Equilibrium Bidding Strategies Research Articles

Risk aversion in first-price security-bid auctions

We study how risk aversion affects equilibrium bidding strategies in a first-price security-bid auction. We show that if the bidder's expected utility is concave and their marginal expected utility decreases as the degree of risk aversion increases – conditions that are shown to be satisfied under some conditions for four prominent securities (cash, equity, debt, and call option) –, the increase in the degree of risk aversion leads to lower bids in equilibrium.

Toeholds and information quality in common‐value takeover auctions

AbstractIn this article I analyze the effect of the sensitivity of firm value on the information available to potential acquirers in common‐value takeover auctions with toeholds. I show that the quality of information does not affect equilibrium when bidders have equal toeholds but has a significant effect when toeholds are different. My article demonstrates that increasing the relative information quality of the bidder with a smaller toehold makes both bidders bid more aggressively and leads to a higher price. I also analyze the combined effect of toeholds and information quality on equilibrium bidding strategies and discuss ways target shareholders can increase the expected final price.

Auction-based parking mechanisms considering withdrawal behaviors

Auction-based parking mechanisms (ABPM) have been recognized as promising solutions to the parking problem. As the operator of the ABPM, the parking platform should not only be well-operated to provide parking services for demanders in need but also maintain commercial viability for sustainable operations. Motivated by this, we propose three ABPMs, namely, the ABPM without withdrawal right, the ABPM with costless withdrawal right, and the ABPM with non-free withdrawal right. Particularly, we characterize the parking demanders' withdrawal behaviors in the ABPMs. This behavior is motivated by the demanders' negative utilities derived from the auction sequence, where the demanders will first submit bid prices and are then allowed to withdraw from the parking market after the announcement of auction results when they are aware of their true valuations. We derive the equilibrium bidding strategies and the platform's expected revenues under the above three mechanisms and compare them in terms of the analytical results. It has been found that the introduction of withdrawal right will stimulate demanders to raise their bids for winning and the ABPM with non-free withdrawal right generates the highest revenue under certain conditions. A case study of parking in Beijing Financial Street has been conducted to illustrate the findings and explore managerial implications, including adapting the ABPM with non-free withdrawal right, narrowing the parking supply-demand gap, and enhancing travel experiences on the way to parking.

Comparisons of standard royalty auctions with seller post-auction effort

This study considers standard royalty auctions, in which each bidder’s valuation is determined endogenously by his type and the seller’s post-auction effort. We first characterize the unique symmetric separating D1 PBE of both sealed bid and ascending bid royalty auctions and compare them in terms of allocative efficiency, production efficiency, and seller revenue. We then show that no symmetric (semi-) pooling PBE survives the D1 criterion in these auctions (under mild conditions). Furthermore, we explore each bidder’s equilibrium bidding strategy, the associated efficiency and revenue, under a second-price royalty auction without disclosing the winning bid. We also consider a scenario where the seller adopts an entry fee instead of a reserve price to maximize their expected payoff. Finally, we examine the robustness of revenue ranking with respect to general forms of value creation.

Contextual Standard Auctions with Budgets: Revenue Equivalence and Efficiency Guarantees

The internet advertising market is a multibillion dollar industry in which advertisers buy thousands of ad placements every day by repeatedly participating in auctions. An important and ubiquitous feature of these auctions is the presence of campaign budgets, which specify the maximum amount the advertisers are willing to pay over a specified time period. In this paper, we present a new model to study the equilibrium bidding strategies in standard auctions, a large class of auctions that includes first and second price auctions, for advertisers who satisfy budget constraints on average. Our model dispenses with the common yet unrealistic assumption that advertisers’ values are independent and instead assumes a contextual model in which advertisers determine their values using a common feature vector. We show the existence of a natural value pacing–based Bayes–Nash equilibrium under very mild assumptions. Furthermore, we prove a revenue equivalence showing that all standard auctions yield the same revenue even in the presence of budget constraints. Leveraging this equivalence, we prove price of anarchy bounds for liquid welfare and structural properties of pacing-based equilibria that hold for all standard auctions. In recent years, the internet advertising market has adopted first price auctions as the preferred paradigm for selling advertising slots. Our work, thus, takes an important step toward understanding the implications of the shift to first price auctions in internet advertising markets by studying how the choice of the selling mechanism impacts revenues, welfare, and advertisers’ bidding strategies. This paper was accepted by Itai Ashlagi, revenue management and market analytics. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2023.4719 .

