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 a complete characterization for the existence of $Δ$-regret equilibria by introducing the concept of a parameterized social welfare problem, where the right-hand side of the original social welfare problem is changed. Our characterization then relates the achievable regret value with the associated duality/integrality gap of the parameterized social welfare problem. For the special case of monotone valuations this translates to regret bounds recovering the duality/integrality gap of the original social welfare problem. We further 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.

翻译：本文研究组合多物品市场，提出Δ-遗憾瓦尔拉斯均衡的概念。该均衡包含物品对参与者的分配方案及物品价格体系，需同时实现以下目标：价格使市场出清，分配方案满足容量可行性，且参与者策略产生的总遗憾值为Δ。此处遗憾值定义为各参与者遗憾之和，通过参与者实际效用与给定价格下最优物品组合的效用差距进行度量。通过引入参数化社会福利问题（即改变原始社会福利问题右侧参数），我们建立了Δ-遗憾均衡存在性的完整刻画。该刻画将可实现的遗憾值与参数化社会福利问题的对偶/整数间隙相关联。对于单调估值函数的特殊情况，该结论可推导出恢复原始社会福利问题对偶/整数间隙的遗憾界。我们进一步建立了与线性优化灵敏度理论的有趣联系，证明通过改变右侧参数的两个（配置型）线性规划最优值函数的灵敏度间隙，可确定遗憾值的可实现上界。最后，我们运用这些通用结构结论，将社会福利优化问题的已知近似算法转化为计算低遗憾瓦尔拉斯均衡的算法，并展示如何基于整数间隙、对偶间隙及NP复杂性理论推导出强下界。