{\em Distortion} is a well-established notion for quantifying the loss of social welfare that may occur in voting. As voting rules take as input only ordinal information, they are essentially forced to neglect the exact values the agents have for the alternatives. Thus, in worst-case electorates, voting rules may return low social welfare alternatives and have high distortion. Accompanying voting rules with a small number of cardinal queries per agent may reduce distortion considerably. To explore distortion beyond worst-case conditions, we introduce a simple stochastic model, according to which the values the agents have for the alternatives are drawn independently from a common probability distribution. This gives rise to so-called {\em impartial culture electorates}. We refine the definition of distortion so that it is suitable for this stochastic setting and show that, rather surprisingly, all voting rules have high distortion {\em on average}. On the positive side, for the fundamental case where the agents have random {\em binary} values for the alternatives, we present a mechanism that achieves approximately optimal average distortion by making a {\em single} cardinal query per agent. This enables us to obtain slightly suboptimal average distortion bounds for general distributions using a simple randomized mechanism that makes one query per agent. We complement these results by presenting new tradeoffs between the distortion and the number of queries per agent in the traditional worst-case setting.

翻译：{\em 失真}是衡量投票中社会福利损失的一个成熟概念。由于投票规则仅以序数信息作为输入，它们本质上被迫忽略代理人对备选方案的确切价值。因此，在最坏情况下的选民群体中，投票规则可能返回低社会福利的备选方案，并导致高失真。通过为每个代理人配备少量基数查询，投票规则可以显著降低失真。为了探索超越最坏情况条件的失真，我们引入了一个简单的随机模型，其中代理人对备选方案的价值独立地从一个共同的概率分布中抽取。这产生了所谓的{\em 无偏文化选民}。我们重新定义了失真，使其适用于这种随机设定，并表明，相当令人惊讶的是，所有投票规则{\em 平均}都具有高失真。在积极方面，针对代理人具有随机{\em 二元}备选方案价值的基本情况，我们提出了一种机制，通过每个代理人进行{\em 单一}基数查询，实现了近似最优的平均失真。这使我们能够使用一种每个代理人进行一次查询的简单随机机制，为一般分布获得略微次优的平均失真界。我们通过展示传统最坏情况设定中失真与每个代理人查询次数之间的新权衡，对这些结果进行了补充。