SAGT 2017: Algorithmic Game Theory pp 131-143

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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10504)

## 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.

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## Authors and Affiliations

1. 1.Department of Computer ScienceUniversity of CyprusNicosiaCyprus