Combinatorial Contract Design & Ultra Functions

• Combinatorial contract design is a framework that models optimal delegation by designing contracts over exponentially many action combinations.
• It utilizes discrete convex analysis and valuated matroid intersection techniques to efficiently compute incentive schemes under strict computational constraints.
• The approach highlights Ultra (M♮-concave) reward functions as expanding the tractable frontier beyond Gross Substitutes, setting new directions in mechanism design.

Combinatorial contract design addresses the computation and structure of optimal contracts in delegative environments where a principal motivates an agent (or agents) to exert costly, hidden, and combinatorial effort on her behalf. The core setting involves exponentially many possible combinations of costly actions and a reward function that maps each set of actions onto the principal’s expected payoff. The primary research challenge is to characterize classes of reward and cost functions for which the principal can efficiently compute an optimal incentive scheme, usually under informational or computational constraints, and to develop algorithms that compute (exact or approximate) optimal contracts. The recent identification of Ultra (M^♮-concave) functions as the tractability frontier, surpassing the previously conjectured Gross Substitutes (GS) class, constitutes a central advance in this field (Feldman et al., 22 Jun 2025).

1. Combinatorial Principal-Agent Model

Consider a principal who delegates project execution to an agent faced with a ground set of actions $N = \{1, 2, ..., n\}$. The agent may select any subset $S \subseteq N$ to perform, incurring cost $c(S)$. The outcome is observed by the principal in terms of a set-function reward $f : 2^N \to \mathbb{R}_{\geq 0}$, which is monotone and normalized: $f(\emptyset) = 0$. The agent is incentivized by a contract—a share $\alpha \in [0,1]$ of the reward—so his utility for set $S$ is $\alpha f(S) - c(S)$. The agent selects $S$ to maximize his own utility. The principal receives the residual $(1-\alpha)f(S)$ and designs the contract to maximize her own expected utility: $$\max_{\alpha \in [0,1]} U(\alpha) \quad\text{where}\quad U(\alpha) = \max_{S \subseteq N} \left[ f(S) - \alpha^{-1} c(S) \right].$$ The contract problem is algorithmically nontrivial since optimizing over exponentially many subsets and anticipating the agent's best-response as $\alpha$ varies is, a priori, intractable except for special function classes.

2. Functional Structure and Tractability Frontiers

The complexity of the optimal contract problem hinges on the reward and cost structures:

• Gross Substitutes (GS): $f$ is GS if for any $S,T \subseteq N$ and $i \in T \setminus S$, there exists $j \in S \setminus T$ (possibly $j = \emptyset$) such that

$$f(S) + f(T) \leq f\big( S \cup \{i\} \setminus \{j\} \big) + f\big( T \cup \{j\} \setminus \{i\} \big).$$

The GS property guarantees that increasing the price (or cost-share) of an item never decreases demand for other items. For additive costs, prior work established a polynomial-time algorithm for optimal contracts if $f$ is GS (Duetting et al., 2021, Dütting et al., 2023).

• Ultra Functions (M^♮-concave): A function $f$ is Ultra if for any $S \subseteq N$ and any distinct $i, j, k \notin S$,

$$f(S+i+j) + f(S+k) \leq \max \{ f(S+i+k) + f(S+j),\, f(S+j+k) + f(S+i)\}.$$

Ultra strictly generalizes GS: every GS function is Ultra, but Ultra contains symmetric and non-submodular cases not captured by GS (e.g., functions depending symmetrically and concavely on $|S|$). The intersection of Ultra and submodular is precisely GS.

• Submodular: For $S \subseteq T \subseteq N$ and $i \notin T$, $f(S+i)-f(S) \geq f(T+i)-f(T)$. Submodularity expresses diminishing marginal returns but is insufficient for tractability.

With costs of type "additive + symmetric" ($c(S) = \sum_{i \in S} c_i + g(|S|)$ for some $g: \mathbb{N} \to \mathbb{R}_{\geq 0}$), tractability for Ultra functions extends beyond GS, overturning prior assumptions that submodularity is an essential ingredient of efficient contract computation (Feldman et al., 22 Jun 2025).

