On the optimal design of all-pay auctions

This paper examines the optimal design of contests in the presence of negative prizes and establishes the optimality of a modified all-pay auction with entry fee and reserve. The entry fee always equals the contestants’ liability, and the reserve is weakly higher than in contests without negative prizes. The modification involves awarding all contestants a strictly positive prize if none meet the reserve. This optimal contest better incentivizes high-ability contestants by offering them a higher prize augmented by entry fees, while still ensuring full participation from low-ability contestants. Theoretical analysis demonstrates that when contestants’ liability is sufficiently high, the same contest maximizes both the expected total effort and winner’s effort, with both measures increasing with liability. Numerical simulations show that even with low liability, predictions from the two optimal contests are closely aligned. To test these predictions, we conduct an experiment comparing optimal contests across different liability levels, confirming the “killing-two-birds-with-one-stone” prediction. This paper was accepted by Elena Katok, operations management. Funding: The authors gratefully acknowledge the financial support of the UTS Business School Research Grant, UTS Behavioural Lab Grant and National Natural Science Foundation of China [Grants 72192842, 72203099, 72422018]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.03580 .

Contests are ubiquitous but do not simply arise in a vacuum. Competitors make conscious decisions before the fighting stage. This paper looks at the interplay between the decision to enter and then undertake a pre-contest investment to enhance the chance of winning the prize. We use an all-pay auction to model the contest stage; investment cost is private information and its return is stochastic. We characterize equilibrium in terms of threshold strategies on the cost parameter, both for the entry and the subsequent investment decision. We show that the stochastic nature of the investment outcome has a non-monotonic effect on players decisions in equilibrium. A contest designer can use our results to directly achieve goals related to the maximization of contest effort, entry fee revenue and investment propensity. In all cases, we demonstrate that limiting entry may be optimal. • We consider the interplay between entry and investment decisions in a contest. • We derive threshold strategy equilibrium for entry and investment. • Conditions for full entry and investment, and limited entry/investment. • Stochastic return to investment has a non-monotonic effect on equilibrium actions. • An entry fee can be set to achieve design goals. Full entry is not always optimal.

In an auction, each party bids a certain amount, and the one who bids the highest is the winner. Interestingly, auctions can also be used as models for other real-world systems. In an all-pay auction all parties must pay a forfeit for bidding. In the most commonly studied all-pay auction, parties forfeit their entire bid, and this has been considered as a model for expenditure on political campaigns. Here, we consider a number of alternative forfeits that might be used as models for different real-world competitions, such as preparing bids for defense or infrastructure contracts.

Damaging conflict: All-pay auctions with negative spillovers and bimodal bidding

Providing information in financing: Experimental evidence from all-pay auctions

Spiteful bidding in first-price all-pay auctions

Abstract Consider a war of attrition game in continuous time with complete information, in which N ≥ 2 players compete for N − K prizes. I focus on the equilibria in which the strategies follow exponential distributions, which are memoryless. When K = 1, such an equilibrium can be explicitly characterized. The equilibrium certainly exists if N = 2. If N ≥ 3, it exists as long as the weakest player is not too weak compared to the average. If it exists, the equilibrium is unique under some conditions. When K ≥ 2, the game typically has nondegenerate equilibria in which K − 1 relatively weak players concede at the beginning. The model can be extended to the case in which the players have loser-dependent valuations. The model helps to solve a generalized exit game in a “nature oligopoly” and an all-pay auction with ascending bids.

We consider group all-pay auctions with a wide range of effort complementarity in which teammates coordinate efforts via a group-specific correlation device. Under mild regularity conditions, there is a unique equilibrium effort distribution, independent of the distribution of the correlation device. We characterize this unique distribution of efforts and analyze the effects of value dispersion, degree of complementarity, and group size. We show that this effort distribution can be achieved without correlation devices as the equilibrium outcome of a cheap talk game in which players engage in costless unmediated preplay communication.

Corrigendum to: “Ridge distributions and information design in simultaneous all-pay auction contests”

All-Pay Auctions with Discrete Private Information: Regularization and Optimal Design

Providing Information in Financing: Experimental Evidence from All-Pay Auctions

The Regularization and Optimal Design of All-Pay Auctions

Ridge distributions and information design in simultaneous all-pay auction contests

Resource pricing is an important issue in mobile edge computing, how to determine the bid of end user (EU) appropriately is an incentive factor for edge cloud (EC) to offer service. In this paper, we propose an equilibrium pricing scheme based on all-pay auction model in end-to-end collaboration environment, wherein the EUs can acquire service at a lower price than the own value of the required resource. In addition, we propose a set allocation algorithm to divide all bidders into different sets according to their resource valuations, which averts the situation of the higher bidder monopolizing the EC resources. Extensive simulation results demonstrate that the proposed scheme effectively maximizes the total profit of ECs while guaranteeing the EU can access services at a lower price.

Disclosure policies in all-pay auctions with affiliated values

All-pay vs. standard auctions when competing for budget-constrained buyers

I study a multi-stage all-pay auction in which the lowest bidder in each stage is eliminated. Elimination continues until only two bidders remain, one of whom wins the auction. I analyse optimal bidding behaviour and the seller’s expected revenue when bidders have independent and private values. In contrast to typical bidding strategies, the optimal bid in each stage is strictly decreasing in the number of bidders. For a fixed number of bidders, however, bids increase as bidders progress through the stages of the auction. Despite independent values, this multi-stage auction yields less expected revenue to the seller than its single-stage counterpart when there are more than three bidders.

We study the power of price discrimination via an intermediary in bilateral trade, when there is a revenue-maximizing seller selling an item to a buyer with a private value drawn from a prior. Between the seller and the buyer, there is an intermediary that can segment the market by releasing information about the true values to the seller. This is termed signaling, and enables the seller to price discriminate. In this setting, Bergemann et al. [ 7 ] showed the existence of a signaling scheme that simultaneously raises the optimal consumer surplus, guarantees the item always sells, and ensures the seller’s revenue does not increase. Our work extends the positive result of Bergemann et al. to settings where the type space is larger, and where the optimal auction is randomized, possibly over a menu that can be exponentially large. In particular, we consider two settings motivated by budgets: The first is when there is a publicly known budget constraint on the price the seller can charge [ 12 ] and the second is the FedEx problem [ 19 ] where the buyer has a private deadline or service level (equivalently, a private budget that is guaranteed to never bind). For both settings, we present a novel signaling scheme and its analysis via a continuous construction process that recreates the optimal consumer surplus guarantee of Bergemann et al. and further subsumes their signaling scheme as a special case. In effect, our results show settings where even though the optimal auction is randomized over a possibly large menu, there is a market segmentation such that for each segment, the optimal auction is a simple posted price scheme where the item is always sold. The settings we consider are special cases of the more general problem where the buyer has a private budget constraint in addition to a private value. We finally show that our positive results do not extend to this more general setting, particularly when the budget can bind in the optimal auction, and when the seller’s mechanism allows for all-pay auctions. Here, we show that any efficient signaling scheme necessarily transfers almost all the surplus to the seller instead of the buyer.

Bayesian Nash equilibrium in all-pay auctions with interdependent types