Equilibrium Bidding Research Articles

A common‐value auction with state‐dependent participation

This paper analyzes a common‐value, first‐price auction with state‐dependent participation. The number of bidders, which is unobservable to them, depends on the true value. For participation patterns with many bidders in each state, the bidding equilibrium may be of a “pooling” type—with high probability, the winning bid is the same across states and is below the ex ante expected value—or of a “partially revealing” type—with no significant atoms in the winning bid distribution and an expected winning bid increasing in the true value. Which of these forms will arise is determined by the likelihood ratio at the top of the signal distribution and the participation across states. We fully characterize this relation and show how the participation pattern determines the extent of information aggregation by the price.

Entry in First-Price Auctions with Signaling

We study the optimal entry fee in a symmetric private value first-price auction with signaling, in which the participation decisions and the auction outcome are used by an outside observer to infer the bidders’ types. We show that this auction has a unique fully separating equilibrium bidding function. The expected revenue maximizing entry fee is the maximal fee that guarantees full participation.

Spite vs. risk: Explaining overbidding in the second-price all-pay auction: A theoretical and experimental investigation

We use an experiment to compare a theory of risk-aversion and a theory of spite as an explanation for overbidding in auctions. As a workhorse we use the second-price all-pay auction. Both risk and spite are used to rationalize deviations from risk-neutral equilibrium bids. We exploit that equilibrium predictions in the second-price all-pay auctions for spite are different than those for risk-aversion. We find that spite is a convincing explanation for bidding behavior for the second-price all-pay auction. Not only can spite rationalize observed bids, also our measure for spite is consistent with observed bids.

Effects of risk aversion in auctions without and with default

We study the second‐price auctions without and with default, where bidders are assumed to be risk averse and have uncertain valuations. We show that without default both the equilibrium bid and the seller's expected revenue decrease with risk aversion, and with default the equilibrium bid still decreases but the seller's expected revenue may decrease or increase as risk aversion increases. These results imply that without default the seller prefers risk‐neutral bidders to risk‐averse bidders, but with default the seller's preference is ambiguous, which contrast with the well‐known results obtained in auctions with certain valuations.

Does jump bidding increase sellers’ revenue? Theory and experiment

We consider an environment in which bidders decide whether to jump bids in a simple two-bidder ascending auction with independent private values. We compare two types of equilibria, one that involves jump bidding and another that does not. We show that the revenue in the jump bidding equilibrium dominates that in the no-jump equilibrium when bidders are risk averse. Isolating the revenue impact of jump bidding from that of overbidding, our experimental design allows us to demonstrate that sellers’ revenue increases due to jump bidding but only insignificantly so.

The bid orchestration and competitions in scoring procurement auctions

In this paper, we focus on the issue of bid orchestration in scoring procurement auctions, which is realized by a supplier, who receives preferential evaluation of the respective bid in exchange of a bribe to the auctioneer. For first‐score and second‐score auctions, with an honest (noncorrupted) and a dishonest (corrupted) supplier, we investigate the equilibrium bidding strategy and find that an asymmetry between the two bidders is created by the presence of quality manipulation. We also explore how bidding is affected compared with the symmetric case where both opponents are honest. Our main findings indicate that both bidders (honest and dishonest) find the first‐score auction more profitable compared with the second‐score auction format. Contrary to the symmetric case, the asymmetry of quality manipulation differentiates the utility of the buyer and translates to a preference of the buyer for a second‐score auction. Finally, we also provide some discussions on policy implications of our results.

Entry Deterrence and Free Riding in License Auctions: Incumbent Heterogeneity and Monotonicity

Abstract We examine free riding for entry deterrence in license auctions with heterogeneous incumbents. We establish the monotonicity of randomized preemptive bidding equilibria: an incumbent with a higher entry-loss rate has greater free-riding incentive, choosing a lower deterring probability. We then identify conditions for the existence of a series of fully or partially participating equilibria such that two or more incumbents with bounded heterogeneity in their entry-loss rates participate in randomized preemptive bidding. As an application, we examine a simple case of a bipartite group of participating incumbents consisting of one “leader” and many “followers”. We show that the policy of limiting the leader’s participation (set-asides for entrants, limiting participation of incumbents with excessive market shares, etc.) may or may not increase entry probability.