3. Polynomial-Time Algorithm for Ultra Functions

The principal’s problem under Ultra $f$ and additive+symmetric costs can be solved in polynomial time using discrete convex analysis and valuated matroid intersection techniques. The approach proceeds as follows:

• For any scalar $\lambda > 0$, solve

$$\max_{S \subseteq N} \left[ f(S) - \lambda c(S) \right].$$

If $f$ is Ultra and $c$ is additive+symmetric, then $f - \lambda c$ remains M$^\natural$-concave. The maximization is performed using Murota’s value-oracle based strongly polynomial algorithm, running in $O(n^5 T_f)$, where $T_f$ is the time to compute $f(S)$.

• The outer maximization over $\alpha$ is equivalently a maximization over $\lambda=1/\alpha$. The principal’s utility as a function of $\lambda$ is piecewise-concave with $O(n^2)$ breakpoints, as the optimal set $S$ transitions at these critical values.
• A parametric search over these breakpoints and the corresponding $S$ allows exact identification of the global optimum.
• The same framework produces, as a byproduct, a decomposition of the agent’s demand correspondence at each $\alpha$, enabling the principal to compute an incentive-compatible payment scheme.

The total complexity is $O(n^5 T_f + n^3 \mathrm{polylog}(C))$, where $C$ is a bound on the bit-size of the input, including all $c_i$ and the univariate cost $g$.

4. Beyond Additive Costs and the Role of Discrete Convexity

The fundamental structural insight is that tractability for the contract problem is governed by M$^\natural$-concavity—a form of discrete concavity introduced by Murota—which encompasses Ultra functions. The algorithm and the reduction to valuated matroid intersection do not require either full submodularity or GS; the critical property is that $f$ is Ultra and $c$ is the sum of an M-convex function of $|S|$ and a linear term, which includes numerous cost structures of economic relevance beyond simple additivity (e.g., costs with bundle-size penalties).

Gross substitutes, identified previously as the “tractable frontier,” are now understood to be only the intersection of two properties—Ultra and submodular—while it is Ultra that drives tractability. Examples of Ultra but non-GS functions include certain symmetric concave-up functions of set size and coverage functions on laminar families. The Ultra property allows economies and diseconomies of scale in reward functions, provided the Ultra-exchange inequality holds.

5. Connections, Implications, and Open Directions

• Connections to Discrete Convex Analysis: The theory of M$^\natural$-concave (Ultra) and M-convex functions, and the associated maximization algorithms, form a deep link between combinatorial contract design and discrete convex analysis (see Murota 2003, Frank–Murota 1999).
• Breakdown of Previous Beliefs: The tractability of the contract problem for Ultra functions demonstrates that submodularity, traditionally viewed as crucial for efficient combinatorial optimization, is insufficient—Ultra stands as the true structural requirement (Feldman et al., 22 Jun 2025).
• Approximation Frontiers: For more general classes (e.g., XOS), the optimal contract problem remains NP-hard and does not admit constant-factor approximations unless further structure is imposed. The contract design landscape thus closely mirrors that of combinatorial auctions, where GS is the classical boundary for efficient algorithmic markets.
• Extensibility: The algorithmic framework accommodates cost functions that are general M-convex plus linear, not just additive + symmetric, further demonstrating the power of discrete convexity in economic mechanism design.
• Open Problems: Precise characterization of the optimal contract problem's tractable frontier for cost functions that deviate from additive + symmetric, and reward functions beyond Ultra (e.g., general subadditive or XOS), remains unresolved. Additional research into parametric maximization for other discrete convex function classes and the development of efficient approximate algorithms for intractable cases form important future work.

6. Summary Table: Function and Cost Classes

Reward Class	Cost Structure	Tractability	Algorithmic Tool
Additive	Additive	Poly-time	DP / sorting
GS	Additive	Poly-time	Matroid intersection
Ultra	Additive + sym	Poly-time	Valuated matroid intersection
Submodular	Additive	NP-hard	-
XOS	Additive	Exponential	-

Key: "Additive + sym" denotes costs of the form $c(S) = \sum_{i} c_i + g(|S|)$. "Valuated matroid intersection" refers to the strongly polynomial combinatorial algorithm for M$^\natural$-concave maximization.

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The principled extension of contract design tractability from GS to Ultra (M$^\natural$-concave) reward functions, coupled with additive plus symmetric (M-convex + linear) costs, provides powerful new capabilities for designing contracts in combinatorial environments and establishes new directions for both theory and applications in economic mechanism design (Feldman et al., 22 Jun 2025).