Paper: arXiv 2406.02712

Authors: Mario Ghossoub, Michael Boyuan Zhu

Abstract

We study Pareto efficiency in a pure-exchange economy where agents’ preferences are represented by risk-averse monetary utilities. These coincide with law-invariant monetary utilities, and they can be shown to correspond to the class of monotone, (quasi-)concave, Schur concave, and translation-invariant utility functionals. This covers a large class of utility functionals, including a variety of law-invariant robust utilities. We show that Pareto optima exist and are comonotone, and we provide a crisp characterization thereof in the case of law-invariant positively homogeneous monetary utilities. This characterization provides an easily implementable algorithm that fully determines the shape of Pareto-optimal allocations. In the special case of law-invariant comonotone-additive monetary utility functionals (concave Yaari-Dual utilities), we provide a closed-form characterization of Pareto optima. As an application, we examine risk-sharing markets where all agents evaluate risk through law-invariant coherent risk measures, a widely popular class of risk measures. In a numerical illustration, we characterize Pareto-optimal risk-sharing for some special types of coherent risk measures.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is a rigorous mathematical work in theoretical quantitative finance, establishing existence and characterizations of Pareto-optimal allocations for law-invariant monetary utilities using dual representations and sup-convolutions. However, it is purely theoretical with minimal empirical data validation, relying primarily on algorithmic and stylized numerical illustrations.

Research Flowchart

  flowchart TD
    A[Goal: Characterize Pareto Efficiency & Risk-Sharing in Pure-Exchange Economies] --> B[Model Setup: Risk-Averse Monetary Utilities<br><i>Monotone, Schur Concave, Translation-Invariant</i>]
    B --> C[Core Analysis: Prove Existence & Comonotonicity of Pareto Optima]
    C --> D[Special Case: Law-Invariant Positively Homogeneous & Yaari-Dual Utilities]
    D --> E[Computational Algorithm & Closed-Form Solutions]
    E --> F[Application: Numerical Simulation of Coherent Risk Measure Markets]
    F --> G[Outcome: Easily Implementable Framework for Optimal Risk Allocations]

Cited References (73)

