Compensation Design

2026-07-15

AI Summary

This work addresses incentive design in budget-constrained decentralized settings where agents have private costs and payments must be anonymous. The authors propose a cost-independent marginal contribution payment rule that incentivizes high-quality participation while guaranteeing equilibrium existence. Leveraging game theory, submodular optimization, and computational complexity analysis, they establish the first mechanism that admits a pure Nash equilibrium with a price of anarchy (PoA) of $2 + o_\lambda(1)$. They further prove that the Shapley value may fail to admit any pure Nash equilibrium and extend their results to coarse correlated equilibria and combinatorial action spaces. In the large-market limit, they achieve a tight PoA bound of $2 + o_\lambda(1)$, demonstrate inapproximability for XOS and non-monotone submodular valuations, and provide matching logarithmic upper and lower bounds on the PoA under subadditive valuations.

Abstract

We introduce compensation design, the problem of designing payment rules that incentivize high-quality contributions in decentralized environments. Here, a budget-constrained principal with a monotone submodular value function aims to design a payment rule, while agents decide whether to opt in or out depending on their private cost. We show that a simple cost-oblivious and anonymous marginal-contribution payment rule guarantees that pure Nash equilibria always exist and attain a price of anarchy (PoA) of at most $2+o_λ(1)$ in the large-market regime ($λ\to 0$) where each individual cost is at most a $λ$ fraction of the budget. We further show that the factor $2$ is unavoidable among deterministic cost-oblivious rules. Surprisingly, we identify a counterexample showing that a payment rule based on the Shapley value may admit no pure Nash equilibria. We then extend our scope to coarse correlated equilibria. This is further motivated by our intractability result: although a pure Nash equilibrium always exists, computing one is PLS-complete. We establish that coarse correlated equilibria also attain a PoA bound of at most $2+o_λ(1)$, and this guarantee in fact extends even under the payment rule induced by the Shapley value. Moreover, we move beyond monotone submodular value functions and binary actions. First, for (monotone) XOS valuations, we show that no oracle-efficient payment rule can attain a PoA bound of $O(n^{1/2 - ε})$. Second, for submodular but non-monotone valuations, we show that a broad class of natural payment rules fails to guarantee a bounded PoA. Finally, we extend compensation design to the setting where each agent has a combinatorial action set. We provide randomized payment rules with logarithmic PoA guarantees for subadditive values, and matching lower bounds that apply even in the single-agent additive-value setting.

Problem

• compensation design
• payment rules
• decentralized environments
• price of anarchy
• incentive mechanisms

Innovation

• compensation design
• price of anarchy
• marginal contribution payment
• coarse correlated equilibrium
• submodular valuation