[Paper Review] On risk averse competitive equilibrium

Henri Gérard, Vincent Leclère+1 more|arXiv (Cornell University)|Jun 26, 2017

Risk and Portfolio Optimization19 references44 citations

TL;DR

This paper investigates risk-averse competitive equilibrium in incomplete markets using coherent risk measures, demonstrating that even with strictly concave objectives, multiple equilibria can exist. It shows that standard solvers like PATH converge to an unstable equilibrium, while tâtonnement algorithms find multiple stable equilibria, challenging the uniqueness and reliability of numerical solutions in risk-averse market models.

ABSTRACT

We discuss risked competitive partial equilibrium in a setting in which agents are endowed with coherent risk measures. In contrast to socialplanning models, we show by example that risked equilibria are not unique, even when agents' objective functions are strictly concave. We also show that standard computational methods find only a subset of the equilibria, even with multiple starting points.

Motivation & Objective

• To analyze the existence and uniqueness of risk-averse competitive equilibria when risk markets are incomplete.
• To investigate the reliability of standard numerical methods (e.g., PATH, tâtonnement) in computing equilibria under risk aversion.
• To demonstrate that multiple equilibria can coexist even when the social planner problem has a unique solution.
• To compare the stability and convergence properties of different equilibrium computation methods in risk-averse settings.
• To highlight the implications of non-uniqueness for market design and policy justification in electricity and risk markets.

Proposed method

• Models a two-stage stochastic market with a single good, using scenario-based uncertainty and coherent risk measures.
• Formulates risk-averse social planner problems using polyhedral risk measures defined over a set of probability distributions.
• Defines risk-averse competitive equilibrium via complementarity conditions linking production, consumption, and market clearing.
• Employs the PATH solver in GAMS and a tâtonnement algorithm to compute equilibria from multiple starting points.
• Derives analytical conditions for optimal decisions based on price regimes and probability weights.
• Uses geometric analysis of excess supply manifolds to identify and classify equilibria in parameter space.

Experimental results

Research questions

• RQ1 Can risk-averse competitive equilibria be non-unique even when the social planner problem has a unique solution?
• RQ2 Why does the PATH solver converge to an unstable equilibrium in a risk-averse market with incomplete risk trading?
• RQ3 Do tâtonnement algorithms converge to different equilibria depending on initial prices, and are these equilibria stable?
• RQ4 Is there a convex combination relationship between distinct equilibria in the risk-averse setting?
• RQ5 Can multiple equilibria coexist with non-dominated welfare outcomes for producers and consumers?

Key findings

• The paper presents a counterexample where the risk-averse social planner problem has a unique solution, yet three distinct equilibria exist in the competitive market.
• The PATH solver consistently converges to a single equilibrium (π = (1.23578, 2.10953)), which is analytically shown to be unstable under tâtonnement dynamics.
• Two additional equilibria are found via tâtonnement: (1.2256, 2.0698) and (1.2478, 2.1564), both stable and non-dominated.
• The risk-adjusted welfare outcomes differ across equilibria: producer welfare is 2.152 and 2.113, while consumer welfare is 0.798 and 0.845, respectively.
• The unstable equilibrium found by PATH lies on the convex hull of the two stable equilibria, as confirmed by analytical geometry.
• The set of parameters yielding three distinct equilibria has non-zero Lebesgue measure, indicating this behavior is not rare or pathological.