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

M. Mavronicolas, B. Monien, in: Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017), 2017, pp. 131--143.

ConditionalValue-at-RiskStruct.pdf 352.37 KB
Conference Paper | English
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Abstract
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.
Publishing Year
Proceedings Title
Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017)
Volume
10504
Page
131--143
Conference
10th International Symposium on Algorithmic Game Theory (SAGT 2017)
Conference Location
L'Aquila, Italy
Conference Date
2017-09-12 – 2017-09-14
LibreCat-ID

### Cite this

Mavronicolas M, Monien B. Conditional Value-at-Risk: Structure and Complexity of Equilibria. In: Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017). Vol 10504. LNCS. ; 2017:131--143. doi:10.1007/978-3-319-66700-3_11
Mavronicolas, M., & Monien, B. (2017). Conditional Value-at-Risk: Structure and Complexity of Equilibria. In Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017) (Vol. 10504, pp. 131--143). L’Aquila, Italy. https://doi.org/10.1007/978-3-319-66700-3_11
@inproceedings{Mavronicolas_Monien_2017, series={LNCS}, title={Conditional Value-at-Risk: Structure and Complexity of Equilibria}, volume={10504}, DOI={10.1007/978-3-319-66700-3_11}, booktitle={Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017)}, author={Mavronicolas, Marios and Monien, Burkhard}, year={2017}, pages={131--143}, collection={LNCS} }
Mavronicolas, Marios, and Burkhard Monien. “Conditional Value-at-Risk: Structure and Complexity of Equilibria.” In Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017), 10504:131--143. LNCS, 2017. https://doi.org/10.1007/978-3-319-66700-3_11.
M. Mavronicolas and B. Monien, “Conditional Value-at-Risk: Structure and Complexity of Equilibria,” in Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017), L’Aquila, Italy, 2017, vol. 10504, pp. 131--143.
Mavronicolas, Marios, and Burkhard Monien. “Conditional Value-at-Risk: Structure and Complexity of Equilibria.” Proceedings of the 10th International Symposium on Algorithmic Game Theory (SAGT 2017), vol. 10504, 2017, pp. 131--143, doi:10.1007/978-3-319-66700-3_11.
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