Effects of competition in first-price auctions

This paper analyzes the effects of industrial concentration on bidding behavior and expected revenues. These effects are studied under the CIAPI model, an affiliated value set-up that nests a variety of valuation and information environments. We formally decompose the revenue effects coming from less competition into five types. The properties of these effects are discussed and conditions for (non)monotonicity of both the equilibrium bid and revenue are stated. Our results suggest that it is more likely that the seller benefits from less competition in markets with more complete valuation and information structures.

Balancing Cost Effectiveness and Incentive Properties in Conservation Auctions: Experimental Evidence from Three Multi-award Reverse Auction Mechanisms

Government agencies are increasingly using economic incentives to encourage landowners to adopt conservation practices. Auctions enable agencies to identify land conservation practices with low opportunity costs. At the same time, landowners’ opportunity costs contain useful information for government agencies to rank conservation priorities. This paper introduces a new reverse auction mechanism that performs well both from the cost effectiveness and cost-revelation perspectives and compares three multi-award reverse auction mechanisms. The first mechanism is called the Uniform Price Reverse (UPR) auction, where each winning bidder is paid the lowest rejected bid. The second mechanism is called the First Price Reverse (FPR) auction, where winning bidders are paid their submitted bids. The third, novel, mechanism is called the Generalized Second Price Reverse (GSPR) auction, where each winning bidder is paid the bid that is immediately higher. Theoretically, I derive the equilibrium bidding strategy for each auction mechanism and show that a symmetric equilibrium strategy may not exist under the GSPR auction. Empirically, lab experiment results show that UPR and GSPR auctions lead to a higher efficiency level compared to FPR, while UPR auction yields the lowest auctioneer surplus and is the least cost effective. As a result, GSPR maintains good incentive properties similar to UPR and presents potential large cost-saving opportunities to the auctioneer.

Entry Deterrence and Free Riding in License Auctions: Incumbent Heterogeneity and Monotonicity

We examine free riding for entry deterrence in license auctions with heterogeneous incumbents. We establish the monotonicity of randomized preemptive bidding equilibria: an incumbent with a higher entry-loss rate has greater free-riding incentive, choosing a lower deterring probability. We then identify conditions for the existence of a series of fully or partially participating equilibria such that two or more incumbents with bounded heterogeneity in their entry-loss rates participate in randomized preemptive bidding. As an application, we examine a simple case of a bipartite group of participating incumbents consisting of one “leader” and many “followers”. We show that the policy of limiting the leader’s participation (set-asides for entrants, limiting participation of incumbents with excessive market shares, etc.) may or may not increase entry probability.

Discrete Richman-bidding scoring games

We study zero-sum combinatorial games within the framework of so-called Richman auctions (Lazarus et al., in Games No Chance 29:439–449, 1996). We modify the alternating play scoring ruleset Cumulative Subtraction (Cohensius et al., in Electron J Combin 26(P4):52, 2019), to a discrete bidding scheme, similar to Develin and Payne (Electron J Combin 17(1):85, 2010). Players bid to move, and the player with the highest bid wins the move and hands over the winning bid amount to the other player. The new game is dubbed Bidding Cumulative Subtraction. In so-called unitary games, players remove exactly one item out of a single heap of identical items, until the heap is empty, and their actions contribute to a common score, which increases or decreases by one unit depending on whether the maximizing player wins the turn or not. We show that there is a unique bidding equilibrium for a much larger class of games that generalize standard scoring play. We prove that for all sufficiently large heap sizes, the equilibrium outcomes of unitary games are eventually periodic, with period 2. We show that the periodicity appears at the latest for heaps of sizes quadratic in the total budget.

