This paper defines the notion of a local equilibrium of quality $(r , s)$, $0 \leq r , s$, in a discrete exchange economy: a partial allocation and item prices that guarantee certain stability properties parametrized by the numbers $r$ and $s$. The quality $( r , s )$ measures the fit between the allocation and the prices: the larger $r$ and $s$ the closer the fit. For $r , s \leq 1$ this notion provides a graceful degradation for the conditional equilibria of [10] which are exactly the local equilibria of quality $( 1 , 1 )$. For $1 < r , s $ the local equilibria of quality $( r , s )$ are {\em more stable} than conditional equilibria. Any local equilibrium of quality $( r , s )$ provides, without any assumption on the type of the agents' valuations, an allocation whose value is at least $\frac{r s} { 1 + r s }$ the optimal fractional allocation. In any economy in which all agents' valuations are $a$-submodular, i.e., exhibit complementarity bounded by $a \: \geq \: 1$, there is a local equilibrium of quality $( \frac{1} {a} , \frac{1}{a} )$. In such an economy any greedy allocation provides a local equilibrium of quality $( 1 , \frac{1}{a} ) $. Walrasian equilibria are not amenable to such graceful degradation.

中文翻译：

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离散交换经济中局部均衡的质量

本文定义了离散交换经济中质量 $(r , s)$, $0 \leq r , s$ 的局部均衡的概念：部分分配和项目价格保证由数字 $r 参数化的某些稳定性属性$ 和 $s$。质量 $( r , s )$ 衡量分配和价格之间的拟合：$r$ 和 $s$ 越大，拟合越接近。对于 $r , s \leq 1$ 这个概念为 [10] 的条件均衡提供了优雅的退化，这正是质量 $( 1 , 1 )$ 的局部均衡。对于 $1 < r , s $ 质量 $( r , s )$ 的局部均衡比条件均衡{\em 更稳定}。质量 $( r , s )$ 的任何局部均衡都提供了，无需对代理人的估价类型进行任何假设，其值至少为 $\frac{rs} { 1 + rs }$ 的分配是最佳分数分配。在所有代理人的估值都是 $a$-submodular 的任何经济体中，即表现出以 $a \: \geq \: 1$ 为界的互补性，存在质量 $( \frac{1} {a} , \frac{1}{a} )$。在这样的经济中，任何贪婪的分配都会提供质量 $( 1 , \frac{1}{a} ) $ 的局部均衡。瓦尔拉斯均衡不适合这种优雅的退化。