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Optimal Crowdsourcing Contests
Shuchi Chawla, Jason D. Hartline, Balasubramanian Sivan
TL;DR
The paper asks how to design crowdsourcing contests that maximize the quality of the best submission and how their inefficiency compares with conventional procurement. It models contests as all-pay auctions, develops virtual-value-based optimal mechanisms, and shows that contests are within factors of 2 or 4 of conventional methods. Static winner-take-all is optimal, while dynamic optimal contests allocate rewards using transformed submission qualities.
Problem
The paper studies which crowdsourcing competition format maximizes winning-submission quality and how inefficient crowdsourcing is relative to conventional contracting.
Method
The paper models crowdsourcing as an all-pay auction and characterizes optimal contests using ironed virtual values based on contestant skills and contest size.
Results
Crowdsourcing contests are 2-approximations to conventional procurement for regular distributions and 4-approximations more generally; winner-take-all is optimal among static allocations.
Takeaways & Limitations
Crowdsourcing can achieve near-conventional procurement performance despite wasted losing effort, while optimal reward sharing may depend dynamically on submission qualities.
Takeaways & Limitations
The utilization-ratio bound does not hold for arbitrary symmetric all-pay auctions, particularly when ironing creates large tied bid intervals.
Abstract
from arXiv · showhide
We study the design and approximation of optimal crowdsourcing contests. Crowdsourcing contests can be modeled as all-pay auctions because entrants must exert effort up-front to enter. Unlike all-pay auctions where a usual design objective would be to maximize revenue, in crowdsourcing contests, the principal only benefits from the submission with the highest quality. We give a theory for optimal crowdsourcing contests that mirrors the theory of optimal auction design: the optimal crowdsourcing contest is a virtual valuation optimizer (the virtual valuation function depends on the distribution of contestant skills and the number of contestants). We also compare crowdsourcing contests with more conventional means of procurement. In this comparison, crowdsourcing contests are relatively disadvantaged because the effort of losing contestants is wasted. Nonetheless, we show that crowdsourcing contests are 2-approximations to conventional methods for a large family of "regular" distributions, and 4-approximations, otherwise.
1 Introduction
The paper frames crowdsourcing contests as all-pay auctions whose objective is maximizing the best submission, then characterizes their efficiency and optimal reward structures. It shows that crowdsourcing can approach conventional procurement while favoring winner-take-all among static allocations and virtual-value-based dynamic allocation.
- Motivation: Crowdsourcing contests map to all-pay auctions, but the principal values the maximum submission quality rather than total payments.Contestant submissions correspond to bids, contestant skills to private values, and effort is paid upfront by all entrants.
- Comparison with procurement: The paper compares crowdsourcing with conventional procurement because losing contestants’ effort is wasted, even though all-pay and conventional auctions can have equal total revenue.This creates a loss when the principal values only the winning submission.
- Approximation guarantees: 2-approximation follows because the maximum agent payment in highest-bid-wins all-pay auctions is at least half of total revenue.For regular distributions, minimum-quality contests therefore remain 2-approximations to conventional procurement; more generally, they are 4-approximations.
- Static rewards: Winner-take-all is optimal among static reward allocations, despite mechanisms such as TopCoder’s 2/3-to-1/3 division.The result concerns maximizing the quality of the best submission, not total submission quality.
- Dynamic rewards: Optimal dynamic contests are ironed virtual-value optimizers that divide rewards among contestants tied under a transformed submission quality.The number of reward recipients is determined dynamically, and the transformation depends on the number of contestants.
2 Preliminaries
The preliminaries establish the auction-theoretic model underlying crowdsourcing contests and distinguish maximum-payment optimization from revenue optimization. They define equilibrium and virtual-value tools used to analyze reserves, regularity, and optimal allocation.
- Auction model: A standard auction specifies allocation and payment rules for agents with private values, with Bayes-Nash equilibrium requiring truthful strategy choices to be optimal against others’ strategies.The model assumes independently drawn values from a continuous distribution and interim allocation and payment rules.
- Auction formats: Revenue equivalence states that mechanisms with the same equilibrium allocation generate the same expected revenue.This principle connects first-price, second-price, and all-pay formats despite their different payment timing.
- Optimal auctions: Virtual valuations transform values so that revenue-optimal auctions maximize virtual value subject to a monotone allocation rule.For regular distributions, the optimal auction awards the item to the highest positive virtual value, equivalently using a reservation value.
- All-pay implementation: In an all-pay auction, the highest bidder wins while every bidder pays, and equilibrium bids match expected payments from an equivalent second-price auction.A value reserve r is implemented with the all-pay reserve bid rF(r)^(n−1).
- Irregular distributions: Irregular distributions require ironing, which converts a non-monotone virtual valuation into a monotone function whose pointwise maximization implements the optimal allocation.This provides the auction-theoretic basis for handling non-regular skill distributions.
- Crowdsourcing model: Crowdsourcing models submission quality as p_i=v_ie_i and evaluates a contest by the expected maximum submission rather than the sum of submissions.Conventional first- and second-price procurement can extract the winner’s full payment, whereas all-pay contests waste losing effort.
