部分可观测风险敏感马尔可夫决策过程
Partially Observable Risk-Sensitive Markov Decision Processes
摘要：
本文研究了由部分可观测马尔可夫决策过程（POMDP）在有限和无限时间范围内最小化总成本或折扣成本的确定性等价问题。确定性等效由$U^{-1}(EU(Y))$定义，其中$U$是增函数。与风险中性的决策者相比，这种优化标准考虑了成本的可变性。它包含作为一个特殊的情况下，经典的风险敏感的优化准则与指数效用。我们表明，这个最优化问题可以解决嵌入到一个完全可观察的马尔可夫决策过程与扩展的状态空间的问题，并给出条件下，最优策略存在。状态空间必须由当前未观测状态和累积代价的联合条件分布来扩展。在指数效用的情况下，问题大大简化，我们重新发现在以前的文献中被命名为信息状态。然而，由于我们在这里没有使用任何度量变化技术，我们的方法更简单。一个小的数值例子，即经典的重复赌场游戏与未知的成功概率被认为是说明的确定性当量及其参数的影响。
英文摘要：
We consider the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite time horizon which is generated by a Partially Observable Markov Decision Process (POMDP). The certainty equivalent is defined by $U^{-1}(EU(Y))$ where $U$ is an increasing function. In contrast to a risk-neutral decision maker this optimization criterion takes the variability of the cost into account. It contains as a special case the classical risk-sensitive optimization criterion with an exponential utility. We show that this optimization problem can be solved by embedding the problem into a completely observable Markov Decision Process with extended state space and give conditions under which an optimal policy exists. The state space has to be extended by the joint conditional distribution of current unobserved state and accumulated cost. In case of an exponential utility, the problem simplifies considerably and we rediscover what in previous literature has been named information state. However, since we do not use any change of measure techniques here, our approach is simpler. A small numerical example, namely the classical repeated casino game with unknown success probability is considered to illustrate the influence of the certainty equivalent and its parameters.
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