Compensation Design

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 $2+o_{\lambda}(1)$ in the large-market regime ($\lambda \to 0$) where each individual cost is at most a $\lambda$ 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 $2+o_\lambda(1)$, 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 $O(n^{1/2 - \epsilon})$. 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.