Consistency between transitive relations and between cones

TL;DR

This paper explores the conditions under which transitive relations and cones can be extended consistently, with implications for preference orders, market arbitrage, and economic allocations.

Contribution

It establishes a characterization of when finite circular chains of transitive relations exist and links this to the existence of total preorders and applications in economics.

Findings

No finite circular chain of two transitive relations with asymmetric links exists iff a total preorder extends both.

Extension of relations is unique if the union's reflexive transitive closure is total.

Applications include conditions for no arbitrage and Pareto optimality in economic models.

Abstract

A relation extends another relation consistently if its symmetric, respectively its asymmetric, part contains the corresponding part of the smaller relation. It is shown that there exists no finite circular chain made from two transitive relations $\mathbf{A}$ and $\mathbf{B}$ with at least one link from their asymmetric parts if and only if there exists a total preorder which consistently extends both. Additionally, this extension is uniquely determined if and only if the reflexive transitive closure of the union of $\mathbf{A}$ and $\mathbf{B}$ is total. Applications: (1) If the steps of a walk come from two positive cones, with at least one step from one of the cones' non-linear parts, then returning to the origin is impossible if and only if there exists a third cone of which the linear part contains each of the linear parts of the two original cones, and of which the non-linear part contains each of the two non-linear parts. (2) Reminiscent of the Fundamental Theorem of Asset Pricing, absence of arbitrage is equivalent to the existence of a complete preference order which consistently extends a market's fair exchange relation and its objective strict preference order. (3) Another impossibility in microeconomics: For two agents, or an agent and a market, with additive positively homogeneous preferences, an allocation can never be Pareto optimal if they are not `cut from the same cloth'.