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

Conditional Value-at-Risk, denoted as CVaRα, is becoming the prevailing measure of risk over two paramount economic domains: the insurance domain and the financial domain; α∈(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 CVaRα -equilibrium, the mixed strategy of each player is a best-response. We establish two significant properties of CVaRα 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 CVaRα in the involved probabilities, which we establish. (2) The Crawford property: For every α, there is a 2-player game with no CVaRα-equilibrium. The property is established using the Optimal-Value property and a new functional property of CVaRα, called Weak-Equilibrium-for- VaRα, we establish. On top of these properties, we show, as one of our two main results, that deciding the existence of a CVaRα-equilibrium is strongly NP-hard even for 2-player games. As our other main result, we show the strong NP-hardness of deciding the existence of a V-equilibrium, over 2-player games, for any valuation 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.