Equivalence and Design Daniel R. 1 1 Department of Economics University of Maryland, College Park. September 2017 / Econ415
IPV, Total Surplus
Background the mechanism designer The fact that there are many potential auction formats raises an immediate question, is one better than the other? It obviously depends on what the goals of the auctioneer are. Two obvious goals are maximizing overall efficiency (getting the objects in the hands of the bidders who value them the most) and maximizing expected revenue. One can think of the role of the mechanism designer as to select the best auction so as to achieve the goals of her principal. If we can predict the play of bidders in the various auctions (solution concept appropriate?) then we can use this analysis as a guide. Other goals equity, antitrust interests, national security.
Outcome related results efficiency Note that, in many of our auctions, monotonic bids imply that we can predict who will win. In IPV auctions, if bids are monotonic in type then highest value bidder will win. In IPV auctions, if highest value bidder wins, then outcome automatically is efficient (why?)
Outcome related results Vickrey (1961) and later Myerson (1981) showed a surprising but powerful related result with respect to revenue. In IPV auctions, total expected revenue raised in an auction depends only on two things, how the good is allocated and the payoff of the lowest type of bidder. Thus, Equivalence: Every auction type that generates the same bidder and yields zero payoff to the lowest type of bidder yields the same expected revenue. Since SPA, FPA, ascending and descending clock auctions all yield zero payoff to the lowest type and all result in the highest type bidder obtaining the object (in the synmetric IPV environment), they all generate the same expected revenue. What happens if the auction has a reserve price?
Affiliated, non-private values: Linkage Principal The Equivalence result does not hold outside of the risk neutral IPV model. A result by Milgrom and Weber (1982) showed that with affiliated and non-private values, between two auction mechanisms, the one auction that more strongly links the price paid to more private information held by bidders in the auction, the higher the expected revenue. Since in SPA, the final price depends not only on the information of the winner but also the second highest bidder, with affiliated non-private values, SPA generally will raise more expected revenue than FPA. Since ascending bid auctions also potential incorporate information of all other (losing) bidders, ascending bid auctions raise more money than SPAs. FPA and descending clock auctions?
IPV auctions with Risk Averse bidders Observe that even if expected payments of bidders were the same, actual payments of bidders in auctions (say FPA vs. SPA) can differ. Consider a bidder who is concerned about randomness. A SPA imposes two types of randomness will the bidder win, and how much the bidder pays. A FPA removes one of those types of uncertainty. A consequence is that risk averse bidders in FPAs will tend to be willing to bid more aggressively in FPAs than in SPAs.
Multi-unit auctions In one sense, the Equivalence Theorem stills holds (with IPV environment). Auctions that generate the same allocation will generate the same expected revenue. A problem posed by multi-unit auctions is that we typically do not yet understand what are equilibria in these complicated mechanisms (excepting Vickrey auctions). Thus, on revenue grounds, it is not known which auction format is the best.
A Narrative We can conceive of an auction specialist hired by a principal (government, seller, agency) to design a mechanism to sell one or more objects. FCC spectrum auctions are a natural case study, but also electrical power auctions, carbon emission auctions, internet auctions etc. The principal ideally communicates to the specialist the criteria that matters most. The specialist (mechanism designer) then designs a mechanism to best achieve these goals.
Direct s Analytically, a classic way to achieve this is via direct mechanisms and the Revelation Principal. A direct mechanism is itself an auction game where the players strategies are simply to report their types to the auctioneer. See SPA, Vickrey auction, what does type mean? It is important to note that in a context of private information, there is no direct way to ensure bidders report their true type. Part of the goal of the Designer, is to design a mechanism where bidders own incentives are to report their true types. In general, though, the need to induce agents to tell the truth limits the scope of outcomes the mechanism designer can achieve.
Revelation Principal Why such an artificial approach? The Revelation Principal is the guiding principle underlying it. The Revelation Principle states that if an outcome in an auction can be implemented as a Bayesian Nash Equilibrium of some auction game, then there is a direct mechanism which implements the same outcome via truthful strategies as a BNE. The advantage of this result is that we can narrow down our search over mechanisms to only direct mechanisms for a lot of purposes. Much of the theoretical work in auction theory takes this approach. Note parallel to proxy bidder idea.
Remaining Problems The Revelation Principle is deceptively simple particularly in implementing the MD in practice. As we have seen, for more general auction settings, we do not know what equilibrium outcomes might look like (multi-unit, eg.) Additionally, the standard model avoids some other basic questions (a partial list): How do we determine what are the objects to sell? (Spectrum license design, power contract length etc.) How can we determine who are the bidders (bidder qualifications, entry, merger etc.) What about other criteria (anti-trust, equity etc.) In the FCC spectrum auction case study we examine some of these issues.