Revealed Preference Dimension via Matrix Sign Rank

TL;DR

This paper characterizes which graphs can be represented as revealed preference graphs in fixed-dimensional markets, linking the problem to the Matrix Sign Rank of their adjacency matrices, and relates it to the order-dimension of preference relations.

Contribution

It provides an exact characterization of attainable revealed preference graphs in fixed dimensions using Matrix Sign Rank, solving an open problem.

Findings

Feasible graphs are characterized by their Matrix Sign Rank.

Graphs from posets with order-dimension k are realizable in k-dimensional markets.

The work establishes a direct link between market dimension and graph properties.

Abstract

Given a data-set of consumer behaviour, the Revealed Preference Graph succinctly encodes inferred relative preferences between observed outcomes as a directed graph. Not all graphs can be constructed as revealed preference graphs when the market dimension is fixed. This paper solves the open problem of determining exactly which graphs are attainable as revealed preference graphs in $d$-dimensional markets. This is achieved via an exact characterization which closely ties the feasibility of the graph to the Matrix Sign Rank of its signed adjacency matrix. The paper also shows that when the preference relations form a partially ordered set with order-dimension $k$, the graph is attainable as a revealed preference graph in a $k$-dimensional market.