Abstract:

The English auction has many desirable properties when a single object is to be sold. However, the properties of generalized versions of the English auctions, which could be used by the owner of several objects facing buyers who are willing to pay for more than one object, are much less understood. In particular, the possibility of collusion among bidders is a central issue in evaluating the effectiveness of English auctions with multiple goods in terms of seller's expected revenue, and social efficiency. The following conjectures are usually held about collusion in ascending-open auctions with multiple objects:

• The buyers collude by signaling their willingness to abstain from competing over certain objects, provided they are not challenged on others. This collusion mechanism is thus genuine to multiple auctions environments.

• The scope for collusion increases with the ratio of objects to bidders.

• The presence of strong complementarities between objects in the buyers' preferences, limit the incentive to collude.

In this paper, we put these conjectures under formal scrutiny. We show that, in a variety of generalized versions of the English auctions which can be used to sell multiple objects the bidders can successfully coordinate on equilibria that yield a low seller's expected revenue and socially inefficient allocations of the objects.

The model is as follows. There are n bidders and a set K = {1,...,k} of objects, with k,n finite. The bidders have quasi-linear utility functions, and the willingness to pay of bidder i = 1,...,n for bundle any subset of object in K i's private information. The rest of the world only knows that such values are drawn from a joint distribution with a density f.

We assume that the k objects are sold with a simultaneous ascending English auction, which we call the generalized English auction (GEA). The auction proceeds in rounds. At the initial round, each bidder submits up to k bids, one for each object. After the first round, the highest bid on each object is taken as the outstanding bid on that object. At the next round, each bidder can submit a new bid on any object as long as it is higher than the current outstanding one. The auction continues in this fashion until no bid, on any of the k objects, is increased. Thus, no object is assigned until all bids are the same in two consecutive rounds. Once the auction is over each object j is assigned to the highest bidder, that is the agent who has made the highest bid on j (Tie-breaking rules are not crucial).

We start studying the benchmark case of purely additive values, we focus initially on the case in which the willingness to pay of any bidder for any object is independent of how many other objects he is buying. For this case we establish the existence of equilibria in which bidders collude by softening the degree of competition. The equilibria can be roughly described as follows. Each player starts by bidding the smallest possible amount on a subset of objects, and zero on all other objects, thus indicating her most wanted object(s). If different agents indicate different subsets of objects, then no player revises bids in the next round, hence the bidding terminates. When different bidders place initial bids for same set of objects then the agents keep raising their bids according to some equilibrium strategy. In these equilibria, the final allocation is inefficient but the bidders end up paying less than what would be paid in a `truth-telling' equilibrium in which each bidder bids for each object until the price exceeds her valuation for the object. The reduced payments make up for the loss of efficiency in assigning the objects, so that bidders' surplus is increased. We show however that for this class of equilibria, as the number of bidders increases with respect to the number of objects then the possibility of collusion decreases. The conjecture that collusion is a "low numbers" phenomenon therefore appears to be confirmed.

We then establish that collusive equilibria exist even in the presence of high complementarities. Thus the presence of complementarities does not necessarily destroy collusion. In fact, in some cases the same collusive equilibria found for the case of no complementarities continue to hold in the presence of high complementarities. In this case the loss in social welfare increases with the degree of complementarities. In the class of collusive equilibria that we study, the objects are split among the bidders, so that the possible synergies are lost. The conjecture that the presence of complementarities invariably decreases the possibility of collusion appears therefore not to be necessarily correct.