All-pay Auction Research Articles

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

Two privately informed contestants compete in a contest, and the organizer ex-ante designs a public anonymous disclosure policy to maximize the contestants' total effort. We fully characterize ridge distributions, under which the organizer achieves the first-best outcome in equilibrium: the allocation is efficient, and the entire surplus goes to the organizer. When the prior is a mixture of a ridge distribution and a perfectly correlated distribution, the first-best outcome is achievable by a signal that solely generates ridge distributions as posteriors.

All-Pay Auction Based Profit Maximization in End-to-End Computation Offloading System

All-Pay Auction Based Profit Maximization in End-to-End Computation Offloading System

Disclosure policies in all-pay auctions with affiliated values

We consider all-pay auctions with private and affiliated binary values. The organizer commits ex ante to fully disclosing or concealing bidders’ valuations. We examine the effects of full disclosure on bidders’ welfare and the organizer’s revenue (i.e., expected aggregate bid). We find that full disclosure, as opposed to full concealment, always yields a higher welfare for bidders. On the other hand, full concealment yields a higher revenue if the affiliation in bidders’ valuations is low or if there are only two bidders. Conversely, if the affiliation is high and there are sufficiently many bidders, full disclosure tends to increase revenue.

Sequential elimination in multi-stage all-pay auctions

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.

Optimal Price Discrimination for Randomized Mechanisms

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

We prove the existence of a behavioral-strategy Bayesian Nash equilibrium in all-pay auctions with statistically interdependent types (signals) under quite general assumptions on the values, costs and tie-breaking rules. Moreover, the set of equilibria is shown to be the same for any tie-breaking rule used in the auction.

Competition for Budget-Constrained Buyers: Exploring All-Pay Auctions

Competition for Budget-Constrained Buyers: Exploring All-Pay Auctions

The tripartite auction folk theorem

We formally study two bidder first-price, second-price, and all-pay auctions with known values, deriving the equilibrium payoffs and strategies and showing when all three yield the same equilibrium payoffs to the bidders. This latter result, the tripartite auction theorem, does not hold for all auctions, in particular it can fail for symmetric auctions with high stakes and in auctions with very low stakes.

Optimal disclosure in all-pay auctions with interdependent valuations

We study all-pay auctions with one-sided private information and interdependent valuations. To sharpen the competition and maximize revenue, the auction organizer can design an information disclosure policy through Bayesian persuasion about the bidder with private information. We characterize optimal disclosure and find that optimal disclosure exhibits almost full disclosure, where the uninformed bidder can always narrow the informed bidder's private information down to at most two types. We also illustrate our characterization in a simple binary-type setting and investigate issues such as comparative statics, welfare, and efficiency.

The n-player Hirshleifer contest

While the game-theoretic analysis of conflict is often based on the assumption of multiplicative noise, additive noise such as considered by Hirshleifer (1989) may be equally plausible depending on the application. In this paper, we examine the equilibrium set of the n-player difference-form contest with heterogeneous valuations. For high and intermediate levels of noise, the equilibrium is in pure strategies, with at most one player being active. For small levels of noise, however, we find a variety of equilibria in which some but not necessarily all players randomize. In the case of homogeneous valuations, we obtain a partial uniqueness result for symmetric equilibria. As the contest becomes increasingly decisive, at least two contestants bid up to the valuation of the second-ranked contestant, while any others ultimately drop out. Thus, in the limit, equilibria of the Hirshleifer contest share important properties of equilibria of the corresponding all-pay auction.

Correlated play in weakest-link and best-shot group contests

We explore public randomization (Harris et al., 1995) in group contests and introduce group public randomization equilibria (GPRE). We consider group all-pay auctions with weakest-link and best-shot impact functions. While best-shot contests without public randomization are known for their multiplicity of equilibria, introducing public randomization results in a unique GPRE in which only one of the strongest players in each group is active. However, in the weakest-link case, the well-known multiplicity of equilibria becomes even more pronounced with public randomization. Despite this multiplicity, a refinement that selects GPRE immune to coalitional deviations reduces the gamut of GPRE to a unique equilibrium group-effort distribution, which, for identical groups composed of identical agents, features the highest expected total effort among all equilibria of the full information game without correlation devices. The (weak) increase in expected efforts due to correlated strategies is also observed in the best-shot case.

Two-stage elimination games

We study two-stage elimination games with four heterogeneous players. In the first stage, the players are allocated into two contests, each of which includes two players competing against each other in an all-pay auction. The winners of the two contests then interact with each other in the second stage. The outcomes of the interaction in the second stage are given by a general form of the players’ payoffs as functions of their types (abilities). Then, we examine the effect of the timing of the competitions in the first stage on the players’ expected payoffs and their total effort. In particular, we show that if the players’ payoff functions in the second stage are separable in the players’ types, then the timing of play in the first stage of either simultaneous or sequential contests has no effect on the players’ expected payoff as well as their total effort in the elimination games.