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 show that a pure strategy Nash equilibrium need not exist, and then look for mixed Nash equilibrium bidding strategies. The implications of these strategies, and more broadly blockchain, on systemic risk are considered.

A Conceptual Analysis of Equilibrium Bidding Strategy in a Combined Oligopoly and Oligopsony Wholesale Electricity Market

This paper proposes a semi-analytical method to obtain the equilibrium bidding strategies for generation and demand units in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">combined</i> oligopoly and oligopsony wholesale electricity market. Such market structure is the outcome of the increasing deployment of demand response programs that facilitate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">active participation of demand-side players</i> in the price-setting process. In this analysis, the concept of supply function equilibrium (SFE) is used to investigate the oligopolistic competition among generation units. The SFE model is extended and the demand function equilibrium (DFE) is obtained to study the oligopsonistic competition among demand units. The economic behavior of a market participant is formulated as a bi-level programming (BLP) problem. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">imperfect competition</i> among generation units, as well as among demand units, are modeled as a non-cooperative game. Next, a direct method is developed to calculate all candidate equilibriums of the market, and the locational marginal prices (LMPs) in terms of the bidding strategies of the market participants. The BLP problem is solved by obtaining the coordinated Pareto-dominant Nash equilibrium of the market participants’ non-cooperative games. Finally, the proposed analysis is examined in case studies. Accordingly, we report <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">insightful observations</i> with respect to the impact of the changes in the new market structure, at firm-level and market-level, such as in terms of mitigating market power of generation units, the market clearing prices and quantities, surplus for generation units and demand units, and potential impact on market efficiency.

The Strange Role of Information Asymmetry in Auctions—Does More Accurate Value Estimation Benefit a Bidder?

We study the second-price auction in which bidders have asymmetric information regarding the item’s value. Each bidder’s value for the item depends on a private component and a public component. While each bidder observes their own private component, they hold different and asymmetric information about the public component. We characterize the equilibrium of this auction game and study how the asymmetric bidder information affects their equilibrium bidding strategies. We also discover multiple surprisingly counter-intuitive equilibrium phenomena. For instance, a bidder may be better off if she is less informed regarding the public component. Conversely, a bidder may sometimes be worse off if she obtains more accurate estimation about the auctioned item. Our results suggest that efforts devoted by bidders to improve their value estimations, as widely seen in today’s online advertising auctions, may not always be to their benefit.

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.

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.

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.

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.

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.

Reference-Dependent Preferences and Overbidding in Private and Common Value Auctions

This paper proposes a model of reference dependent preferences to explain overbidding in private and common value auctions. It is assumed that the reference point is proportional to the value of the object and that losses are weighed more heavily than gains in the utility function. Equilibrium bidding strategies are derived for first- and second-price private and common value auctions. I find that this model fits the data of all experiments analyzed, both private and common value, better in terms of the Bayesian Information Criterion than a standard risk neutral model; moreover, it explains overbidding in all private value and some common value auctions better than other alternative models. These results suggest that reference dependence, among other factors, might play a role in the widespread tendency of subjects to overbid in most experimental auctions.

