Quality of local equilibria in discrete exchange economies

TL;DR

This paper introduces a new concept of local equilibria in discrete exchange economies, parameterized by quality measures, which generalizes and stabilizes traditional equilibrium notions, providing bounds on allocation efficiency.

Contribution

It defines the notion of local equilibria of quality (r, s), extending conditional equilibria, and analyzes their stability and efficiency bounds in discrete exchange economies.

Findings

Local equilibria of quality (r, s) guarantee a fraction of the optimal allocation.

In economies with a-submodular valuations, such equilibria exist with specific quality parameters.

Greedy allocations can achieve certain local equilibrium qualities in submodular valuation settings.

Abstract

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.