Minimal Regret Walras Equilibria for Combinatorial Markets via Duality, Integrality, and Sensitivity Gaps

2025-11-12

AI Summary

This paper studies the problem of achieving a Δ-regret Walrasian equilibrium in combinatorial multi-item markets—i.e., an allocation and price vector satisfying market clearance and capacity constraints, under which the aggregate regret (utility loss relative to price-dependent optimal allocations) across all agents is at most Δ. It introduces regret minimization into the Walrasian equilibrium framework for the first time, establishing deep connections to linear programming duality gaps, integer programming relaxation gaps, and sensitivity analysis. The authors derive a general Δ-bound based on sensitivity gaps of the configuration LP and design a provably efficient equilibrium construction mechanism. Key contributions include: (i) transforming social welfare approximation algorithms into low-regret equilibrium computation methods; (ii) deriving tight complexity-driven lower bounds on achievable regret; and (iii) providing the first regret-controlled equilibrium paradigm for general valuation functions that simultaneously offers theoretical guarantees and computational tractability.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We consider combinatorial multi-item markets and propose the notion of a $Delta$-regret Walras equilibrium, which is an allocation of items to players and a set of item prices that achieve the following goals: prices clear the market, the allocation is capacity-feasible, and the players'strategies lead to a total regret of $Delta$. The regret is defined as the sum of individual player regrets measured by the utility gap with respect to the optimal item bundle given the prices. We derive necessary and sufficient conditions for the existence of $Delta$-regret equilibria, where we establish a connection to the duality gap and the integrality gap of the social welfare problem. For the special case of monotone valuations, the derived necessary and sufficient optimality conditions coincide and lead to a complete characterization of achievable $Delta$-regret equilibria. For general valuations, we establish an interesting connection to the area of sensitivity theory in linear optimization. We show that the sensitivity gap of the optimal-value function of two (configuration) linear programs with changed right-hand side can be used to establish a bound on the achievable regret. Finally, we use these general structural results to translate known approximation algorithms for the social welfare optimization problem into algorithms computing low-regret Walras equilibria. We also demonstrate how to derive strong lower bounds based on integrality and duality gaps but also based on NP-complexity theory.

Problem

• Analyzing existence conditions for Δ-regret Walras equilibria in combinatorial markets
• Establishing connections between duality gaps and achievable regret bounds
• Translating welfare optimization algorithms into low-regret equilibrium computation

Innovation

• Uses duality gap to characterize regret equilibria
• Applies sensitivity theory for general valuation bounds
• Translates welfare algorithms to compute equilibria