses to sell the item after the auction, he then decides upon a price p \in \mathbb{R}_{+} after the auction. Any of the other bidder can then choose to accept the item at price p or to reject this item. Note that the possibility of post-auction resale is known to all bidders a priori. We seek to determine how the winning bidder should set the price p, as well as how the losing bidder should respond.

In order to analyze this mechanism, we consider the post-auction resale mechanism as a dynamic game. We apply dynamic programming, which is referred to as backward induction in game theory, to find the equilibrium strategies. In particular, this equilibrium is a subgame perfect Nash equilibrium, which induces a Nash equilibrium at every subgame.

We start by examining the last subgame, the post-auction resale. Let player i be the winning bidder, and let j be a player with the second highest bid. By monotonicity of the bidding strategies, we have v_{i} \geq v_{j}, where v_{i}, v_{j} are the bidders’ respective valuations. Let p \in \mathbb{R}_{+} be the fixed price at which player i chooses to sell the item. As the bids are public information, player i will set p > b_{j}, player j‘s bid. Player j will purchase the item from player i if and only if p \leq v_{j}. If player j purchases the item from player i, then player i‘s profit is p - b_{i}. As player i will not own the item if he resells it, player i does not obtain valuation v from winning the item in the auction. Since p \leq v_{i} \leq v_{j}, it follows that p - b_{i} \leq v_{i} - b_{i}. So player i has no incentive to resell the item.

Since the winning bidder has no incentive to resell the item after the auction, each bidder will submit his or her bid in an attempt to win and keep the item. Thus, each player should bid \beta in the subgame perfect Nash equilibrium, then the winner should opt not to resell the item and a losing bidder should purchase the item if and only if p is less than its valuation.

The above proof holds as well in the second price auction with the possibility of post-auction resale, noting that the winning bidder’s profit upon successful resale is p - b_{j} \leq v_{j} - b_{j} \leq v_{i} - b_{j}. So the subgame perfect Nash equilibrium is for each player to bid his or her valuation, and for the winner not to resell the item after the auction.

We can more strongly say that there exists no incentive compatible, individually rational mechanism for the winning bidder to use to liquidate the item post-auction. This follows from the fact that no player will pay more than his or her valuation for the item, so the maximum payoff the winning bidder can obtain in choosing to liquidate the item post auction is v_{j} - b_{i} in the case of the first price auction or v_{j} - b_{j} in a second price auction, where v_{j} is the valuation of the second highest bidder.

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