A public decision-making problem consists of a set of issues, each with multiple possible alternatives, and a set of competing agents, each with a preferred alternative for each issue. We study adaptations of market economies to this setting, focusing on binary issues. Issues have prices, and each agent is endowed with artificial currency that she can use to purchase probability for her preferred alternatives (we allow randomized outcomes). We first show that when each issue has a single price that is common to all agents, market equilibria can be arbitrarily bad. This negative result motivates a different approach. We present a novel technique called pairwise issue expansion, which transforms any public decision-making instance into an equivalent Fisher market, the simplest type of private goods market. This is done by expanding each issue into many goods: one for each pair of agents who disagree on that issue. We show that the equilibrium prices in the constructed Fisher market yield a pairwise pricing equilibrium in the original public decision-making problem which maximizes Nash welfare. More broadly, pairwise issue expansion uncovers a powerful connection between the public decision-making and private goods settings; this immediately yields several interesting results about public decisions markets, and furthers the hope that we will be able to find a simple iterative voting protocol that leads to near-optimum decisions.

一个公共决策问题包括一组问题，每个问题都有多种可能的选择，以及一组竞争代理，每个问题都有一个针对每个问题的首选选择。我们研究市场经济对这种情况的适应，重点是二元问题。问题具有价格，每个代理商都被赋予人造货币，可以用来购买首选方案的可能性（我们允许随机分配结果）。我们首先证明，当每个发行都具有所有代理商共同的单一价格时，市场均衡可能会任意恶化。这种负面结果促使人们采用了不同的方法。我们提出一种称为成对问题扩展的新颖技术，它将任何公共决策实例转变为等效的费舍尔市场，这是最简单的私人商品市场类型。这是通过将每个问题扩展为多种商品来完成的：对于在该问题上有分歧的每对代理商，每对都有一个。我们表明，在构建的费希尔市场中的均衡价格在原始的公共决策问题中产生了成对的定价均衡，从而使纳什福利最大化。更广泛地说，成对问题扩展揭示了公共决策与私人物品设置之间的强大联系；这立即产生了一些有关公共决策市场的有趣结果，并进一步希望我们能够找到一个导致接近最优决策的简单迭代投票协议。