Revenue Equivalence Theorem Research Articles

OPTIMIZATION OF THE AUCTION DURATION IN THE PRESENSE OF TIME-DEPENDANT COSTS

This paper examines the influence of the duration of auctions or tenders on the expected gain of their organizer. Extending the duration of bidding affects auction results in two ways. On the one hand, it allows attracting a larger number of participants to the auction, and the competition between them increases the chances of the auctioneer to get a better price. On the other hand, delaying bids delays the receipt of money (for auctions) or required goods or services (for tenders), and time has value in itself. The influence of these two factors, which act in opposite directions, suggests the existence of an optimal duration of the bidding process. The paper develops a mathematical model of bidding, which formalizes these considerations and provides an algorithm to determine their optimal duration. The arrival of bidders willing to participate in the auction is modeled as a Poisson process. Each participant is characterized by his own assessment of the value of the object put up for auction. These estimates are assumed to be independent identically distributed random variables drawn from some parametric distribution. Under these assumptions, Myerson's revenue equivalence theorem makes it possible to predict the expected results of the auction as a function of the number of bidders, regardless of the auction rules. On this basis, it is possible to compare the benefits and costs associated with changing the duration of time for accepting applications for participation in bidding, which makes it possible to determine its optimal value. The obtained optimality conditions have a meaningful and intuitive economic interpretation. For practical applications, the use of Monte Carlo methods based on the empirical distribution of bid and ask prices is proposed. The practical implementation of the proposed algorithm can improve the economic performance of the auctioneer, which is especially relevant for the public sector of the economy.

Basic Bidding Formats: Characteristics and Differences

Abstract In standard auction theory, the ‘revenue equivalence theorem’ asserts that the outcomes of the elementary allocation methods coincide. However, bidding processes differ fundamentally with regard to the decision situation of the participants: Is it at all imperative to take into consideration the number of competitors (‘stochastic’ strategy) or not (‘deterministic’ course of action)? Furthermore, established auction theory neglects the operating modes of procurement alternatives under uncertainty. Apart from the lacking knowledge how many rivals have to be beaten, tenderers regularly are ignorant of the buyer’s reserve price. Then it is even more tentative to calculate an offer based on probability theory. Consequently, the suppliers’ propensity to collude increases.

Optimal Auctions for Spiteful Bidders

Designing revenue-optimal auctions for various settings is perhaps the most important, yet sometimes most elusive, problem in mechanism design. Spiteful bidders have been intensely studied recently, especially because spite occurs in many applications in multiagent system and electronic commerce. We derive the optimal auction for such bidders (as well as bidders that are altruistic). It is a generalization of Myerson’s (1981) auction. It chooses an allocation that maximizes agents’ virtual valuations, but for a generalized definition of virtual valuation. The payment rule is less intuitive. For one, it takes each bidder’s own report into consideration when determining his payment. Moreover, bidders pay even if the seller keeps the item; a similar phenomenon has been shown in other settings with neg- ative externalities (Jehiel, Moldovanu, and Stacchetti 1996; Deng and Pekec 2011). On the other hand, a novel aspect of our auction is that it sometimes subsidizes losers when the item is sold to some other bidder. We also derive a revenue equivalence theorem for this setting. Using it, we generate a short proof of (a slight generalization of) the previously known result that, in two-bidder settings with independently uniformly drawn valuations, second-price auctions yield greater expected revenue than first-price auctions. Finally, we present a template for comparing the expected revenues of any two auction mechanisms that have the same allocation rule (for the valuations distributions at hand).

A non-expected utility model and its application in practical auctions

ABSTRACT This article presents a non-expected utility decision model which is nonlinear in the winning probabilities. The model not only explicitly expresses bidders’ attitudes to risk, but also addresses their preference over the bidding criteria. To demonstrate how to apply the model in the practical auction design, the first- and second-price auctions with both commission rate and reserve price are examined, respectively. For nonrisk-neutral bidders, the equilibrium bidding strategies are characterized, in which the commission rate has a significant influence on the bidding strategy through the critical valuation. However, the existence of the optimal commission rate is uncertain, but once it exists, it depends on the information rent of the highest or second highest order valuation in terms of the inverse hazard rate. With risk-aversion bidders, the only difference to the optimal reserve price is a constant between the first- and second-price auctions. The revenue comparisons show that the classical Revenue Equivalence Theorem fails in practical auctions with the commission rate. This article extends the application of the decision-making model in the auction design in theory and provides some guidance for the auction house and the seller to make their decisions in reality.

