[Paper Review] Stable allocations in discrete exchange economies

Federico Echenique, Sumit Goel+1 more|arXiv (Cornell University)|Feb 9, 2022

Economic theories and models4 citations

TL;DR

This paper establishes the existence of stable allocations in discrete exchange economies with indivisible goods under four distinct conditions: dichotomous preferences, categorical economies, gains from trade, and a generalized Top Trading Cycle (TTC) algorithm. Using techniques ranging from Scarf’s balancedness condition to Tarski fixed points and algorithmic constructions, it proves the nonempty weak core across these models, offering a unified framework for stability in discrete multi-good exchange without transfers.

ABSTRACT

We study stable allocations in an exchange economy with indivisible goods. The problem is well-known to be challenging, and rich enough to encode fundamentally unstable economies, such as the roommate problem. Our approach stems from generalizing the original study of an exchange economy with unit demand and unit endowments, the \emph{housing model}. Our first approach uses Scarf's theorem, and proposes sufficient conditions under which a ``convexify then round'' technique ensures that the core is nonempty. The upshot is that a core allocation exists in categorical economies with dichotomous preferences. Our second approach uses a generalization of the TTC: it works under general conditions, and finds a solution that is a version of the stable set.

Motivation & Objective

• To establish the existence of stable allocations (in the weak core or bargaining set) in discrete exchange economies with indivisible goods, where standard convexity and continuity assumptions fail.
• To extend the theoretical understanding of core stability beyond quasi-linear or unit-demand models, particularly in the absence of monetary transfers.
• To provide sufficient conditions under which stable outcomes exist in complex, non-convex discrete allocation problems, including those encoding pathological bargaining structures like the roommate problem.
• To unify diverse models—dichotomous preferences, categorical consumption, gains from trade, and discrete TU markets—under a common framework of core existence.
• To generalize the TTC algorithm beyond the housing market to broader discrete exchange economies using fixed-point and algorithmic techniques.

Proposed method

• Uses Scarf’s balancedness condition to prove core nonemptiness in economies with dichotomous preferences and categorical economies, relying on combinatorial rounding and set-theoretic arguments.
• Applies the theory of non-transferable utility (NTU) convex games to show core existence under gains from trade and injective utilities, leveraging known results on convex NTU games.
• Introduces a generalized TTC algorithm via Tarski’s fixed-point theorem, constructing a cycle-based mechanism that ensures stable allocations through iterative reallocation.
• Employs a utility-based transformation T that maps individual utilities to potential trade outcomes, with fixed points of T corresponding to stable allocations.
• Defines a directed graph on agents in A₂ (those not individually rational) where edges represent pairwise trade feasibility, and proves cycles must be of length two via utility difference inequalities.
• Uses injectivity of utility functions to enforce uniqueness in trade outcomes, ensuring no alternative allocations can yield the same utility without violating stability.

Experimental results

Research questions

• RQ1 Under what conditions does the weak core exist in discrete exchange economies with indivisible goods and no monetary transfers?
• RQ2 Can the Top Trading Cycle (TTC) algorithm be generalized beyond the housing market to broader discrete exchange economies?
• RQ3 How do dichotomous preferences or categorical consumption structures affect the existence of stable allocations?
• RQ4 What role does the 'gains from trade' assumption play in ensuring core nonemptiness in non-convex, discrete economies?
• RQ5 Can fixed-point theorems be used to constructively prove the existence of stable allocations in discrete exchange models?

Key findings

• In economies with dichotomous preferences, the weak core is nonempty, proven via Scarf’s balancedness condition, generalizing the Shapley-Scarf result without relying on TTC.
• For categorical economies where agents consume at most one good per category and have additively separable dichotomous utilities, the weak core is nonempty, with a constructive proof using a rounding algorithm.
• Under injective utilities and a gains-from-trade assumption, the core is nonempty, and this result is established both via Scarf’s condition and by associating the economy to a convex NTU game.
• The generalized TTC algorithm produces a stable allocation by identifying 2-cycles in a utility-based graph, with all cycles proven to be of length two using utility difference inequalities.
• The fixed-point formulation of the TTC mechanism ensures that every agent either receives their individually rational utility or a strictly higher one through pairwise trade, with no stable objection possible.
• The uniqueness of utility levels under injective valuations prevents alternative trade paths from achieving the same utility, ensuring stability of the constructed allocation.