Short vs. long-term procurement contracts when supplier can invest in cost reduction

In dynamic markets where cost of components changes fast, buyers typically auction off regular short-term contracts to fully leverage supplier competition in each period to continuously source from the lowest-cost supplier. However, too much competition through short-term contracts does not incentivize the incumbent supplier to make relation-specific investments in reducing costs, as future business is not assured. We investigate this trade-off, between leveraging supplier competition in each period versus incentivizing incumbent’s investment, with a stylized two-period model in which the buyer decides whether to auction off short-term contracts in each period or auction off a single long-term contract spanning both periods. In both cases, we characterize the optimal incumbent supplier’s investment, the suppliers’ equilibrium bidding strategy and the buyer’s expected cost. Our analysis shows that the supplier always invests more in a long-term contract. However, the buyer’s cost depends on supply base size: it prefers short-term contracts for large supply base size, otherwise it prefers long-term contract. Moreover, we find that system cost is typically lower with short-term contracts and that the suppliers are always better off with short-term contracts. Finally, adding non-discriminatory or discriminatory reserve prices to our model does not fundamentally modify the trade-off, but we find that auctions with discriminatory reserve price are better at balancing this trade-off compared to long or short-term contracts.

Auctions with endogenous opting‐out fees and recursive winning procedures from the Talmud

AbstractThis article analyzes open ascending and first‐price sealed‐bid auctions with uncertain value, recursive winning procedure, and cooling‐off right in return for an endogenous opting‐out fee. I show that although equilibrium bidding strategies in these auctions tend to be less aggressive than in auctions with costless withdrawals, expected revenues are usually higher. Both auctions are spurious‐bidding free and the sealed‐bid auction is almost shill‐bidding robust.

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.

An ascending auction with multi-dimensional signals

This paper examines a single-unit ascending auction where agents observe two-dimensional Gaussian signals. The model combines the pure private-values model with the pure common-values model. The challenge is to characterize how the multi-dimensional signals observed by an agent are aggregated onto that agent's one-dimensional bid. The challenge is solved by projecting an agent's private signals onto a one-dimensional equilibrium statistic; the equilibrium bidding strategies are constructed as if each agent observed only his own equilibrium statistic. An agent's equilibrium statistic aggregates this agent's private signals while taking into account the additional information deduced from the other agents' bids. In contrast to one-dimensional environments, an ascending auction may have multiple symmetric equilibria that yield different social surpluses.

Learning in Repeated Auctions with Budgets: Regret Minimization and Equilibrium

In online advertising markets, advertisers often purchase ad placements through bidding in repeated auctions based on realized viewer information. We study how budget-constrained advertisers may compete in such sequential auctions in the presence of uncertainty about future bidding opportunities and competition. We formulate this problem as a sequential game of incomplete information, in which bidders know neither their own valuation distribution nor the budgets and valuation distributions of their competitors. We introduce a family of practical bidding strategies we refer to as adaptive pacing strategies, in which advertisers adjust their bids according to the sample path of expenditures they exhibit, and analyze the performance of these strategies in different competitive settings. We establish the asymptotic optimality of these strategies when competitors’ bids are independent and identically distributed over auctions, but also when competing bids are arbitrary. When all the bidders adopt these strategies, we establish the convergence of the induced dynamics and characterize a regime (well motivated in the context of online advertising markets) under which these strategies constitute an approximate Nash equilibrium in dynamic strategies: the benefit from unilaterally deviating to other strategies, including ones with access to complete information, becomes negligible as the number of auctions and competitors grows large. This establishes a connection between regret minimization and market stability, by which advertisers can essentially follow approximate equilibrium bidding strategies that also ensure the best performance that can be guaranteed off equilibrium. This paper was accepted by Noah Gans, stochastic models and simulation.

Efficient Multi-Attribute Auctions Considering Supply Disruption

Although supply disruption is ubiquitous because of natural or man-made disasters, many firms still use the price-only reverse auction (only the cost is considered) to make purchase decisions. We first study the suppliers’ equilibrium bidding strategies and the buyer’s expected revenue under the first- and second-price price-only reverse auctions when the suppliers are unreliable and have private information on their costs and disruption probabilities. We show that the two auctions are equivalent and not efficient. Then we propose two easily implementable reverse auctions, namely the first-price and second-price format announced penalty reverse auction (APRA), and show that the “revenue equivalence principle” holds, i.e., the two auctions generate the same ex ante expected profit to the buyer. We further show that the two reverse auctions are efficient and “truth telling” is the suppliers’ dominant strategy in the second-price format APRA. We conduct numerical studies to assess the impacts of some parameters on the bidding strategies, the buyer’s profit and social profit.