Extended Abstract:

The notion of optimality is central to economics. Yet, economic theory typically makes only a dichotomous distinction between optimal and sub-optimal outcomes. By and large, there is no attempt to quantify how far from optimality are sub-optimal outcomes, allocations, or institutions.

In this paper we attempt to make a preliminary step toward quantification of optimality in the context of auction theory. We define a new notion -- effectiveness -- which measures proximity to optimality. We argue that the prevalence of the English auction (and perhaps other auction forms) is due to (1) its high effectiveness, that is, the fact that it performs reasonably well in a wide class of environments; and (2) the fact that it is "simple, " or can be readily applied in many environments. While an optimal auction (see Myerson (1981), Crémer and McLean (1988) and McAfee and Reny (1992)) may outperform the English auction in terms of the expected revenues it generates for the seller, it does not outperform it by much and it is difficult to apply in practice. Furthermore, even for relatively simple environments such as those studied in this paper (single object, private non-independent values, and risk averse buyers) finding an optimal auction is still an open problem.

We focus on two particular simple single object auctions -- the English auction and the modified Vickrey auction, which is a sealed-bid second-price auction where the seller may set a reserve price -- and demonstrate their effectiveness in private values environments. There are several reasons why the English and modified Vickrey auction are particularly compelling economic institutions on which to perform our analysis. First, they have the advantage of inducing bidders to adopt simple dominant strategies, namely bidding their valuations for the object. Thus, our conclusions do not depend on the plausibility of the common prior assumption or of Bayesian-Nash equilibrium as the appropriate solution concept for auction games. Second, as Milgrom and Weber (1982) have shown, the English auction maximizes the seller's revenues among all commonly used auction forms for a wide class of environments. In particular, this implies that the English auction is also more effective than the Dutch and sealed bid first price auctions in these environments. Third, if the seller sets a positive reserve price, he may retain the object in spite of it being less valuable to him than to the buyers. But, conditional on the seller selling the object, the English auction is efficient. Unlike the first price or Dutch auction, it awards the object to the buyer that values it most. Finally, both the English and the second price auctions are "simple" in the sense of satisfying the easy applicability criterion above.

For a fixed environment, define the effectiveness of an auction as the ratio between the expected revenue it generates for the seller and the expected valuation of the object to the bidder with the highest valuation (total surplus). If the bidders and seller cannot be coerced into participating in the auction, its effectiveness will be a number between zero and one. Generally, the effectiveness of any auction form will depend on the environment considered; the closer effectiveness is to one, the closer the auction is to being optimal for this particular environment. We are interested in demonstrating the effectiveness of the English auction for general environments. We therefore seek to determine the effectiveness of the English auction in the worst-case environment, that is, in the environment in which it is the lowest. A seller that relies on the English auction can be sure that his expected revenue from selling the object (as a proportion of total surplus) will be at least as high as this worst-case effectiveness, and in general will be higher. Indeed, even a seller who is uncertain about which environment he is facing is guaranteed an expected revenue that is higher than worst-case effectiveness. Thus, if the worst-case effectiveness of the English auction is "high," a seller who is uncertain about the environment, is unable to figure out the optimal auction, and even if he is able, is suspicious about whether bidders understand the optimal auction's rules and is doubtful whether bidders employ Bayesian Nash equilibrium strategies, is well advised to employ an English auction; even in the worst-case, his losses from not doing otherwise will be small.

We establish the following results. We parameterize all possible private values environments by the number of bidders, n, and their expected valuations of the object as a percent of the maximum possible valuation, a . We obtain a lower bound on the ratio between the expected revenue generated by an English and second price auction and expected total surplus under three different assumptions on the seller's behavior: (1) the seller does not set a reserve price, (2) the seller sets an optimal reserve price given his belief about the expectation of buyers' valuations for the object, and (3) the seller sets an optimal reserve price given his belief about the distribution of buyers' valuations for the object. These three assumptions describe three different modes of "seller's rationality." From a seller that fails to recognize the fact that setting a positive reserve price may increase his expected revenue, to a seller who recognizes the usefulness of a reserve price but is unable to articulate a belief about buyers' valuations beyond a specification of their expected valuations, to a seller who can fully articulate his beliefs and set an optimal reserve price accordingly, but cannot compute an auction form that would yield a higher expected revenue than the English auction.

For each mode of seller's rationality and every given n and a , we ask how effective are the English and second price auctions in the worst possible private values environment. We show that the bound that we identify is tight by presenting examples of environments that attain it. We then repeat this exercise for environments where bidders' valuations are non-negatively correlated (we assume, specifically, that buyers' valuations of the object are conditionally independent and identically distributed) and for environments where bidders' valuations are independent and identically distributed. We thus obtain nine results. However, when bidders' valuations are either non-negatively correlated or independent, worst-case performance of the English and second price auction under either one of our assumptions about the seller's rationality is identical. That is, on the distributions that attain the worst-case bound, being able to set a positive reserve price does not help the seller. Obviously, the ability to set a positive reserve price is generally advantageous.

As expected, the worst case effectiveness of the English and second price auctions improves as the number of bidders and their expected valuations for the object increase. What is surprising is how high this worst-case bound sometimes is. For example, when n = 12 and a = 60%, even in the worst possible case, an English or a second price auction with no reserve price sells the object at an expected price that is almost 85% of the expected value of the object to the highest valuation bidder. When bidders' valuations are independent and identically distributed, as Myerson (1981) have shown, the English and second price auction are optimal and the expected price, under the same parameters (n = 12 and a = 60%), increases to 99% in the worst possible case.

We propose in this paper a new research agenda. Instead of demonstrating the optimality of simple mechanisms in special environments and arguing that this explains their prevalence in general environments, we replace the notion of optimality with that of effectiveness and argue that simple auction mechanisms, while perhaps strictly sub-optimal in most environments, are nevertheless highly effective in most plausible environment. As we argue above, it is likely that a complete explanation of the phenomenon of the persistence of sub-optimal institutions would involve both effectiveness, that is, how much is being lost by sub-optimality, and complexity or simplicity, that is, how hard or costly it would be to design optimal institutions. Indeed, one may interpret our results as an attempt to establish the cost of simplicity in the case of the English and second price auctions.

We believe that similar reasoning to that presented here could also help explain the prevalence of other economic and political institutions as well as that of "behavioral rules of thumb" and other modes of boundedly rational behavior.