We consider 2-players minimization games with very few cost values. Players are risk-averse and play mixed strategies. The players care about minimizing some function other than expectation or minimizing expectation with additional properties: Expectation plus Variance (E+Var), or Extended Sharpe Ratio (ESR), or Expectation (E) with the additional property that Variance is zero ($\mathsf{Var}=0$). These give rise to ($\mathsf{E}+\mathsf{Var}$)-equilibria, to $\mathsf{\text{ESR}}$-equilibria, and to SuperE-equilibria, respectively: in an $(\mathsf{E}+\mathsf{Var})$-equilibrium, no player could unilaterally reduce her ($\mathsf{E}+\mathsf{Var}$)-cost; in an $\mathsf{\text{ESR}}$-equilibrium, no player could unilaterally reduce her $\mathsf{\text{ESR}}$-cost; in a SuperE-equilibrium, Var =0 and no player could unilaterally reduce her E-cost. We show two complexity results:

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Deciding the existence of an ($\mathsf{E}+\mathsf{R}$)-equilibrium is strongly $\mathcal{NP}$-hard for 3-values games, where $\mathsf{R}$ is a general risk valuation, assuming that $\mathsf{E}+\mathsf{R}$ is strictly quasiconcave and satisfies certain technical properties. $\mathcal{N}P$-hardness is inherited to $\mathsf{E}+\mathsf{Var}$ and to $\mathsf{ESR}$, shown to have the properties.

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Deciding the existence of a SuperE-equilibrium is strongly $\mathcal{NP}$-hard for 3-values games, but computing one is in $\mathcal{P}$ for 2-values games. These results identify a complexity separation between 2-values and 3-values games. We also identify certain combinatorial properties of $(\mathsf{E}+\mathsf{Var})$-equilibria for 2-values games.

我们认为成本极少的2人最小化游戏。玩家会规避风险，并且会采取多种策略。玩家关心的是使期望以外的其他功能最小化或使用其他属性将期望最小化：期望加方差（E + Var）或扩展夏普比率（ESR）或期望（E）具有附加属性，即方差为零（$\mathsf{Var}=0$）。这些引起了$\mathsf{Ë}+\mathsf{Var}$） -平衡，以$\mathsf{\text{ESR}}$-equilibria和SuperE -equilibria：$（\mathsf{Ë}+\mathsf{Var}）$平衡，没有玩家可以单方面减少她（$\mathsf{Ë}+\mathsf{Var}$）-成本; 在一个$\mathsf{\text{ESR}}$平衡，没有玩家可以单方面减少她 $\mathsf{\text{ESR}}$-成本; 在SuperE平衡中，Var = 0，并且没有玩家可以单方面降低其E成本。我们显示了两个复杂性结果：

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确定是否存在（$\mathsf{Ë}+\mathsf{[R}$）平衡很强 $\mathcal{NP}$-难于进行三值游戏 $\mathsf{[R}$是一般风险评估，假设$\mathsf{Ë}+\mathsf{[R}$ 是严格拟凹的，并且满足某些技术性能。 $\mathcal{ñ}P$-硬度是继承给 $\mathsf{Ë}+\mathsf{Var}$ 并 $\mathsf{ESR}$，显示为具有属性。

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决定是否存在SuperE平衡$\mathcal{NP}$-适用于3值游戏，但需要计算一个 $\mathcal{P}$适用于2值游戏。这些结果确定了2值游戏和3值游戏之间的复杂度分离。我们还确定了某些组合特性$（\mathsf{Ë}+\mathsf{Var}）$2值游戏的均衡。