Risk-neutral Bidders Research Articles

Quantile regression methods for first-price auctions

The paper proposes a quantile-regression inference framework for first-price auctions with symmetric risk-neutral bidders under the independent private-value paradigm. It is first shown that a private-value quantile regression generates a quantile regression for the bids. The private-value quantile regression can be easily estimated from the bid quantile regression and its derivative with respect to the quantile level. This also allows to test for various specification or exogeneity null hypothesis using the observed bids in a simple way. A new local polynomial technique is proposed to estimate the latter over the whole quantile level interval. Plug-in estimation of functionals is also considered, as needed for the expected revenue or the case of CRRA risk-averse bidders, which is amenable to our framework. A quantile-regression analysis to USFS timber is found more appropriate than the homogenized-bid methodology and illustrates the contribution of each explanatory variable to the private-value distribution. Linear interactive sieve extensions are proposed and studied in the Appendices.

Corporate Bonds: Uniform Auction under Risk and Budget Limits

A perfectly divisible corporate bond is allocated to a set of bidders characterized by limits both to their budget, but most importantly to the risk entailed in their portfolio. Bidders possess symmetric information concerning the secondary market's yield. We choose to use a uniform pricing mechanism contrary to discriminatory as the former generates more revenues and reduces the winner's curse. As a first step, we prove the existence of symmetric Bayesian Nash equilibrium when risk-neutral bidders respond to an exogenous secondary market providing the necessary comparative statics. In the second stage, we confirm the existence of equilibrium when bidders' types are affiliated with the secondary market by a copula.

Identification in Auction Models with Interdependent Costs

This paper provides a nonparametric identification result for procurement models with asymmetric bidders, dependent private information, and interdependent costs. For risk-neutral bidders, the model’s payoff-relevant primitives are the joint distribution of private information and each bidder’s full-information expected cost. The joint distribution of bids identifies the joint distribution of signals. First-order conditions identify the expected cost conditional on tying with at least one competitor for the lowest bid. I show identification of each bidder’s full-information cost, using variation in competitors’ cost shifters that are excludable from bidders’ own full-information costs, and generate variation in the set of competitors’ signals that induce a tie for the lowest bid. I estimate the relevant payoff primitives using data from Michigan highway procurements and evaluate policies that affect the winner’s curse’s severity.

Partnerships of Bidders with Constant Relative Risk Aversions

In this paper, we study a dynamic auction for allocating a single indivisible project while different participants have different bid values for the project. When the price rises continuously, the bidders can retreat the auction and obtain the compensation by the difference between the price at retreating time and the previous bid price. The final successful bidder achieves the project and pays compensations to others. We show that the auction of bidders with constant relative risk aversion (CRRA) has a unique equilibrium. While the relative risk aversion coefficient approaches to zero, the equilibrium with CRRA bidders would approach to the equilibrium with risk-neutral bidders.

A model of parallel contests

We develop a model of two parallel contests, asymmetric in quantity of homogeneous prizes open to contest, with a finite number of homogeneous risk-neutral bidders. Whether the bidder upon entry into a particular contest is aware of the realized number of competing contestants in the contest is irrelevant to the expected effort at equilibrium. At equilibrium the expected effort per capita in the larger contest (the contest with more prizes) is greater than that in the smaller one. The larger contest nonetheless does not attract enough contestants to achieve optimum in rent extraction from the bidders.

Scale effects in multi-unit auctions

We argue that auctions with multi-unit demand can be used to experimentally conduct new and novel tests of auction theory. The focus of the paper is on changes in bidding behavior as the auction is “scaled up” by adding more bidders and increasing both the demand per bidder and supply of the auctioned good. We identify uniquely tractable environments and obtain clear cut theoretical comparative statics in these settings. This leads to the construction of uniform price auctions of different scales where the prediction is that risk neutral bidders' bids on the last unit they demand are independent of scale. In our experiment, bidders on average bid much more aggressively than predicted by theory. There is also evidence of a scale effect, with more aggressive bidding in the small-scale treatment compared to the large-scale treatment. It is argued that these features are consistent with joy of winning and anticipated regret.

Pure-strategy equilibria in Bayesian games

A general condition called “coarser inter-player information” is introduced and shown to be necessary and sufficient for the validity of several fundamental properties on pure-strategy equilibria in Bayesian games, such as existence, purification from behavioral strategies, and convergence for a sequence of games. Our sufficiency results cover various earlier results on pure-strategy equilibria in Bayesian games as special cases. New applications are presented as illustrative examples, including auctions with externalities and risk-neutral bidders, and Bertrand pricing games with asymmetric information.

A Non-Game-Theoretic Approach to Bidding in First-Price and All-Pay Auctions

We propose a novel approach to the modelling of behavior in first-price and all-pay auctions that builds on the use of a heuristic to achieve an Impulse Balance Equilibrium. The resulting individual decision-making model, nIBE, assumes no expected profit-maximization and accommodates any distribution of private values. Its parameter-free variant entails the Symmetric Bayes Nash Equilibrium bidding strategy for risk neutral bidders. Assuming impulse weighting may lead to under- or overbidding and organizes the effect of end-of-round information feedback on behavior. We assess nIBE’s explanatory power with experimental data and find that it usually outperforms the available regret models of bidding in first-price and all-pay auctions.

