We introduce the theoretical study of a Platform Equilibrium in a market with unit-demand buyers and unit-supply sellers. Each seller can join a platform and transact with any buyer or remain off-platform and transact with a subset of buyers whom she knows. Given the constraints on trade, prices form a competitive equilibrium and clears the market. The platform charges a transaction fee to all on-platform sellers, in the form of a fraction of on-platform sellers' price. The platform chooses the fraction to maximize revenue. A Platform Equilibrium is a Nash equilibrium of the game where each seller decides whether or not to join the platform, balancing the effect of a larger pool of buyers to trade with, against the imposition of a transaction fee. Our main insights are: (i) In homogeneous-goods markets, pure equilibria always exist and can be found by a polynomial-time algorithm; (ii) When the platform is unregulated, the resulting Platform Equilibrium guarantees a tight $\Theta(log(min(m, n)))$-approximation of the optimal welfare in homogeneous-goods markets, where $n$ and $m$ are the number of buyers and sellers respectively; (iii) Even light regulation helps: when the platform's fee is capped at $\alpha\in[0,1)$, the price of anarchy is 2-$\alpha$/1-$\alpha$ for general markets. For example, if the platform takes 30 percent of the seller's revenue, a rather high fee, our analysis implies the welfare in a Platform Equilibrium is still a 0.412-fraction of the optimal welfare. Our main results extend to markets with multiple platforms, beyond unit-demand buyers, as well as to sellers with production costs.

翻译：本文首次从理论层面研究了单位需求买家与单位供给卖家市场中的平台均衡问题。每个卖家可选择加入平台并与任意买家交易，或保持离线状态仅与已知的部分买家交易。在交易约束下，价格形成竞争均衡并实现市场出清。平台对所有入驻卖家按交易价格的一定比例收取交易费用，并通过调整该比例实现收益最大化。平台均衡是指每个卖家在权衡扩大交易对象范围与支付交易费用之间的利弊后，决定是否加入平台的纳什均衡。我们的核心发现包括：（一）在同质商品市场中，纯策略均衡始终存在，且可通过多项式时间算法求解；（二）在无监管条件下，平台均衡能实现同质商品市场最优社会福利的紧界$\Theta(\log(\min(m, n)))$近似，其中$n$和$m$分别代表买家与卖家数量；（三）适度监管即能改善效率：当平台费率上限设为$\alpha\in[0,1)$时，一般性市场的无政府价格比为$(2-\alpha)/(1-\alpha)$。例如当平台抽取卖家30%收益（较高费率）时，平台均衡下的社会福利仍能达到最优水平的0.412倍。本研究结论可进一步拓展至多平台市场、非单位需求买家市场以及存在生产成本的卖家市场。