Charity Auctions as Assets: All-Pay vs. Winner-Pay Mechanisms in Mean-Variance Space

Auctions that generate public good benefits for bidders through revenue, such as those run by a charity, do not adhere to the revenue equivalence theorem. Instead, theory predicts that all-pay auctions will generate greater expected revenues than winner-pay mechanisms. However, this paper demonstrates these greater expected revenues are always accompanied by greater (and in some cases infinite) revenue variance. While a risk-neutral seller would be indifferent to this revenue variance, we show through a mean-variance portfolio approach that plausible levels of risk aversion would compel them to generally avoid all-pay auctions and instead use a combination of winner-pay designs.

Sequential First-Price Auction with Randomly Arriving Buyers

AbstractIn this paper we reanalyze Said’s (2011) work by retaining all his assumptions except that we use the first-price auction to sell differentiated goods to buyers in dynamic markets instead of the second-price auction. We conclude that except for the expression of the equilibrium bidding strategy, all the results for the first-price auction are exactly the same as the corresponding ones for the second-price auction established by Said (2011). This implies that the well-known “revenue equivalence theorem” holds true for Said’s (2011) dynamic model setting.

Existence of Equilibria in Procurement Auctions

This paper investigates symmetric equilibria in first- and second-price auctions with multidimensional types. The constructed model mirrors the information structure of actual procurement auctions. We demonstrate by a counterexample that symmetric and continuous type distribution is not a sufficient condition for the existence of a pure-strategy equilibrium, but it is guaranteed if distributions of all components are log-concave. We state a new Revenue Equivalence Theorem applied to first- and second-price auctions and conclude that the two standard auction formats yields the same expected price to the auctioneer.

Selling to Advised Buyers

In many cases, agents that make purchase decisions are uninformed and rely on the advice of biased experts. For example, when contemplating an acquisition, the board of the bidder relies on the advice of the managerial team when deciding what offer to make for the target. We study how to sell assets to such “advised buyers” if the goal is (i) to maximize revenues; (ii) to maximize allocative efficiency. In static mechanisms, such as first- and second-price auctions, advisors communicate a coarsening of information and a version of the revenue equivalence theorem holds. In contrast, dynamic mechanisms, such as multiple-round auctions, result in more informative communication between buyers and their advisors, which leads to more efficient allocations. Whether this leads to higher revenues depends on the direction of the bias. When advisors are biased for overbidding, an ascending-price auction dominates static formats in both efficiency and expected revenues. When advisors are biased for underbidding, a descending-price auction dominates static mechanisms efficiency but often results in lower revenues.

Wars of Attrition with Unrecoverable Upfront Investments

This paper models a war of attrition where unrecoverable upfront investments decide for how long a participant can compete. In a two-stage setting, participants competing for a single prize first privately choose a resource investment to utilize through the contest, then second choose for how long they are willing to compete given those resources. Cases where unused resources from the contest have zero, partial, and full repurpose value (e.g., for future contests) are considered, which allows for a direct comparison to the first price all-pay auction (zero repurpose value) and the standard war of attrition (full repurpose value). The theoretical contribution of this model has direct implications for wars of attrition where upfront costs are a prerequisite to competing, such as for R&D endeavors, market exit scenarios, and contract fulfillments. In addition to these applications, the model is illustrated in a charity auction setting where the revenue equivalence theorem is violated and the effect of varying repurpose values on expected revenue is discussed.

A Brief Survey of the Theory of Auction

This is a brief survey of the theory of auction. We first discuss the types of agents who participate in an auction and their respective objectives. In the next section we briefly explain the concepts of standard auction, revenue equivalence theorem (perhaps the most cited result in auction theory, along with the result of Myerson on optimal auction), optimal auction, interdependent valuations, risk and some other commonly used concepts. Finally we discuss the theory of contest, as contests are essentially auctions where a part of the payment function is exogenously given to the contestants. JEL Classification: D44 (Auctions)

Auctions with contingent payments — An overview

I survey a literature on auctions with contingent payments, that is auctions in which payments are allowed to depend on an ex-post verifiable variable, such as revenues in oil lease auctions. Based on DeMarzo et al. (2005), I describe a partial ranking of auction revenues for auctions that differ in terms of contract forms, pricing rules and seller commitment and why the revenue equivalence theorem does not apply even in an independent private values setup. I discuss models that incorporate adverse selection, moral hazard, competition between auctioneers, common values and the sale of multiple units.