Participation Costs and Inefficiency in Takeover Contests

We consider a takeover in which risk neutral bidders incur private costs when they participate in the auction. Supposing that valuations for a target firm are common knowledge, we study the optimal strategy of bidders and analyze the takeover result when they get or do not get toeholds in the target firm. We find that bidders’ decision of participation is endogenous. By analyzing bidders’ condition of participation, we find that there is a probability that the potential bidder with the highest valuation does not participate in the control. We show that this probability increases with the size of toeholds possessed by the bidder with a low valuation. Nevertheless, a larger size of toeholds possessed by the bidder with the highest valuation increases the probability of his participation in the auction. Asymmetric toeholds between potential bidders can accentuate inefficient allocation when participation costs are private information. However, if the bidder with toeholds deters completely bidders without toeholds in private valuation setting (Ettinger [Ettinger D (2009) Takeover contests, toeholds and deterrence. Scandinavian J. Econom. 111(1):103–124.]), the results of the current papers show that the phenomenon of fully deterring is destroyed when valuations are common knowledge.

Strategies for Making Best Offers on eBay

We model for “Buy-It-Now or Best Offer” auctions on eBay using two different models. In the first model, risk-neutral bidders submit bids in serial and try to surpass a stochastic seller threshold while taking into account how many previous failed bids were made by other bidders. We compute optimal strategies for this model and show that bidder expected surplus decreases in the number of previous failed bids. In the second model we assume bidders do not know how many previous failed bids have been made, and instead use a first-price sealed-bid mechanism with a buy-out price where bidders serially submit bids with the knowledge that no previous bidders have used the buy-out price. We derive a unique equilibrium bidding strategy for risk-neutral bidders in this serial model, show that any equilibrium in a similar parallel bidding model is the same as the equilibrium in the serial model, and compute seller revenue. In particular, under certain circumstances, bidders will bid more in this format than they would in a standard first-price sealed-bid auction, but that a seller maximizes expected revenue by setting a buy-out price higher than any bidder is willing to pay thereby making the auction essentially a first-price auction. KEYWORDS: Auction Theory; eBay; Buy-It-Now or Best Offer; Symmetric Bayesian Nash Equilibrium; Buy-Out Price; First-Price Sealed-Bid

A Model of Parallel Contests

We develop a model of parallel contests, asymmetric in quantity of homogeneous prizes open to contest, with a finite number of homogeneous risk-neutral bidders assumed to have a linear effort-cost function. We show that at equilibrium the expected effort per capita in the larger contest is greater than in a smaller one. The larger contest nonetheless does not attract enough contestants to achieve the optimum in extracting the rent from the bidders. Such an optimum can be achieved by a careful design of the asymmetric entry fees across the contests. The asymmetry between the contest-specific entry fees decreases as the number of the bidders increases, and approaches zero in the limit.

Sequential Auctions with Generalized Interdependent Values

A common assumption in the analysis of symmetric auctions is that the bidders' value estimates exhibit positive informational externalities (PIE). This assumption implies upward drifting price sequences at sequential auctions, which is challenged by an empirical regularity, known as the price anomaly, that observed price sequences at real sequential auctions tend to be downward-drifting. This paper extends the existing analysis to a generalized interdependent values environment, in which the bidders' values can exhibit both PIE and NIE (negative informational externalities). The case of NIE can arise naturally when competing bidders are also competitors in the same product market. If a bidder's type is related to his or his firm's ensuing competitive advantage, then an increase of a bidder's type increases his own but may decrease other bidders' expected values.We consider a general sequential auction mechanism that sells m identical objects through K (≤m) consecutive rounds, each round involving possibly a different number of objects for sale and a different payment rule. For risk neutral bidders having unit demand and independent types, we obtain two major results. First, the direct sequentially incentive compatible auction mechanisms, which implement the performance of essentially all standard auctions, are feasible under both PIE and NIE. Second, while the total expected revenue is invariant to sequencing and payment rules, the expected selling prices from different rounds of the auction are not the same. In a PIE environment the expected price sequence tends to be upward drifting, whereas in an NIE environment the expected price sequence is strongly downward drifting: the expected lowest price in round k exceeds the expected highest price in round k 1. The declining price anomaly could, therefore, be evidence of bidders' values featuring NIE or post-auction competition.

