We discuss a model of a two-person, non-cooperative stochastic game, inspired by the discrete version of the red-and-black gambling problem presented by Dubins and Savage. Assume that two players hold certain amounts of money. At each stage of the game they simultaneously bid some part of their current fortune and the probability of winning or loosing depends on their bids. In many models of the red-and-black game it is assumed that the win probability is a function of the quotient of the bid of the first player and the sum of both bids. In the literature some additional properties, like concavity or super-multiplicativity are assumed in order to ensure that bold and timid strategy is the Nash equilibrium. Our aim is to provide a generalization in which the probability of winning is a two-variable function which depends on both bids. We introduce two new functional inequalities whose solutions lead to win probability functions for which a Nash equilibrium is realized by the bold-timid strategy. Since both inequalities have not easy intuitive meanings, we discuss them in a separate section of the paper and we give there some illustrating examples.

中文翻译：

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两人红黑游戏中概率函数的新不等式

我们讨论了一个两人非合作随机博弈模型，其灵感来自于 Dubins 和 Savage 提出的离散版本的红黑赌博问题。假设两个玩家持有一定数量的钱。在游戏的每个阶段，他们同时出价他们当前财富的一部分，获胜或失败的概率取决于他们的出价。在许多红黑游戏模型中，假设获胜概率是第一个玩家的出价与两个出价之和的商的函数。在文献中，假设了一些额外的属性，如凹度或超乘性，以确保大胆和胆小的策略是纳什均衡。我们的目标是提供一个概括，其中获胜的概率是一个取决于两个出价的双变量函数。我们引入了两个新的函数不等式，它们的解决方案导致获胜概率函数，其中通过大胆胆怯策略实现了纳什均衡。由于这两个不等式都没有直观的含义，我们在论文的单独部分中讨论它们，并在那里给出了一些说明性的例子。