Contracting with a Mechanism Designer

This paper explores the economic interactions within modern crowdsourcing markets. In these markets, employers issue requests for tasks, platforms facilitate the recruitment of crowd workers, and workers complete tasks for monetary rewards. Recognizing that these roles serve distinct functions within the ecosystem, we introduce a three-party model that distinguishes among the principal (the requester), the intermediary (the platform), and the pool of agents (the workers). The principal, unable to directly engage with agents, relies on the intermediary to recruit and incentivize them. This interaction unfolds in two stages: first, the principal designs a profit-sharing contract with the intermediary; second, the intermediary implements a mechanism to select an agent to complete the delegated task. We analyze the proposed model as an extensive-form Stackelberg game. Our contributions are threefold. First, we fully characterize the subgame perfect equilibrium of our model. In particular, the principal’s contract design problem can be represented as virtual value pricing, a novel auction-theoretic formulation. We identify the optimality of linear contracts, even when the task has multiple outcomes and agents’ cost distributions are asymmetric. Second, to quantify the principal’s utility loss from delegation and information asymmetry, we introduce the price of double marginalization (PoDM) and the classical price of anarchy (PoA). We derive tight or nearly tight bounds on both ratios under regular and monotone hazard rate distributions. Finally, we extend our analysis to two natural variants of the base model: (i) the intermediary is restricted to anonymous pricing mechanisms, and (ii) the principal lacks precise information about the market size.

💡 Research Summary

The paper develops a three‑party model of modern crowdsourcing markets, consisting of a principal (the requester), an intermediary (the platform), and a pool of agents (the workers). The principal cannot interact directly with workers and therefore contracts with the intermediary, offering a profit‑sharing agreement that pays the platform a fraction of the realized task outcome. After the contract is signed, the intermediary runs a Bayesian mechanism to recruit and select a worker, who then completes the task and generates a stochastic outcome that yields a fixed reward to the principal.

The interaction is formalized as an extensive‑form Stackelberg game. The principal moves first by choosing the contract parameters (essentially a share α of the reward), and the intermediary moves second by selecting a mechanism that maximizes its own expected utility given the contract. Workers’ private types are their costs of execution, drawn independently from known distributions; the outcome distribution of the task is common knowledge.

Using backward induction, the authors fully characterize the subgame‑perfect Nash equilibrium (SPNE). They show that, for any contract, the intermediary’s optimal mechanism is equivalent to Myerson’s revenue‑optimal single‑item auction, where each worker’s cost is transformed into a virtual value. The optimal mechanism assigns the task to the lowest‑cost worker whose cost is below a reserve price derived from the contract, and pays that worker the larger of the reserve and his cost. This mechanism satisfies Bayesian incentive compatibility and individual rationality.

Given the intermediary’s optimal mechanism, the principal’s contract design problem reduces to maximizing expected payoff subject to the induced reserve price. The paper distinguishes three settings: (i) asymmetric outcome and cost distributions, (ii) identical outcome distributions but asymmetric cost distributions, and (iii) identical outcome and cost distributions. In the most general case (i) the optimal contract may be nonlinear. However, in cases (ii) and (iii) the authors prove that a simple linear contract—paying a fixed fraction of the reward conditional on success—is optimal. The intuition is that when all workers share the same outcome distribution, the principal’s payoff depends only on whether the task is completed, not on which worker is selected; thus any contract that induces the same expected payment can be linearized without loss of optimality.

When both outcomes and costs are identical (case iii), the contract problem can be reformulated as a “virtual‑value pricing” problem: the principal effectively sells a single item to two virtual buyers whose values are uniformly distributed, but receives the virtual value of the posted price rather than the price itself. This captures the double‑marginalization effect: the intermediary’s markup reduces the principal’s effective revenue to the virtual value, which is always less than or equal to the actual price under regularity conditions.

To quantify efficiency loss, the authors introduce two ratios. The Price of Double Marginalization (PoDM) compares the principal’s utility when using the intermediary to the utility she would obtain by contracting directly with workers. The Price of Anarchy (PoA) compares the equilibrium utility to the socially optimal utility when both parties coordinate. Under regular and monotone hazard‑rate (MHR) distributions, they derive tight (or nearly tight) bounds of Θ(log κ) for both ratios, where κ is the ratio of the highest to lowest value in the support of the cost distribution. As the number of workers n grows, the loss remains logarithmic rather than linear, indicating that the inefficiency does not explode in large markets.

The paper also studies two natural extensions. First, when the intermediary is restricted to anonymous pricing (a single posted price to all workers), the optimal mechanism simplifies to a take‑it‑or‑leave‑it offer, and the PoDM/PoA bounds worsen modestly but retain the same logarithmic order. Second, when the principal lacks precise knowledge of the market size n, she must design a contract based on a prior distribution over n. The authors show that linear contracts remain approximately optimal and that the same logarithmic loss bounds continue to hold.

Overall, the work bridges contract theory and auction theory to analyze the strategic interaction between requesters and platforms in crowdsourcing. It demonstrates that even with hidden mechanisms and asymmetric information, simple linear contracts are often optimal, and that the efficiency loss due to double marginalization is bounded and manageable. The virtual‑value pricing insight provides a novel connection between profit‑sharing contracts and classic auction pricing, opening avenues for further research on dynamic contracts, multi‑task allocation, and non‑monetary incentives in platform‑mediated labor markets.