Intransitivity often emerges when ranking three or more alternatives. Condorcet paradox and Arrow's theorem are key examples of this phenomena in the social sciences, and non-transitive dice are a fascinating aspect of games of chance. In this paper, we study intransitivity in natural random models of dice and voting. First, we follow a recent thread of research that aims to understand intransitivity for three or more $n$-sided dice (with non-standard labelings), where the pairwise ordering is induced by the probability, relative to 1/2, that a throw from one die is higher than the other. Conrey, Gabbard, Grant, Liu and Morrison studied, via simulation, the probability of intransitivity for a number of random dice models. Their findings led to a Polymath project studying three i.i.d. random dice with i.i.d. faces drawn from the uniform distribution on $1,\ldots,n$, and conditioned on the average of faces equal to $(n+1)/2$. The Polymath project proved that the probability that three such dice are intransitive is asymptotically 1/4. We study some related models and questions. Among others, we show that if the uniform dice faces are replaced by any other continuous distribution (with some mild assumptions) and conditioned on the average of faces equal to zero, then three dice are transitive with high probability, in contrast to the unique behavior of the uniform model. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $H\in(0,1)$. We study analogous questions in social choice theory, where we define a notion of almost tied elections in the standard voting model, and show that the probability of Condorcet paradox for those elections approaches 1/4, in contrast to the unconditioned case. We also explore voting models where methods other than simple majority are used for pairwise elections.

中文翻译：

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骰子和势均力敌选举中不及物的概率

当对三个或更多选择进行排名时，经常会出现不及物性。孔多塞悖论和阿罗定理是社会科学中这种现象的关键例子，非传递骰子是机会游戏的一个迷人方面。在本文中，我们研究了骰子和投票的自然随机模型中的不传递性。首先，我们遵循最近的研究线索，旨在了解三个或更多 $n$ 面骰子（带有非标准标签）的不传递性，其中成对排序是由概率引起的，相对于 1/2，a一个骰子的投掷量高于另一个。Conrey、Gabbard、Grant、Liu 和 Morrison 通过模拟研究了许多随机骰子模型的不传递性概率。他们的发现导致了一个 Polymath 项目，研究具有 iid 的三个 iid 随机骰子 从 $1,\ldots,n$ 上的均匀分布绘制的人脸，并以等于 $(n+1)/2$ 的人脸平均值为条件。Polymath 项目证明三个这样的骰子不传递的概率是渐近的 1/4。我们研究了一些相关的模型和问题。其中，我们表明，如果均匀骰子面被任何其他连续分布（带有一些温和的假设）替换并以面数的平均值为零为条件，那么三个骰子以高概率传递，与独特的行为相反的统一模型。我们还将我们的结果扩展到固定高斯骰子，例如，其面可以是带有 Hurst 指数 $H\in(0,1)$ 的分数布朗增量。我们研究社会选择理论中的类似问题，我们在标准投票模型中定义了几乎绑定选举的概念，并表明与无条件情况相比，这些选举的孔多塞悖论的概率接近 1/4。我们还探索了投票模型，其中使用简单多数以外的方法进行配对选举。