Reserve price of risk-averse search engine in keyword auctions with advertisers’ endogenous investment

Motivated by vigorous development of keyword auctions, this paper analyzes the reserve price policies in keyword auction with advertisers’ endogenous investment and risk-averse search engine. We explore advertisers’ optimal investment and equilibrium bidding strategies and derive the determination functions where utility-maximizing reserve price and efficient reserve price which maximizes the social welfare satisfy respectively. The results show that advertisers’ equilibrium bidding is monotonously increasing in bidders’ valuations, the number of advertisers, as well as the reserve price. Meanwhile, advertisers’ optimal investment is negatively correlated with reserve price and the number of advertisers. By numerical examples, the utility-maximizing reserve price decreases with the risk aversion parameter and the number of advertisers. Search engine’s expected utility increases with risk aversion parameter and decreases with the number of advertisers. Moreover, we declare that search engine can use reserve price as a regulatory tool to increase the utility. But there is an upper bound on search engine’s utility. It is interesting to find the efficient reserve price equals to zero. Hence there is a trade-off between total efficiency and search engine’s utility by search engine that has incentive to withhold reserve price that would benefit social welfare.

Property rights in the Crypto age: NFTs and the auctioning of limited edition artwork

Non-fungible tokens (NFTs) on blockchains have recently emerged as a means of certifying the originality of digital properties, such as artwork. In this paper, we examine limited-edition auctions for the sale of digital artwork using NFTs. We study two types of limited-edition auctions that have been used in practice: the `silent' and `ranked' auctions. We argue that the silent limited-edition auction as currently used in NFT markets is a variant of the well-known discriminatory price auction. We derive the Bayesian Nash equilibrium of this auction, and show that it is revenue equivalent to a VCG auction. Our analysis suggests that bidding behavior in a silent limited-edition auction is more aggressive than a standard discriminatory price auction without editioning; consequently, equilibrium bids are higher in the former. We also study the generalized English auction as an outcome equivalent auction to the ranked limited-edition auction, and show that this does not have a truthful equilibrium. Finally, we examine the uniform-price auction as a potential alternative mechanism for conducting an auction with editioning, and establish the absence of a truthful equilibrium in this instance as well. Our paper represents one of the first attempts to formally model the allocation of property rights using auctions in the digital environment, which is at the forefront of current innovation.

Decentralized Payment Clearing using Blockchain and Optimal Bidding

In this paper, we construct a decentralized clearing mechanism which endogenously and automatically provides a claims resolution procedure. This mechanism can be used to clear a network of obligations through blockchain. In particular, we investigate default contagion in a network of smart contracts cleared through blockchain. In so doing, we provide an algorithm which constructs the blockchain so as to guarantee the payments can be verified and the miners earn a fee. We, additionally, consider the special case in which the blocks have unbounded capacity to provide a simple equilibrium clearing condition for the terminal net worths; existence and uniqueness are proven for this system. Finally, we consider the optimal bidding strategies for each firm in the network so that all firms are utility maximizers with respect to their terminal wealths. We first look for a mixed Nash equilibrium bidding strategies, and then also consider Pareto optimal bidding strategies. The implications of these strategies, and more broadly blockchain, on systemic risk are considered.

Auctions with signaling concerns

AbstractWe study a symmetric private value auction with signaling, in which the auction outcome is used by an outside observer to infer the bidders' types. We elicit conditions under which an essentially unique D1 equilibrium bidding function exists in the second‐price and the English auctions. We establish there is no equivalence between these two auction designs, neither in bidding strategies nor in expected revenue. This is because the presence or the absence of an increasing price clock affects signaling incentives differently in both auction formats, and thereby also the bidders' incentives to overbid their types. This leads to a strictly higher expected revenue in the second‐price auction than in the English auction. Our analysis is completed by a comparison with other disclosure policies. Applications include art auctions and charity auctions.