Persuasion with ambiguous receiver preferences

AbstractI describe a Bayesian persuasion problem where Receiver has a private type representing a cutoff for choosing Sender’s preferred action, and Sender has maxmin preferences over all Receiver type distributions with known mean and bounds. This problem can be represented as a zero-sum game where Sender chooses a distribution of posterior mean beliefs that is a mean-preserving contraction of the prior over states, and an adversarial Nature chooses a Receiver type distribution with the known mean; the player with the higher realization from their chosen distribution wins. I formalize the connection between maxmin persuasion and similar games used to model political spending, all-pay auctions, and competitive persuasion. In both a standard binary-state setting and a new continuous-state setting, Sender optimally linearizes the prior distribution over states to create a distribution of posterior means that is uniform on a known interval with an atom at the lower bound of its support.

Mapping an information design game into an all-pay auction

Mapping an information design game into an all-pay auction

Information sharing decisions in all-pay auctions with correlated types

In many real-life competitions, contestants may not be aware of their own type (e.g., value or ability) prior to the contest. Furthermore, contestants’ types, which are observed privately after entering the competition, are frequently correlated with one another. We examine a two-stage competition that involves two players with correlated (binary) types. In the first stage, players decide simultaneously or sequentially on the probabilities they use to disclose or conceal their type, which will become their private information later on. In the second stage, each player privately observes their own type and commits to disclosing or concealing it, after which they compete in an all-pay auction. We discover that information sharing does not occur when players’ types are negatively correlated. However, when players’ types are positively correlated, information is partially shared in all equilibria examined in this study. In an asymmetric pure strategy equilibrium, one player shares his information with probability one and the other player with probability zero. In a symmetric mixed strategy equilibrium, each player shares his information with the same positive probability.

Correlated equilibria and communication equilibria in all-pay auctions

Correlated equilibria and communication equilibria in all-pay auctions

Subsidy and taxation in all-pay auctions under incomplete information

We study all-pay auctions under incomplete information with n contestants who have non-linear cost functions. The designer may award two kinds of subsidy (taxation): one that decreases (increases) each contestant's marginal cost of effort and another that increases (decreases) each contestant's value of winning. The designer's expected payoff is the contestants' expected total effort minus the cost of subsidy or, alternatively, plus the tax payment. We show that when the resource of subsidy (the marginal taxation rate) is relatively small and the cost function is concave (convex), the designer's expected payoff in all-pay auctions with both kinds of subsidy (taxation) is higher than in the same contest without any subsidy (taxation).

Disclosure Policies in All-Pay Auctions with Bid Caps

In this paper, we study the optimal disclosure of bidders’ types in a two-bidder all-pay auction in the presence of a common and exogenously given bid cap. We focus on the set of disclosure policies that are contingent on realizations of bidders’ types and fully characterize the unique symmetric equilibrium under each of them. We find that, allowing only for full disclosure and full concealment, the ex ante expected revenue is higher under full disclosure only for intermediate levels of the bid cap. Allowing also for partial disclosure policies: (i) the ex ante expected revenue under full disclosure is higher than under partial disclosure to low-type bidders but lower than under partial disclosure to high-type bidders, regardless of the bid cap; (ii) full concealment could rank from the highest to the third highest, depending on the bid cap, the prior type distribution, and the type values. These results are due to the different sensitivities of bidders’ bids across bidders’ types and disclosure policies.

The economics of China’s between-city height competition: A regression discontinuity approach

In this article, I use a regression discontinuity design to quantify the race between Chinese cities to build taller skyscrapers. I begin by characterizing this height competition with a game-theoretic model for multi-stage, two-player, all-pay auctions with carryover. The dominant mixed strategy equilibrium of this dynamic game implies that a city would be more inclined to construct a taller skyscraper if its current tallest building is merely shorter than the tallest in the rival city, and this inclination is discontinuous at which the heights of these two buildings are equal. Utilizing a large data set of China’s skyscrapers constructed between 2004 and 2019, I confirm the presence of this between-city contest by empirically identifying the discontinuity in a city’s probability of constructing a taller building. I also find that this height competition has been more prevalent among the cities with fewer historic amenities or with leaders that have lower promotion likelihoods prior to the start of office.

The role of the second prize in all-pay auctions with two heterogeneous prizes

We study complete information all-pay contests with n players and two heterogeneous prizes with distinct values. Among the players, n−1 are symmetric (i.e., they evaluate the prizes in a similar manner), whereas the remaining player has different valuations for each of the prizes. Our analysis focuses on the equilibrium profiles and expected payoffs for the case of three players. We also partially extend our analysis for cases with additional players. Our results show that in all-pay auctions with heterogeneous prizes, the ordering of the players according to their expected payoffs in equilibrium might vary significantly, depending on both prizes. Furthermore, although the values for the first (larger) prize have the greatest effect on the identity of the players with positive expected payoffs, the value of the second prize might have a non-negligible effect.