  1. O n O ptimal A llocation of R isk V ectors (2010)
  2. O ptimal R isk A llocation for C onvex R isk F unctionals in G eneral R isk D omains (2014)
  3. Comonotonicity and Pareto Optimality with Application to Collaborative Insurance (2023)
  4. Non- A dditive M easure and I ntegral (1994)
  5. P areto E fficient I nsurance C ontracts when the I nsurer’s C ost F unction is D iscontinuous (2003)
  6. R earrangement I nequalities in N on-convex I nsurance M odels (2005)
  7. Quantile-Based Risk Sharing (2018)
  8. Stochastic Orders (2007)
and 65 more
  1. Actuarial Theory for Dependent Risks: Measures, Orders and Models (2006)
  2. Multi-Agent Investment in Incomplete Markets (2004)
  3. I nfinite D imensional A nalysis - $3^ rd $ edition (2006)
  4. A Representation Result for Concave Schur Concave Functions (2005)
  5. P areto E fficiency for the C oncave O rder and M ultivariate C omonotonicity (2012)
  6. C ore of C onvex D istortions of a P robability (2003)
  7. Risk Sharing for Capital Requirements with Multidimensional Security Markets (2019)
  8. Pareto Optimal Allocations for Law Invariant Robust Utilities on $L^1$ (2014)
  9. Competitive Equilibria in a Comonotone Market (2021)
  10. Insurance with Multiple Insurers: A Game-Theoretic Approach (2018)
  11. Pareto-Efficient Risk Sharing in Centralized Insurance Markets with Application to Flood Risk (2024)
  12. P rospect T heory: A n A nalysis of D ecision U nder R isk (1979)
  13. A dvances in P rospect T heory: C umulative R epresentation of U ncertainty (1992)
  14. The P robability W eighting F unction (1998)
  15. T wo-persons E fficient R isk-sharing and E quilibria for C oncave L aw-invariant U tilities (2008)
  16. Inf-Convolution and Optimal Risk Sharing with Countable Sets of Risk Measures (2024)
  17. T he D ual T heory of C hoice under R isk (1987)
  18. Law-Invariant Functionals on General Spaces of Random Variables (2021)
  19. Convexity and Optimization in Banach Spaces (2012)
  20. I ntroduction to the M athematics of A mbiguity (2004)
  21. L inear O perators, P art 1: G eneral T heory (1958)
  22. Coherent Measures of Risk (1999)
  23. Insurance Pricing and Increased Limits Ratemaking by Proportional Hazards Transforms (1995)
  24. A Class of Distortion Operators for Pricing Financial and Insurance Risks (2000)
  25. Co-monotone Allocations, Bickel-Lehmann Dispersion and the Arrow-Pratt Measure of Risk Aversion (1994)
  26. Modelling Agents’ Preferences in Complete Markets by Second Order Stochastic Dominance (2003)
  27. Optimal Capital and Risk Allocations for Law-and Cash-Invariant Convex Functions (2008)
  28. Inf-Convolution of Risk Measures and Optimal Risk Transfer (2005)
  29. L aw I nvariant C oncave U tility F unctions and O ptimization P roblems with M onotonicity and C omonotonicity C onstraints (2006)
  30. Schur Convex Functionals: Fatou Property and Representation (2012)
  31. Optimal Risk Sharing with Non-Monotone Monetary Functionals (2007)
  32. O n L aw I nvariant C oherent R isk M easures (2001)
  33. Pareto Optimal Allocations and Optimal Risk Sharing for Quasiconvex Risk Measures (2015)
  34. Continuity Properties of Law-Invariant (Quasi-)Convex Risk Functions on $L^\infty$ (2010)
  35. Law Invariant Risk Measures have the Fatou Property (2006)
  36. Optimal Risk Sharing for Law Invariant Monetary Utility Functions (2008)
  37. S tochastic F inance: A n I ntroduction in D iscrete T ime – $4^ th $ ed. (2016)
  38. A T heory of A nticipated U tility (1982)
  39. Equilibrium Prices for Monetary Utility Functions (2008)
  40. Schur Convex Functionals: Fatou Property and Representation (2012)
  41. Ambiguity Aversion, Robustness, and the Variational Representation of Preferences (2006)
  42. Marginal Indemnification Function Formulation for Optimal Reinsurance (2016)
  43. Measure Theory (1994)
  44. Financial Markets in Continuous Time (2003)
  45. Weighted Comonotonic Risk Sharing under Heterogeneous Beliefs (2020)
  46. Risk Aversion in Regulatory Capital Principles (2020)
  47. Elementary Proof for Sion’s Minimax Theorem (1988)
  48. Monetary Utility Functions (2012)
  49. Optimal Risk-Sharing Rules and Equilibria with Choquet-Expected-Utility (2000)
  50. Ambiguity, Uncertainty Aversion and Equilibrium Welfare (2004)
  51. On Comonotonicity of Pareto Optimal Risk Sharing (2008)
  52. Inf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures (2022)
  53. Equilibrium in a Reinsurance Market (1962)
  54. The Theory of Syndicates (1968)
  55. Ambiguity Aversion and Trade (2011)
  56. Optimal Allocations with $\alpha$-MaxMin Utilities, Choquet Expected Utilities, and Prospect Theory (2023)
  57. Maxmin Expected Utility with Non-Unique Prior (1989)
  58. Subjective Probability and Expected Utility Without Additivity (1989)
  59. I ncreasing R isk: I . A D efinition (1970)
  60. Rearrangement Invariant Banach Function Spaces (1967)
  61. Risk Exchange with Distorted Probabilities (2006)
  62. Generalized Expected Utility Theory: The Rank-Dependent Model (2012)
  63. T heory of G ames and E conomic B ehavior (1947)
  64. Rules for Ordering Uncertain Prospects (1969)
  65. Risk Sharing under Heterogeneous Beliefs Without Convexity (2021) — link

0 of 73 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.