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 ${\mathsf{CVaR}}_{\mathit{\alpha }}$-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 α, 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-${\mathsf{VaR}}_{\mathit{\alpha }}$, 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 minimization 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.

条件风险值，表示为${\mathsf{变异系数}}_{\alpha }$，已成为两个最重要经济领域的主要风险度量：保险领域和金融领域； $\alpha \in （0，\mathrm{1个}）$是置信度。在这项工作中，我们研究了一个以博弈为模型的经济系统的战略均衡，在这种均衡中，规避风险的参与者试图将其成本的有条件风险最小化。具体来说，在${\mathsf{变异系数}}_{\mathit{\alpha }}$均衡，每个参与者的混合策略都是最佳响应。我们建立了两个重要的特性${\mathsf{变异系数}}_{\alpha }$在平衡状态下：（1）所述的优化-值属性：对于演奏者的任何最优反应，在支持各混合策略给出相同的成本给玩家。这直接是由于${\mathsf{变异系数}}_{\alpha }$我们确定的相关概率。（2）所述的克劳福德属性：对于每一个α，有一个2人游戏，没有${\mathsf{变异系数}}_{\alpha }$-平衡。该属性是使用Optimal-Value属性和的新功能属性建立的${\mathsf{变异系数}}_{\alpha }$，称为“弱均衡”${\mathsf{风险价值}}_{\mathit{\alpha }}$，我们建立。在这些属性之上，作为两个主要结果之一，我们表明决定是否存在${\mathsf{变异系数}}_{\alpha }$平衡很强 $\mathcal{NP}$-即使对于2人游戏也很难。作为我们的其他主要结果，我们展示了强大的$\mathcal{NP}$-确定一个存在的难度 $\mathsf{V}$平衡，超过2人参与的最小化游戏，任何估值 $\mathsf{V}$具有“最佳价值”和“克劳福德”属性。由于我们证明了非常重要和广泛的一类严格拟凹估值具有“最优价值”属性，因此该结果具有巨大的潜力。