When Contracts Get Complex: Information-Theoretic Barriers

TL;DR

This work establishes intrinsic information-theoretic barriers to tractable optimal contract design in the combinatorial-action model when reward and cost functions are submodular or supermodular in various combinations. The authors construct equal-revenue instances with additive costs that yield exponentially many incentivizable action-sets all achieving the same principal payoff, and they pair this with a polynomial-bounded, yet sharply structured, sparse-demand property to prove exponential hardness for value and demand queries, respectively. They extend these insights to communication complexity through a two-party model with best-response oracles, showing that even with BR access the optimal contract demands exponential communication in most cases, except in a narrow tractable regime. The results are framed as tight within the existing BR-based approximation landscape, and the techniques (equal revenue, sparse demand, and DISJ-based reductions) illuminate fundamental barriers to efficient contract design for a broad class of combinatorial objectives. Together, the findings delineate the information-theoretic limits of tractable contract design and guide future work toward identifying and exploiting tractable subfamilies or alternative contract forms.

Abstract

In the combinatorial-action contract model (Dütting et al., FOCS'21) a principal delegates the execution of a complex project to an agent, who can choose any subset from a given set of actions. Each set of actions incurs a cost to the agent, given by a set function $c$, and induces an expected reward to the principal, given by a set function $f$. To incentivize the agent, the principal designs a contract that specifies the payment upon success, with the optimal contract being the one that maximizes the principal's utility. It is known that with access to value queries no constant-approximation is possible for submodular $f$ and additive $c$. A fundamental open problem is: does the problem become tractable with demand queries? We answer this question to the negative, by establishing that finding an optimal contract for submodular $f$ and additive $c$ requires exponentially many demand queries. We leverage the robustness of our techniques to extend and strengthen this result to different combinations of submodular/supermodular $f$ and $c$; while allowing the principal to access $f$ and $c$ using arbitrary communication protocols. Our results are driven by novel equal-revenue constructions when one of the functions is additive, immediately implying value query hardness. We then identify a new property -- sparse demand -- which allows us to strengthen these results to demand query hardness. Finally, by augmenting a perturbed version of these constructions with one additional action, thereby making both functions combinatorial, we establish exponential communication complexity.

Figures (6)

• Figure 1: The utilities of the players for $f$ and $\{c_i\}_{i\in [n]}$ as defined above, for the case of $n=3$. Observe that each of the critical values yields the principal the same expected utility of 1.
• Figure 2: The agent's utility in an equal-revenue instance $\langle n, \bar{f}, \bar{c} \rangle$ (solid), and the utility for the perturbed instance $\langle n, \bar{f}_k, \bar{c} \rangle$ (dashed). Note that the perturbed critical values satisfy $\alpha'_k < \alpha_k$ and $\alpha_{k+1} < \alpha'_{k+1}$.
• Figure 3: Illustration of ambiguity intervals (top), and minimal ambiguous actions for a fixed price vector $p$ (bottom). The minimal ambiguous action of a set $S_t$ for $t \in [l_3, l_1) \cup (r_1,r_3]$, is $i(S_t) = 3$. By \ref{['lem:RLintervals']} for $S_{t'}$ such that $t'\in (r_1,r_3]$ as in the figure, $S_{t'}$ does not contain action 1, must contain action 2, and may or may not contain action 3.
• Figure 4: Illustration of the proof of \ref{['lem:minAmb']}. Let $S_t$ and $S_{t'}$ have the same minimal ambiguous action $i^*$ and $t'>t$. If both $t$ and $t'$ are on the same side of the ambiguity interval of $i < i^*$, then $S_t$ and $S_{t'}$ agree on action $i$. Otherwise, as in the figure, it must be that $i \in S_t$ and $i \notin S_t$.
• Figure 5: The values and marginal values of $f$ and $c$ in \ref{['eq:sub-sub']} when $|S|\in \{n/2-1,n/2,n/2+1\}$. The possible values of $f(S)$ (left) and $c(S)$ (right) appear inside the circles, the marginals appear in parentheses. The marked circles correspond to sets $S$ such that $S \in x_f$ or $S \in x_c$.

Theorems & Definitions (139)

• Definition 1: Optimal Contract Problem
• Definition 2: Incentivizable Set of Actions
• Definition 3: Critical Value
• Definition 4
• Definition 5: Equal-Revenue Instance
• Theorem 1
• Lemma 1
• Lemma 2
• Proposition 1
• proof