The tension between market shares and profit under platform competition

We introduce asymmetries across platforms in the linear model of competing two-sided platforms with singlehoming on both sides and fully characterize the price equilibrium. We identify market environments in which one platform has a larger market share on both sides while obtaining a lower profit than the other platform. This is compatible with higher price-cost margins on one or both sides, noting that in the latter case one margin must be negative. Our finding raises further doubts on using market shares as a measure of market power in platform markets.