Minimal Regret Walras Equilibria for Combinatorial Markets via Duality, Integrality, and Sensitivity Gaps
Aloïs Duguet, Tobias Harks, Martin Schmidt, Julian Schwarz
TL;DR
The paper introduces $\Delta$-regret Walras equilibria as a robust relaxation for combinatorial markets where exact Walras equilibria may not exist, linking regret to fundamental optimization gaps. For monotone valuations, it provides a tight, duality/integrality-gap characterization that preserves the first welfare theorem within the regret framework, while for general valuations it connects regret to LP sensitivity gaps and integrality gaps, enabling algorithmic translation from welfare approximations. It develops a concrete LP-based method to compute optimal prices for fixed allocations and demonstrates how to bound regret via lifted configurations, yielding instance-dependent and worst-case guarantees. Finally, it presents polynomial-time reduction techniques to compute low-regret equilibria from welfare-approximation algorithms and establishes foundational lower bounds via integrality gaps and $NP$-hardness, highlighting both the potential and limits of regret-based equilibrium computations in combinatorial markets.
Abstract
We consider combinatorial multi-item markets and propose the notion of a $Δ$-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 $Δ$. 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 $Δ$-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 $Δ$-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.