Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning of Nash Equilibria in Risk-Averse Games

Zifan Wang, Yi Shen, Michael M. Zavlanos, Karl H. Johansson

TL;DR

A first-order risk-averse leaning algorithm, in which the CVaR gradient estimate depends on an estimate of the Value at Risk (VaR) value combined with the gradient of the stochastic cost function, is proposed.

Abstract

This paper considers risk-averse learning in convex games involving multiple agents that aim to minimize their individual risk of incurring significantly high costs. Specifically, the agents adopt the conditional value at risk (CVaR) as a risk measure with possibly different risk levels. To solve this problem, we propose a first-order risk-averse leaning algorithm, in which the CVaR gradient estimate depends on an estimate of the Value at Risk (VaR) value combined with the gradient of the stochastic cost function. Although estimation of the CVaR gradients using finitely many samples is generally biased, we show that the accumulated error of the CVaR gradient estimates is bounded with high probability. Moreover, assuming that the risk-averse game is strongly monotone, we show that the proposed algorithm converges to the risk-averse Nash equilibrium. We present numerical experiments on a Cournot game example to illustrate the performance of the proposed method.

Learning of Nash Equilibria in Risk-Averse Games

TL;DR

A first-order risk-averse leaning algorithm, in which the CVaR gradient estimate depends on an estimate of the Value at Risk (VaR) value combined with the gradient of the stochastic cost function, is proposed.

Abstract

This paper considers risk-averse learning in convex games involving multiple agents that aim to minimize their individual risk of incurring significantly high costs. Specifically, the agents adopt the conditional value at risk (CVaR) as a risk measure with possibly different risk levels. To solve this problem, we propose a first-order risk-averse leaning algorithm, in which the CVaR gradient estimate depends on an estimate of the Value at Risk (VaR) value combined with the gradient of the stochastic cost function. Although estimation of the CVaR gradients using finitely many samples is generally biased, we show that the accumulated error of the CVaR gradient estimates is bounded with high probability. Moreover, assuming that the risk-averse game is strongly monotone, we show that the proposed algorithm converges to the risk-averse Nash equilibrium. We present numerical experiments on a Cournot game example to illustrate the performance of the proposed method.
Paper Structure (14 sections, 5 theorems, 33 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 5 theorems, 33 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex Games
  4. CVaR and VaR
  5. Risk-Averse Convex Games
  6. Problem Formulation
  7. Strong Monotonicity Analysis
  8. A Risk-Averse Learning Algorithm
  9. Algorithm Description
  10. Convergence Analysis
  11. Numerical Result
  12. Conclusion
  13. Proof of Lemma \ref{['lemma:FO:sum:epsilon']}
  14. Proof of Theorem \ref{['theorem:FO']}

Key Result

Lemma 1

Let Assumption assump:strong_monotone hold. Then, the risk-averse game def:risk-averse_game is $m$-strongly monotone.

Figures (1)

  • Figure 1: Error to the Nash equilibrium among Algorithm 1, unbiased first-order algorithm (Unbiased FO), and the zeroth-order algorithm with momentum (ZO momentum) in wang2022risk. The solid lines and shades are averages and standard deviations over 20 runs.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof