Price of Fairness in Contract Design

• Price of fairness in contract design quantifies the efficiency loss when fairness constraints replace pure revenue maximization.
• Enforcing envy-free constraints introduces NP-hard challenges, while relaxations like EF1 ensure the revenue loss remains polynomially bounded.
• Dynamic programming and linear programming methods enable FPTAS for fair contracts, offering practical, near-optimal solutions in controlled settings.

The price of fairness in contract design quantifies the efficiency loss—or revenue reduction—incurred when fairness constraints are imposed on contract allocation and compensation, rather than purely optimizing for maximum performance or profit. Recent work in this area formalizes, bounds, and algorithmically studies this trade-off using formal fairness notions such as envy-freeness (EF), its relaxations (ε-EF and EF1), and provides both computational complexity and algorithmic results, alongside asymptotic and explicit price of fairness bounds.

1. Envy-Free Contracts in Contract Design

In the contract design setting, a principal must assign a collection of tasks to a set of agents, each with their own success probabilities and costs for the tasks, and specify a linear contract (sharing rule) α for each task. A contract, defined as a pair (S, α) with S describing the allocation (S_i the set of tasks assigned to agent i) and α the payment schedule, is called envy-free (EF) if, for any pair of agents i ≠ j:

$$\sum_{k \in S_i} (\alpha_k p_{i, k} r_k - c_{i, k}) \geq \sum_{k \in S_j} \max \left\{ \alpha_k p_{i, k} r_k - c_{i, k}, 0 \right\}$$

The EF condition ensures no agent envies the net benefit of another agent’s assigned tasks when evaluated by their own parameters. Relaxations include ε-EF (allowing a small fixed additive gap of at most ε) and EF1 (envy-free up to one item: for each pair i ≠ j, there exists some task k_{ij} in S_j such that the above inequality holds after removing k_{ij} from S_j).

Unlike classic allocation problems, EF contracts are always guaranteed to exist in this model; a trivial full allocation (potentially with minimal or zero payments) will always satisfy fairness (Castiglioni et al., 15 Jul 2025).

2. Computational Complexity of Fair Contract Design

While EF contracts are always guaranteed to exist, computing a revenue-optimal or even constant-factor approximate EF contract is computationally hard in general. It is NP-hard to approximate the optimal EF contract within any constant factor, even when allowing the ε-EF relaxation. This remains true for as few as three agents. The hardness is demonstrated via reductions from classical NP-hard problems, reflecting the intrinsic combinatorial challenge when handling assignments, payments, and agent heterogeneity. Even for just two agents, the best achievable polynomial-time approximation ratio for exact EF contracts is at least 2/5, and the problem is intractable for more agents (Castiglioni et al., 15 Jul 2025).

However, when the number of tasks m is constant, the total number of possible assignments is bounded (O(n^m)), and for each fixed assignment, the optimal payment schedule can be found via linear programming, yielding a polynomial-time solution. For a constant number of agents, the paper presents fully polynomial-time approximation schemes (FPTAS) under relaxed fairness notions (ε-EF and EF1), using dynamic programming with careful utility discretization.

3. Algorithmic Methods for Fair Contract Computation

For a constant number of agents, the FPTAS for ε-EF/EF1 contracts proceeds by:

• Discretizing the set of possible assignment and payment profiles using fine and/or adaptive grids (for ε-EF and EF1, respectively).
• Employing a dynamic programming formulation that tracks cumulative utility vectors, adding each task in turn while maintaining feasibility under incentive compatibility, individual rationality, and fairness (relaxed as appropriate).
• At each stage, selecting the best revenue (or utility) attainable for each grid bucket, building an approximate solution.
• Ensuring the output contract achieves ε-additive proximity to the optimal possible under the fairness constraint.

If m (number of tasks) is constant, the algorithm enumerates all assignments and, for each, solves an LP to determine the optimal contract parameters satisfying EF or its relaxations.

4. Price of Fairness: Bounds and Revenue Implications

The price of fairness in contract design is defined as the ratio of the principal’s revenue without any fairness constraint (OPT^0) to the revenue achievable under a particular fairness condition (e.g., OPT^{EF} or OPT^{EF1}).

For exact EF contracts:

The price of fairness can be unbounded—even in an instance with a single task and two agents, the optimal EF contract may result in zero (or arbitrarily small) revenue if envy-freeness forces allocation to a low-value agent.

For EF1 contracts (envy-free up to one item):

The price of fairness is polynomially bounded: $$\Omega(\sqrt{n}) \leq \text{Price of Fairness} \leq O(n^2)$$ where n is the number of agents. That is, in the worst case, imposing EF1 fairness can reduce revenue by a polynomial (between square-root and quadratic) factor, but does not force it to arbitrarily small values (Castiglioni et al., 15 Jul 2025).

Practical implication: While strict (exact) envy-freeness may completely undermine the principal’s profit, relaxing the requirement to EF1 (accepting a small degree of potential envy associated with one item/task) delivers bounded, and often much more acceptable, efficiency loss.

5. Optimization Formulations

When m (tasks) is constant, the fair contract design problem, for a fixed allocation, reduces to a tractable linear program. For example (with auxiliary variables $t_{i,k}$):

$$\begin{aligned} \max\,\,\,\,\, \sum_{i\in \mathcal{N}} \sum_{k\in S_i}(1-\alpha_k)p_{i,k}r_k\ \text{s.t.} \,\,\,\, & \alpha_j p_{i,j} r_j - c_{i,j} \geq 0,\,\,\forall i,j\in S_i\ & \sum_{k\in S_i} (\alpha_k p_{i,k} r_k - c_{i,k}) \geq \sum_{k\in S_j} t_{i,k},\,\,\forall i\neq j\ & t_{i,k} \geq \alpha_k p_{i,k} r_k - c_{i,k},\,\,\forall i,k\ & t_{i,k} \geq 0, \,\,\forall i,k. \end{aligned}$$

This optimization enforces fair compensation rules while maximizing the principal's revenue.

6. Implications and Trade-offs for Real-World Contracts

Enforcing strict envy-freeness may render contracts unprofitable, especially if fairness forces undesirable allocations. However, EF1 contracts, which allow for a single source of potential envy per agent pair, offer a pragmatic middle ground: fairness is substantially preserved, but the principal can still secure nontrivial (and sometimes near-optimal) revenue, especially when the number of agents is moderate.

Key takeaways for contract designers:

• Fair contracts are always guaranteed to exist in this model, a property not shared by classic fair division scenarios.
• For moderate numbers of agents and tasks, or under relaxed fairness (EF1, ε-EF), efficient or near-optimal computation is practical.
• The “price of fairness” (revenue gap) is deterministically quantifiable when using EF1: although it can grow with n, it is polynomially bounded, in sharp contrast to the potentially unbounded loss under exact EF.
• For real-world contractual platforms, especially where fairness is desirable for regulatory or trust purposes, the findings provide a principled approach for quantifying and managing revenue–fairness trade-offs, with recommended attention to the selection of fairness criteria and problem parameterization.

7. Summary Table: Price of Fairness under Different Notions

Fairness Notion	Existence of Optimal Contracts	Computational Tractability	Price of Fairness Bound
Exact EF	Always exists	NP-hard, even to approximate	Unbounded (can be arbitrarily large)
ε-EF (additive)	Always exists	FPTAS if agents are constant	Additive ε-loss to revenue
EF1	Always exists	FPTAS if agents are constant	Ω(√n) ≤ PoF ≤ O(n²)

• "Fair Contracts" (Castiglioni et al., 15 Jul 2025)