Conditional Value-at-Risk: Structure and Complexity of Equilibria

Abstract

Conditional Value-at-Risk, denoted as ${\mathsf {CVaR}}_{\alpha }$, is becoming the prevailing measure of risk over two paramount economic domains: the insurance domain and the financial domain; $\alpha \in (0,1)$ is the confidence level. In this work, we study the strategic equilibria for an economic system modeled as a game, where risk-averse players seek to minimize the Conditional Value-at-Risk of their costs. Concretely, in a ${\mathsf {CVaR}}_{\alpha }$-equilibrium, the mixed strategy of each player is a best-response. We establish two significant properties of ${\mathsf {CVaR}}_{\alpha }$ at equilibrium: (1) The Optimal-Value property: For any best-response of a player, each mixed strategy in the support gives the same cost to the player. This follows directly from the concavity of ${\mathsf {CVaR}}_{\alpha }$ in the involved probabilities, which we establish. (2) The Crawford property: For every $\alpha$, there is a 2-player game with no ${\mathsf {CVaR}}_{\alpha }$-equilibrium. The property is established using the Optimal-Value property and a new functional property of ${\mathsf {CVaR}}_{\alpha }$, called Weak-Equilibrium-for-${\mathsf {VaR}}_{\alpha }$, we establish. On top of these properties, we show, as one of our two main results, that deciding the existence of a ${\mathsf {CVaR}}_{\alpha }$-equilibrium is strongly ${\mathcal {NP}}$-hard even for 2-player games. As our other main result, we show the strong ${\mathcal {NP}}$-hardness of deciding the existence of a ${\mathsf {V}}$-equilibrium, over 2-player games, for any valuation ${\mathsf {V}}$ with the Optimal-Value and the Crawford properties. This result has a rich potential since we prove that the very significant and broad class of strictly quasiconcave valuations has the Optimal-Value property.

Partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-the-Fly-Computing” (SFB 901), and by funds for the promotion of research at University of Cyprus.