3 Utilization and approximation ratios
The paper bounds how much of an all-pay contest’s total revenue is captured by its highest payment, yielding constant-factor approximations to optimal procurement. These guarantees are tight in relevant settings and do not extend to arbitrary symmetric all-pay auctions.
- Utilization ratio: Rev[A] ≤ 2MP[A] for any highest-bidder-wins reserve-price all-pay auction, so its utilization ratio is at most 2.The proof decomposes revenue into the winning agent’s payment and all other agents’ payments, then shows the former is at least the latter.
- Limitations: The utilization bound can fail for arbitrary symmetric all-pay auctions when ironing creates large intervals of tied bids and low payments.In such cases, many agents contribute equal payments, but only one contributes to the maximum payment.
- Approximation ratios: For regular value distributions, a highest-bid-wins all-pay auction with a suitable reserve bid achieves approximation ratio at most 2.The result uses revenue equivalence and the fact that highest-bidder-wins reserve-price auctions are revenue-optimal for regular distributions.
- Approximation ratios: For all i.i.d. value distributions, a highest-bid-wins all-pay auction with a suitable reserve bid achieves approximation ratio at most 4.The bound combines the utilization result with the factor-2 approximation of anonymous-reserve highest-bidder-wins auctions for irregular distributions.
- Tightness: The factor-2 guarantee is tight for uniform distributions, while the worst-case cost of crowdsourcing can be no smaller than 2.For the uniform example, the expected maximum payment approaches 1/2, and even the optimal all-pay auction only approaches that value.
4 Optimal crowdsourcing contests
The paper characterizes optimal crowdsourcing contests by maximizing a distribution- and contestant-count-dependent virtual value, first for static contests and then for arbitrary symmetric contests. Regular distributions yield reserve-price highest-bid-wins mechanisms, while irregular distributions require ironing and forbidden bid intervals.
- Static contests: Optimal static all-pay contests are highest-bid-wins auctions.
- Symmetric contests: The expected maximum payment of any symmetric all-pay auction can be characterized by a virtual value, making the optimum a virtual value maximizer.
- Regular distributions: For distributions regular with respect to maximum payment, the optimal mechanism is highest-bid-wins with a reserve price.
- Examples: For the uniform distribution on [0,1], the optimal reserve bid is 1/(n + 1), and expected maximum payment approaches 1/2 as n increases.
- Irregular distributions: For irregular distributions, ironing produces forbidden bid intervals, with rewards divided equally among highest bidders above the reserve.
- Irregular distributions: Irregularity increases with n because the value intervals requiring ironing expand, although this need not increase the number of tied winners.
5 Prior-independent approximation
The section shows that a simple highest-bid-wins contest without a reserve approximates the optimal contest for regular distributions, while additional agents cannot improve its factor beyond 2.
- Prior-independent approximation: Bulow and Klemperer’s result implies that the no-reserve highest-value-wins auction on n agents achieves at least a (1 −1/n) fraction of optimal revenue for regular distributions.The argument transfers this approximation benchmark to crowdsourcing contests.
- Prior-independent approximation: 2n/(n −1) is the approximation ratio obtained by a highest-bid-wins all-pay auction without a reserve bid for i.i.d. revenue-regular distributions.The ratio approaches 2 as n grows.
- Prior-independent approximation: Unlike highest-value-wins auctions, all-pay auctions do not improve their approximation ratio beyond 2 when more agents are added.For highest-value-wins auctions without reserves, revenue converges to optimal as the number of agents increases.
A Proof of Theorem 4.1
The proof analyzes symmetric static allocation rules through their induced bid and payment functions, then shows that shifting reward mass toward the highest-ranked bidder improves the maximum-payment objective. Its integral argument establishes the required sign conditions.
- Allocation and symmetry: Symmetry and i.i.d. values imply identical bidding functions across agents and a symmetric allocation as a function of values.This lets the proof study a common bid function induced by the allocation rule.
- Allocation and symmetry: The proof restricts attention to symmetric static allocation rules, where the i-th highest bidder receives reward fraction a_i and all remaining bidders receive zero.The allocation is represented as A = (a_1, ..., a_k, 0, ..., 0), with reward fractions summing to one.
- Payment representation: By revenue equivalence, the bid of an agent with value z equals the expected truthful-auction payment under the same allocation rule.The expected payment of the r-th highest bidder is expressed using conditional expectations of higher order statistics.
- Reward concentration: The proof studies how changing a_k while transferring the corresponding mass to a_1 affects the maximum-payment objective, aiming to show that its derivative is negative.This establishes that the optimal allocation places all reward mass on a_1, the highest-ranked bidder.
- Sign argument: The derivative is bounded above by an expression Q whose relevant integrands are shown to be negative using the monotonicity of g and integration-by-parts transformations.The remaining argument reduces to proving non-negativity of H_n(x), using its endpoint behavior and one-turn shape.