Ascending Combinatorial Auctions with Risk Averse Bidders

Ascending combinatorial auctions are being used in an increasing number of spectrum sales worldwide, as well as in other multi-item markets in procurement and logistics. Much research has focused on pricing and payment rules in such ascending auctions. However, recent game-theoretical research has shown that such auctions can even lead to inefficient perfect Bayesian equilibria with risk-neutral bidders. There is a fundamental free-rider problem without a simple solution, raising the question whether ascending combinatorial auctions can be expected to be efficient in the field. Risk aversion is arguably a significant driver of bidding behavior in high-stakes auctions. We analyze the impact of risk aversion on equilibrium bidding strategies and efficiency in a threshold problem with one global and several local bidders. Due to the underlying free-rider problem, the impact of risk-aversion on equilibrium bidding strategies of local bidders is not obvious. We characterize the necessary and sufficient conditions for the perfect Bayesian equilibria of the ascending auction mechanism to have the local bidders to drop at the reserve price. Interestingly, in spite of the free-riding opportunities of local bidders, risk-aversion reduces the scope of the non-bidding equilibrium. The results help explain the high efficiency of ascending combinatorial auctions observed in the lab.

Sequential Auctions with Waiting Costs

For certain types of goods, the multiple unit auctions have to be conducted sequentially. One probable reason for this is that the different units of the goods are not available together for putting up for sale. This might happen when the objects are available in batches to the auctioneer and do not come together. In such sequential auctions, there may be waiting costs involved for the bidders. This note makes an attempt to look at the bidding behaviour when waiting costs are private information for the bidders and therefore constitute types. The results derived suggest that, in a symmetric, independent, private valuations framework, with risk neutral bidders, there exists a symmetric Perfect Bayesian equilibrium bidding strategy, for which the bids are increasing in types in the first stage and decreasing in types in the second stage.

Equilibrium bids in practical multi-attribute auctions

This article proposes a nonlinear scoring rule which transforms multiple attributes of a bid into comparable dimensionless ones. Practically, the buyer can use it to select the most competitive winner. For risk-neutral bidders, we characterize a symmetric Bayes–Nash equilibrium and find that as the number of bidders increases the equilibrium quality improves, whereas the equilibrium price decreases.

The Role of Risk Preferences in Pay-to-Bid Auctions

We analyze a new auction format in which bidders pay a fee each time they increase the auction price. Bidding fees are the primary source of revenue for the seller but produce the same expected revenue as standard auctions (assuming risk-neutral bidders). If risk-loving preferences are incorporated in the model, expected revenue increases. Our model predicts a particular distribution of ending prices, which we test against observed auction data. The degree of fit depends on how unobserved parameters are chosen; in particular, a slight preference for risk has the biggest impact in explaining auction behavior, suggesting that pay-to-bid auctions are a mild form of gambling. This paper was accepted by Peter Wakker, decision analysis.

Management Insights

Management Insights

Multiple Motivations Consequences on Bidder's Optimal Strategy in Takeover Contests

This paper examines the optimal bidding strategy in takeover contests for a target firm, and the positive correlation between the bidders’ valuation. We consider risk neutral bidders who compete for the control of a target firm in which they get initial shareholdings. The bidder valuation for target firm is correlated with his motivations which determine the bidder’s strategy. We study bidder’s optimal strategy in mixed motivations setting. Since motivations are numerous, hypothesis of affiliated value in auctions allows to study bidder’s strategy. The paper shows that the impact of affiliation degree on bidder’s optimal strategy depends on their private signal and on the ratio between their initial shareholdings. Particularly, we found that if a potential bidder overbids when he gets toeholds in the target firm, the extent of overbidding is not monotonically increasing with the size of toeholds and the level of private signal. There is a threshold of private signal under which the bidder is more aggressive given the size of his toeholds and also in which the bidder is less aggressive. The target firm’s expected revenue increases with the degree of affiliation of bidder’s valuation.When potential bidders get identical size of toeholds, their bid doesn’t depend on the valuation’s affiliation degree. Their optimal offer increases both in their private signal and in toeholds. In this case, the bidder with highest signal wins and competition between bidders yields to an efficient allocation.

Approximately optimal auctions for correlated bidders

We consider the design of dominant strategy incentive compatible, revenue-maximizing auctions for an indivisible good, when bidders' values are drawn from a correlated distribution. For independent distributions, Myerson showed that the optimal auction for risk-neutral bidders remains incentive compatible regardless of bidders' risk attitudes. We show that, for correlated distributions, the same is true when only two bidders are involved, whereas for more bidders, randomization can generate strictly more revenue. However, for risk-neutral bidders, we show this gain is never more than a 53-factor.This is a consequence of two results of independent interest: (1) a polynomial-time derandomization of auctions for two bidders; (2) a polynomial-time computable deterministic auction that 53-approximates the optimal revenue extractable from risk-neutral bidders.Moreover, we give a polynomial-time algorithm to compute optimal auctions for a constant number of bidders, and for any number of bidders we give polynomial-time algorithms with approximation factors arbitrarily close to 32.

The public release of information in first-price sealed-bid auctions

We determine the conditions under which the seller in a first-price sealed-bid auction has an incentive to reveal his private information about the mean of bidders’ valuations in order to increase his expected revenue. With risk-neutral bidders, we show that the seller’s expected revenue is higher when information is revealed. However, when bidders are risk-averse, this result does not necessarily hold; it depends on the bidders’ risk-aversion level and on the number of bidders.