Entropic risk (ERisk) is an established risk measure in finance, quantifying risk by an exponential re-weighting of rewards. We study ERisk for the first time in the context of turn-based stochastic games with the total reward objective. This gives rise to an objective function that demands the control of systems in a risk-averse manner. We show that the resulting games are determined and, in particular, admit optimal memoryless deterministic strategies. This contrasts risk measures that previously have been considered in the special case of Markov decision processes and that require randomization and/or memory. We provide several results on the decidability and the computational complexity of the threshold problem, i.e. whether the optimal value of ERisk exceeds a given threshold. In the most general case, the problem is decidable subject to Shanuel's conjecture. If all inputs are rational, the resulting threshold problem can be solved using algebraic numbers, leading to decidability via a polynomial-time reduction to the existential theory of the reals. Further restrictions on the encoding of the input allow the solution of the threshold problem in NP$\cap$coNP. Finally, an approximation algorithm for the optimal value of ERisk is provided.

翻译：[translated abstract in Chinese] 熵风险（ERisk）是金融领域一种成熟的风险度量方法，通过奖励的指数加权调整来量化风险。我们首次在具有总奖励目标的回合制随机博弈背景下研究熵风险。这催生了一种目标函数，要求以风险规避的方式控制系统。我们证明这类博弈是确定的，且特别允许最优无记忆确定性策略的存在。这与先前在马尔可夫决策过程特例中考虑的风险度量形成对比，后者需要随机化和/或记忆机制。我们针对阈值问题（即熵风险最优值是否超过给定阈值）的可判定性与计算复杂性提供了多项结果。在最一般情形下，该问题在假设沙努埃尔猜想成立时可判定。若所有输入均为有理数，则可利用代数数求解阈值问题，通过多项式时间归约到实数的存在性理论实现可判定性。对输入编码的进一步限制使得阈值问题可在NP∩coNP中求解。最后，我们提出了一个熵风险最优值的近似算法。