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摘要翻译：
竞争环境下的最优行为传统上是通过衡量不同行为的收益的效用函数来确定的。给定收入（收益）空间上的一个排序，冯·诺伊曼和摩根斯坦的经典公理方法建立了适当效用函数的存在性，并将博弈论作为确定最优行为的理论的最突出的物化。尽管这似乎也是风险管理的一种最自然的方法，但关键基础设施中的应用程序经常违反导致确定性后果的行动的隐含假设。从这个意义上说，关键基础设施风险控制竞争中的游戏本质上是随机的，因为行动具有不确定的后果。在数学上，这把我们带到概率分布值的效用函数，在这种情况下，我们松散了报酬空间上的规范（事实上是每一种可能的）排序，冯·诺伊曼和摩根斯坦的原始技术不再适用。这项工作介绍了一种新的博弈，在这种博弈中，不确定性适用于支付函数而不是玩家的行为（这一设置在文献中已经被广泛研究，屈服于像颤抖的手均衡或净化定理这样的著名概念）。详细地，我们给出了如何在概率分布空间上，通过将全集适度地限制为一个可以全序的子集，来确定一个（规范）序的不存在性。我们定义排序和建立基本博弈论的工具是非标准分析和超实数。
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英文标题：
《On Game-Theoretic Risk Management (Part One) -- Towards a Theory of
Games with Payoffs that are Probability-Distributions》
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作者：
Stefan Rass
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最新提交年份：
2020
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分类信息：
一级分类：Economics 经济学
二级分类：General Economics 一般经济学
分类描述：General methodological, applied, and empirical contributions to economics.
对经济学的一般方法、应用和经验贡献。
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一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Quantitative Finance 数量金融学
二级分类：Economics 经济学
分类描述：q-fin.EC is an alias for econ.GN. Economics, including micro and macro economics, international economics, theory of the firm, labor economics, and other economic topics outside finance
q-fin.ec是econ.gn的别名。经济学，包括微观和宏观经济学、国际经济学、企业理论、劳动经济学和其他金融以外的经济专题
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一级分类：Statistics 统计学
二级分类：Applications 应用程序
分类描述：Biology, Education, Epidemiology, Engineering, Environmental Sciences, Medical, Physical Sciences, Quality Control, Social Sciences
生物学，教育学，流行病学，工程学，环境科学，医学，物理科学，质量控制，社会科学
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
Optimal behavior in (competitive) situation is traditionally determined with the help of utility functions that measure the payoff of different actions. Given an ordering on the space of revenues (payoffs), the classical axiomatic approach of von Neumann and Morgenstern establishes the existence of suitable utility functions, and yields to game-theory as the most prominent materialization of a theory to determine optimal behavior. Although this appears to be a most natural approach to risk management too, applications in critical infrastructures often violate the implicit assumption of actions leading to deterministic consequences. In that sense, the gameplay in a critical infrastructure risk control competition is intrinsically random in the sense of actions having uncertain consequences. Mathematically, this takes us to utility functions that are probability-distribution-valued, in which case we loose the canonic (in fact every possible) ordering on the space of payoffs, and the original techniques of von Neumann and Morgenstern no longer apply. This work introduces a new kind of game in which uncertainty applies to the payoff functions rather than the player's actions (a setting that has been widely studied in the literature, yielding to celebrated notions like the trembling hands equilibrium or the purification theorem). In detail, we show how to fix the non-existence of a (canonic) ordering on the space of probability distributions by only mildly restricting the full set to a subset that can be totally ordered. Our vehicle to define the ordering and establish basic game-theory is non-standard analysis and hyperreal numbers.
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PDF链接：
https://arxiv.org/pdf/1506.07368
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