A parametric family of two ranked objects auctions: equilibria and associated risk

This paper deals with simultaneous auctions of two commonly ranked objects following the model studied in Menezes and Monteiro (J. Real Estate Finance Econ., 17(3):219–232, 1998). For these problems we introduce a parametric family of auction mechanisms which includes the three classic auctions (discriminatory-price auction, uniform-price auction and Vickrey auction) and we call it the $\mathcal{DUV}$ family. We provide the unique Bayesian Nash equilibrium for each auction in $\mathcal{DUV}$ and prove a revenue equivalence theorem for the parametric family. Likewise, we study the value at risk of the auctioneer as a reasonable decision criterion to determine which auctions in $\mathcal{DUV}$ may be better taking into account the interests of the auctioneer. We show that there are auction mechanisms in $\mathcal{DUV}$ which are better than the classic auction mechanisms with respect to this criterion.

Mean field equilibria of dynamic auctions with learning

We study learning in a dynamic setting where identical copies of a good are sold over time through a sequence of second price auctions. Each agent in the market has an unknown independent private valuation which determines the distribution of the reward she obtains from the good; for example, in sponsored search settings, advertisers may initially be unsure of the value of a click. Though the induced dynamic game is complex, we simplify analysis of the market using an approximation methodology known as mean field equilibrium (MFE). The methodology assumes that agents optimize only with respect to long run average estimates of the distribution of other players' bids. We show a remarkable fact: in a mean field equilibrium, the agent has an optimal strategy where she bids truthfully according to a conjoint valuation . The conjoint valuation is the sum of her current expected valuation, together with an overbid amount that is exactly the expected marginal benefit to one additional observation about her true private valuation. Under mild conditions on the model, we show that an MFE exists, and that it is a good approximation to a rational agent's behavior as the number of agents increases. We conclude by discussing the implications of the auction format and design on the auctioneer's revenue. In particular, we establish a dynamic version of the revenue equivalence theorem, and discuss optimal selection of reserve prices in dynamic auctions.

A Tale of Two Papers: How Exclusive Auction-Based Approaches Mean the Demise of Unlicensed Spectrum

In 2008 the FCC released two OSP working papers by Bykowsky, Sharkey, and Olson which argued that licensed and unlicensed spectrum were economically inefficiently allocated and for a set of mechanisms to address this problem. This paper analyzes both Bykowsky, et al. papers, concluding that serious problems exist with both. The “tragedy of the commons” approach which they take is examined and rejected as historically unfounded and economically inappropriate for the class of rivalous goods of which spectrum is an example. It demonstrates that Bykowsky, et al. fail to recognize the fundamental differences between licensed and unlicensed spectrum and the ways in which these differences make exclusive-auction mechanisms for the allocation of unlicensed spectrum likely to reduce the economic value of such spectrum and that the market mechanism proposed by Bykowsky, Sharkey, and Olson – auctioning of the rule-making authority over the allocation of licensed and unlicensed spectrum to the highest bidder – is likely to have severely adverse public policy consequences. Bykowsky, Sharkey, and Olson adopt a Coasian approach under precisely the circumstances which Coase himself identifies intractable problems with market approaches. This paper finds that the dependence of Bykowsky, et al.’s model on a very specific auction structure in which firms seeking to acquire unlicensed spectrum outnumber firms seeking to acquire licensed spectrum (and to block the allocation of unlicensed) an unrealistic set of assumptions, renders the model useless for gauging the dynamics of a market in allocation of spectrum allocation rules and of the spectrum itself. Furthermore, their model explicitly violates the two key assumptions of the Revenue Equivalence Theorem. While Bykowsky, et al. are correct about historical provision point and freeridership problems associated with unlicensed spectrum allocation, there is little reason to believe that their model actually provide a remedy. Finally, this paper concludes there is good reason to believe that Bykowsky, Sharkey, and Olson have failed to note developments since 2005 which suggest that incumbents which have traditionally favored licensed spectrum are using significantly more unlicensed spectrum in their operations, particularly for data offloading and underlay sharing, and are now no longer predisposed to continue their historical pattern of opposition to unlicensed spectrum allocation.

