Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and convex polyhedral set, or polytope. Polyhedral combinatorics have thus played a key role in studying the mathematical structure of these models. However, standard methods for characterizing the polytopes of such models are subject to a combinatorial explosion in complexity as the number of choice alternatives increases. Specifically, this is the case for random preference models based on linear, weak, semi- and interval orders. For these, a complete, linear description of the polytope is currently known only for, at most, 5-8 choice alternatives. We leverage the method of extended formulations to break through those boundaries. For each of the four types of preferences, we build an appropriate network, and show that the associated network flow polytope provides an extended formulation of the polytope of the choice model. This extended formulation has a simple linear description that is more parsimonious than descriptions obtained by standard methods for large numbers of choice alternatives. The result is a computationally less demanding way of testing the probabilistic choice model on data. We sketch how the latter interfaces with recent developments in contemporary statistics.

中文翻译：

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通过网络流订购多胞体的扩展公式

数学心理学在通过无分布的随机效用模型和相关的随机偏好模型对概率选择进行建模方面有着悠久的传统。对于此类模型，预测的选择概率通常形成有界凸多面体集或多面体。因此，多面体组合在研究这些模型的数学结构方面发挥了关键作用。然而，随着备选方案数量的增加，用于表征此类模型的多胞体的标准方法的复杂性会出现组合爆炸。具体来说，这是基于线性、弱、半和区间顺序的随机偏好模型的情况。对于这些，完整的、线性的多胞体描述目前只知道最多 5-8 个选项。我们利用扩展公式的方法来突破这些界限。对于四种类型的偏好中的每一种，我们都构建了一个合适的网络，并表明相关的网络流多胞胎提供了选择模型多胞胎的扩展公式。这个扩展公式有一个简单的线性描述，比通过标准方法获得的大量选项的描述更简洁。结果是一种在数据上测试概率选择模型的计算要求较低的方法。我们勾勒出后者如何与当代统计学的最新发展相结合。这个扩展公式有一个简单的线性描述，比通过标准方法获得的大量选项的描述更简洁。结果是一种在数据上测试概率选择模型的计算要求较低的方法。我们勾勒出后者如何与当代统计学的最新发展相结合。这个扩展公式有一个简单的线性描述，比通过标准方法获得的大量选项的描述更简洁。结果是一种在数据上测试概率选择模型的计算要求较低的方法。我们勾勒出后者如何与当代统计学的最新发展相结合。