On a Class of Infinite Games Related to Liar Dice

Abstract

In John Christopher's novel [2], the fate of the whole world is involved in the outcome of a game of Liar Dice. This paper presents a mathematical analysis of a class of zero-sum two-person games related to Liar Dice as described formally in Bell [1]. Karlin [4] provides the necessary background to the theory of games. Player $I$ receives a number $x$ chosen at random from a uniform distribution on the interval (0, 1). He then chooses a number $y$ in (0, 1) and claims that the number $x$ he received is at least $y$. Player $II$, not being informed of $x$, must accept or challenge $I$'s claim. If he accepts $I$'s claim, he loses an amount $b(x, y)$, a given function of $x$ and $y$. If he challenges $I$'s claim, he wins one if in fact $x < y$, and loses one if $x \geqq y$. The function $b(x, y)$ may be understood to represent player $II$'s expected loss in some other game that is played after $II$ accepts $I$'s claim. For example, if when $II$ accepts $I$'s claim, he must draw a number $z$ from the uniform distribution on (0, 1) and win one if $z > y$ and lose one if $z \leqq y$, then $II$'s expected loss is $b(x, y) = 2y - 1$. As another example, if when $II$ accepts $I$'s claim he must draw a number $z$ from the uniform distribution on $(x, 1)$, and win or lose one according as $z > y$ or $z \leqq y$, then $II$'s expected loss is \begin{align*}b(x, y) = -1\quad \text{if} \quad x \geqq y \\ = \frac{2y - x - 1}{1 - x}\quad \text{if} \quad x \leqq y.\end{align*} These two examples are taken later to illustrate the general theory. The use of a general $b(x, y)$ allows treatment of situations wherein the basic game is played again whenever $II$ accepts $I$'s claim, with the roles of the players reversed and with the distribution of the future $x$ dependent upon the past $x$ and $y$. Our solution to the general problem (Theorem 2) requires rather strong conditions on $b(x, y)$. The general problem is therefore not to be considered completely solved. However, when $b(x, y)$ is independent of the variable $x$, a complete solution is possible under the sole requirement that $b(y)$ (no longer a function of $x$) be nondecreasing in $y$. This is presented in Theorem 1 and the subsequent remarks.