Mean Field Equilibria of Dynamic Auctions with Learning

We study learning in a dynamic setting where identical copies of a good are sold over time through a sequence of second price auctions. Each agent in the market has an 'unknown' independent private valuation which determines the distribution of the reward she obtains from the good; for example, in sponsored search settings, advertisers may initially be unsure of the value of a click. Though the induced dynamic game is complex, we simplify analysis of the market using an approximation methodology known as 'mean field equilibrium' (MFE). The methodology assumes that agents optimize only with respect to long run average estimates of the distribution of other players' bids. We show a remarkable fact: in a mean field equilibrium, the agent has an optimal strategy where she bids truthfully according to a 'conjoint valuation'. The conjoint valuation is the sum of her current expected valuation, together with an overbid amount that is exactly the expected marginal benefit to one additional observation about her true private valuation. Under mild conditions on the model, we show that an MFE exists, and that it is a good approximation to a 'rational' agent's behavior as the number of agents increases. Formally, if every agent except one follows the MFE strategy, then the remaining agent's loss on playing the MFE strategy converges to zero as the number of agents in the market increases. We conclude by discussing the implications of the auction format and design on the auctioneer's revenue. In particular, we establish a dynamic version of the revenue equivalence theorem, and discuss optimal selection of reserve prices in dynamic auctions.

A Comparative Study on Pricing Rules and Its Effect on Total Dispatch Cost

A focus of study in auction market design is revenue comparison. Revenue Equivalence Theorem implies that bidders receive the same amount of revenue invariant to the auction type. This paper explores the applicability of this theorem in the context of the electricity market. We develop experimental test cases using agent-based simulation to examine the impact of different pricing rule on total dispatch cost. Using Q-learning in a repetitive trading environment, generator agents can quickly learn the market characteristics and seek to maximize their revenue by adapting their bidding strategies. A look-up table is utilized as a learning mechanism to improve the agent's bidding strategies. We conclude that Revenue Equivalence Theorem holds in a multi-unitmulti- period in symmetric bidding. In asymmetric bidding and when the market share of generator agents differs significantly, the computer simulation allows us to observe the rapid increase in revenue with uniform auction.

Auctions with almost homogeneous bidders

We prove that, around the symmetric case, where the values are identically distributed, the equilibrium of the first price auction is jointly differentiable with respect to general bidder-specific parameters of the value distributions. We show that the revenue equivalence between the first-price and the second-price auctions to the first-order in the size of the parameters is an immediate consequence of this differentiability and the Revenue Equivalence Theorem; thereby formally establishing the first-order equivalence Fibich et al. [G. Fibich, A. Gavious, A. Sela, Revenue equivalence in asymmetric auctions, J. Econ. Theory 115 (2004) 309–321] noticed for their particular perturbation.

A measure of the variability of revenue in auctions: A look at the revenue equivalence theorem

One not-so-intuitive result in auction theory is the revenue equivalence theorem, which states that as long as an auction complies with some conditions, it will on average generate the same revenue to an auctioneer as the revenue generated by any other auction that complies with them. Surprisingly, the conditions are not defined on the payment rules to the bidders but on the fact that the bidders do not bid below a reserve value—set by the auctioneer—the winner is the one with the highest bidding and there is a common equilibrium bidding function used by all bidders. In this paper, we verify such result using extensive simulation of a broad range of auctions and focus on the variability or fluctuations of the results around the average. Such fluctuations are observed and measured in two dimensions for each type of auction: as the number of auctions grows and as the number of bidders increases.

A Test of the Revenue Equivalence Theorem using Field Experiments on eBay

We conducted 80 auctions on eBay. Forty of these auctions were for various popular music CDs while the remain ing 40 auctions were for video games for Microsoft’s Xbox gaming console. The revenue equivalence theorem states that any auction form having the same effective reserve price yields the same expected revenue. The effective reserve price on eBay consists of three components: the opening bid amount, the secret reserve amount, and the shipping and handling charge. We set no secret reserve price and varied the opening bid and the shipping and handling charge to keep the overall reserve level fixed. When the effective reserve was $4, auctions with a low opening bid and high shipping charges attracted more bidders, earlier bidding, and yielded higher revenue than those with a high opening bid and low shipping charges. The same results holds only for Xbox games under the $8 effective reserve. Unlike the other treatments, where the reserve represents less than 30% of the retail price of the item, for CDs, the $8 effective reserve represents over 50% of the retail price of the item. In this treatment, we find no systematic difference in the number of bidders attracted to the auction or revenues as a function of how the effective reserve is allocated between opening bid and shipping charges. We show that these results can be accounted for by boundedly rational bidding behavior.

Equilibrium transformations and the Revenue Equivalence Theorem

I develop a method that transforms an equilibrium strategy profile from one auction to another. The method is constructive and does not require complicated computation. This provides a new approach to revenue equivalence and extends the theorem to domains where it had not previously been known, in particular to simultaneous multiple object auctions with complete information and to auction environments having correlated private values and common values.