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Allocation Mechanisms in Decentralized Exchange Markets with Frictions

Mario Ghossoub, Giulio Principi, Ruodu Wang

TL;DR

This paper develops an axiomatic framework for allocation mechanisms in decentralized pure-exchange economies with transfer frictions, formalizing frictional costs as subadditive transfer costs. It proves that allocation rules satisfying a suite of axioms are representable as robust (worst-case) linear mechanisms or as robust conditional mean allocations, bridging to risk-sharing theory. The authors introduce two concrete mechanisms, Mean-Deviation and Left-Tail Expected Shortfall, to illustrate how a single friction parameter can price and absorb reallocation costs, and they provide detailed mathematical representations via envelope theorems in Dedekind complete Riesz spaces. Overall, the work extends the risk-sharing literature with a rigorous convex-analytic and order-theoretic treatment of frictions, offering both conceptual and practical tools for decentralized risk sharing and friction-aware market design.

Abstract

The classical theory of efficient allocations of an aggregate endowment in a pure-exchange economy has hitherto primarily focused on the Pareto-efficiency of allocations, under the implicit assumption that transfers between agents are frictionless, and hence costless to the economy. In this paper, we argue that certain transfers cause frictions that result in costs to the economy. We show that these frictional costs are tantamount to a form of subadditivity of the cost of transferring endowments between agents. We suggest an axiomatic study of allocation mechanisms, that is, the mechanisms that transform feasible allocations into other feasible allocations, in the presence of such transfer costs. Among other results, we provide an axiomatic characterization of those allocation mechanisms that admit representations as robust (worst-case) linear allocation mechanisms, as well as those mechanisms that admit representations as worst-case conditional expectations. We call the latter Robust Conditional Mean Allocation mechanisms, and we relate our results to the literature on (decentralized) risk sharing within a pool of agents.

Allocation Mechanisms in Decentralized Exchange Markets with Frictions

TL;DR

This paper develops an axiomatic framework for allocation mechanisms in decentralized pure-exchange economies with transfer frictions, formalizing frictional costs as subadditive transfer costs. It proves that allocation rules satisfying a suite of axioms are representable as robust (worst-case) linear mechanisms or as robust conditional mean allocations, bridging to risk-sharing theory. The authors introduce two concrete mechanisms, Mean-Deviation and Left-Tail Expected Shortfall, to illustrate how a single friction parameter can price and absorb reallocation costs, and they provide detailed mathematical representations via envelope theorems in Dedekind complete Riesz spaces. Overall, the work extends the risk-sharing literature with a rigorous convex-analytic and order-theoretic treatment of frictions, offering both conceptual and practical tools for decentralized risk sharing and friction-aware market design.

Abstract

The classical theory of efficient allocations of an aggregate endowment in a pure-exchange economy has hitherto primarily focused on the Pareto-efficiency of allocations, under the implicit assumption that transfers between agents are frictionless, and hence costless to the economy. In this paper, we argue that certain transfers cause frictions that result in costs to the economy. We show that these frictional costs are tantamount to a form of subadditivity of the cost of transferring endowments between agents. We suggest an axiomatic study of allocation mechanisms, that is, the mechanisms that transform feasible allocations into other feasible allocations, in the presence of such transfer costs. Among other results, we provide an axiomatic characterization of those allocation mechanisms that admit representations as robust (worst-case) linear allocation mechanisms, as well as those mechanisms that admit representations as worst-case conditional expectations. We call the latter Robust Conditional Mean Allocation mechanisms, and we relate our results to the literature on (decentralized) risk sharing within a pool of agents.
Paper Structure (32 sections, 19 theorems, 108 equations, 3 figures)

This paper contains 32 sections, 19 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Economy
  3. Allocations
  4. Allocation Mechanisms
  5. Allocation Mechanisms with Frictions
  6. Axioms and Properties
  7. Frictional Participation and Convexity of Costs
  8. Representation of Allocation Mechanisms
  9. Abstract Robust Mechanisms
  10. Robust Conditional Mean Allocation Mechanisms
  11. Conditional Mean Allocation Mechanisms
  12. A Risk-Sharing Perspective
  13. Two Examples
  14. Mean-Deviation Allocation Mechanisms
  15. Expected-Shortfall Allocation Mechanisms
  16. ...and 17 more sections

Key Result

Proposition 3.1

If $H \in \mathcal{AM}$ satisfies FP, OA, SI, and AA, then for all $i,j \in [n]$ and all $\mathcal{G} \in \Sigma$, the mappings $\boldsymbol{X} \mapsto \mathcal{C}^{\boldsymbol{X},\mathcal{G}}_{i,j}(H)$ and $\boldsymbol{X} \mapsto\mathcal{C}^{\boldsymbol{X},\mathcal{G}}(H)$ satisfy and for all $\lambda\in [0,1]$, and all $\boldsymbol{X},\boldsymbol{Y}\in \mathcal{X}^n$ with $S^{\boldsymbol{X}}=S

Figures (3)

  • Figure 1: $\sigma_1=\sigma_2=1$, $\lambda=0.99$, $\rho_{ij}=\rho$ for all $i\neq j$.
  • Figure 2: $n=2$, $\sigma_1=\sigma_2=1$, $\theta_1=1/2$, $\theta_2=3$, $\rho_{ij}=\frac{2}{10}$ for all $i\neq j$.
  • Figure 3: Varying $\rho_{ij}=\left(\frac{1}{2}\right)^{|i-j|}$, fixed $\rho_{ij}=\frac{2}{10}$, $\sigma_i=1$, $\theta_i=\frac{1}{2}$, $\lambda =0.99$.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 3.1
  • Proposition 3.1
  • Definition 5
  • Remark 3.2: Robust Allocation Mechanisms and Convex Costs
  • Theorem 3.1
  • Definition 6
  • ...and 36 more