在偏好不确定的情况下进行分割-选择博弈
Playing Divide-and-Choose Given Uncertain Preferences
AI总结：
本文研究偏好不确定时分割-选择博弈的均衡结构与计算，发现分割者具有多样化动机导致多物品被分割，并证明物品数量较少时选择者更优、较多时分割者更优。
摘要：
我们研究经典的分割-选择方法，用于在两位理性的、自利的贝叶斯智能体之间公平分配可分割物品。两位参与者对物品具有可加性估值，这些估值的先验分布是共同知识。我们同时考虑估值相互独立的情形，以及参与者之间估值相关的情形（例如存在共同价值成分时）。我们描述了分割-选择博弈中最优分割的结构，并识别出若干可以高效计算均衡的情形。针对选择者对每件物品的估值服从正态分布的情形，我们提出了一种近似算法；针对区间上均匀分布的情形，我们提出了一种随机化近似算法。通过分析结果与计算模拟的结合，我们揭示了在偏好已知与偏好未知情形下最优策略之间的若干显著差异。最值得注意的是，在偏好未知时，分割者具有强烈的“多样化”动机来构造选择者的两个选项。这一动机导致多件物品在均衡中被分割，这与偏好已知时分割者的最优策略截然相反。在许多情境中，例如合伙人之间的买卖条款或判断公平性时，评估分割者与选择者的相对期望效用非常重要。我们证明，这些效用取决于参与者对彼此估值的了解程度、参与者估值之间的相关性以及被分割物品的数量。在相当温和的假设下，我们证明当物品数量较少时，选择者严格更优；而当物品数量较多时，分割者严格更优。
英文摘要：
We study the classic divide-and-choose method for equitably allocating divisible goods between two players who are rational, self-interested Bayesian agents. The players have additive values for the goods. The prior distributions on those values are common knowledge. We consider both the cases of independent values and values that are correlated across players (as occurs when there is a common-value component). We describe the structure of optimal divisions in the divide-and-choose game and identify several cases where it is possible to efficiently compute equilibria. An approximation algorithm is presented for the case when the distribution over the chooser's value for each good follows a normal distribution, along with a randomized approximation algorithm for the case of uniform distributions over intervals. A mixture of analytic results and computational simulations illuminates several striking differences between optimal strategies in the cases of known versus unknown preferences. Most notably, given unknown preferences, the divider has a compelling "diversification" incentive in creating the chooser's two options. This incentive leads to multiple goods being divided at equilibrium, quite contrary to the divider's optimal strategy when preferences are known. In many contexts, such as buy-and-sell provisions between partners, or in judging fairness, it is important to assess the relative expected utilities of the divider and chooser. Those utilities, we show, depend on the players' levels of knowledge about each other's values, the correlations between the players' values, and the number of goods being divided. Under fairly mild assumptions, we show that the chooser is strictly better off for a small number of goods, while the divider is strictly better off for a large number of goods.
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