AUCTIONS WITH LEAKS ABOUT EARLY BIDS: ANALYSIS AND EXPERIMENTAL BEHAVIOR

In sequential first‐ and second‐price private value auctions, second movers are informed about the first movers' bid with commonly known probability. Equilibrium bidding in first‐price auctions is mostly unaffected, but there are multiple equilibria in second‐price auctions affecting comparative statics across price rules. We show experimentally that informational leaks in first‐price auctions qualitatively confirm the theoretical predictions. In second‐price auctions, we analyze and experimentally confirm the existence of focal equilibria, and provide evidence for individual consistency in equilibrium selection. (JEL D44, C72, C91)

An indirect evolutionary justification of risk neutral bidding in fair division games

We justify risk neutral equilibrium bidding in commonly known fair division games with incomplete information by an evolutionary setup postulating (i) minimal common knowledge, (ii) optimal responses to conjectural beliefs how others behave and (iii) evolutionary selection of conjectural beliefs with fitness measured by expected payoffs. After justifying the game forms we derive the evolutionary games for first- and second-price fair division and determine the evolutionarily stable conjectures. The latter coincide with equilibrium bidding, irrespectively of the number of bidders, i.e., heuristic belief adaptation can imply the same bidding behavior as equilibrium analysis based on common knowledge and counterfactual bidding.

Renewable auctions: Bidding for real options

Procurement auctions for renewable energy support have become a standard policy instrument to stimulate investment in clean energy. Winning bidders have the right but not the obligation to realize their projects during a grace period following the auction. Currently, the nexus of award prices and the realization rate is not well understood in the literature. We combine auction theory and real options theory to model bidders who view the right to build subsidized renewable capacity as real option. Using asymptotic theory for multi-unit auctions, we derive optimal bidding strategies and analyze how auction design and bidder characteristics impact equilibrium bids, award prices, and realization rates. In particular, we show that bidders who value the flexibility of non-realization higher bid more aggressively and exhibit lower realization rates. We analyze determinants of these effects and illustrate how auction design can trade-off procurement cost and realization rates by adjusting pre-qualification payments and the grace period for construction. Finally, we test our results on real-world auctions in UK and Germany and show that our model explains auction outcomes and observed realization rates.

Optimal reserve prices in sealed-bid auctions with reference effects

The reference effect and loss aversion are incorporated into the buyer’s utility in the symmetric independent private value models of sealed-bid auctions. The buyer’s equilibrium bidding strategy and the seller’s optimal reserve price are derived for the first-price and second-price sealed-bid auctions. In both auction mechanisms, the seller’s optimal reserve price and expected revenue are increasing in the reference point. We compare the seller’s expected revenues as well as the optimal reserve prices in the two auctions. The results show that the seller will set a higher optimal reserve price but obtain lower optimal expected revenue in the second-price auction compared to the first-price auction. Further, we extend the model to the gain-seeking case, and endogenize the reference point as the ex-ante expected price of the item in equilibrium. In contrast to the loss-averse case, the seller will set a lower optimal reserve price but obtain higher optimal expected revenue in the second-price auction compared to the first-price auction if the buyers are gain-seeking. With an endogenous reference point, similar results are obtained in terms of revenue comparison between the two auctions.

Robust bidding and revenue in descending price auctions

We study the properties of Dutch auctions in an independent private value setting, where bidders face uncertainty over the type distribution of their opponents and evaluate their payoffs by the worst case from a set of probabilistic scenarios. In contrast to static auction formats, participants in the Dutch auction gradually learn about the valuations of other bidders. We show that the transmitted information can lead to changes in the worst-case distribution and thereby shift a bidder's payoff maximizing exit price over time. We characterise the equilibrium bidding function in this environment and show that the arriving information leads bidders to exit earlier at higher prices. As a result, the Dutch auction systematically generates more revenue than the first-price auction.