Platform Equilibrium: Analayzing Social Welfare in Online Market Places

We introduce the theoretical study of a Platform Equilibrium. There are unit-demand buyers and unit-supply sellers. Each seller chooses to join or remain off a trading platform, and each buyer can transact with any on-platform seller but only a subset of different off-platform sellers. Given the choices of sellers, prices form a competitive equilibrium, and clear the market considering constraints on trade. Further, 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 $\alpha \in [0, 1]$ 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 possibly larger trading network against the imposition of a transaction fee. Our main insights are:(i) In homogeneous (identical) good 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 ideal welfare in homogeneous good markets; (iii) Even light regulation helps. When the platform's fee is capped at $\alpha$, the price of anarchy is 2-$\alpha$/1-$\alpha$ for general unit demand valuations. 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 ideal welfare. Some of our analysis extends beyond unit-demand buyers as well as to markets where sellers have production costs.