Anonymous Contracts

Abstract

We study a multi-agent contracting problem where agents exert costly effort to achieve individually observable binary outcomes. While the principal can theoretically extract the full social welfare using a discriminatory contract that tailors payments to individual costs, such contracts may be perceived as unfair. In this work, we introduce and analyze anonymous contracts, where payments depend solely on the total number of successes, ensuring identical treatment of agents. We first establish that every anonymous contract admits a pure Nash equilibrium. However, because general anonymous contracts can suffer from multiple equilibria with unbounded gaps in principal utility, we identify uniform anonymous contracts as a desirable subclass. We prove that uniform anonymous contracts guarantee a unique equilibrium, thereby providing robust performance guarantees. In terms of efficiency, we prove that under limited liability, anonymous contracts cannot generally approximate the social welfare better than a factor logarithmic in the spread of agent success probabilities. We show that uniform contracts are sufficient to match this theoretical limit. Finally, we demonstrate that removing limited liability significantly boosts performance: anonymous contracts generally achieve an $O(\log n)$ approximation to the social welfare and, surprisingly, can extract the full welfare whenever agents' success probabilities are distinct. This reveals a structural reversal: widely spread probabilities are the hardest case under limited liability, whereas identical probabilities become the hardest case when limited liability is removed.

Figures (2)

• Figure 1: (a) Under limited liability, the worst-case gap between social welfare $\textup{Sw}$ and anonymous contracts $\textsc{Anonym}$ grows as $\min\{\log(\overline{Q} n), n\}$, where $\overline{Q} = \max_{i,j} {q_i}/{q_j}$. Without limited liability, if $\underline{Q} = \min_{i \neq j} {q_i}/{q_j} = 1$, the worst-case gap stays at $\log n$; once $\underline{Q} > 1$, this gap becomes exactly $1$.
• Figure 2: We portray a cycle composed of two peak walks: One going from set $\{2,3\}$ until set $\{2,3,4,5\}$ (in blue), and the other from set $\{2,3,4,5\}$ until set $\{2,3\}$ (in red).

Theorems & Definitions (70)

• Proposition 2.0: General contracts, CastiglioniM023
• Theorem 3.1
• Lemma 3.1
• Lemma 3.1
• Claim 3.1
• Claim 3.1
• Claim 3.1
• proof : Proof Sketch of Lemma \ref{['lem:nocycle']}
• Proposition 3.2
• proof