Bidding for Network Size: Platform Competition when Quality and Network Size are Complements

We study two platforms competing for members by investing in network quality.  Quality is complementary to the network size: the marginal utility generated by an additional member increases with the network's quality.  Platforms are imperfect substitutes: a share of the potential members are biased toward each of the platforms and some are indifferent ex ante.  We assume that, in case of multiple equilibria, consumers use the investment in quality as a coordination device.  We find that, in equilibrium, platforms randomize over two disconnected intervals of investment levels, corresponding to competing for either the entire population or the mass of ex-ante different members.  While the "prize" of winning the competition for members is identical for both platforms, the value of the outside option "not investing" depends on a platform's share of ex-ante biased members.  The platform with the smallest share of ex-ante biased members bids more aggressively to compensate for its lower outside option and achieves a monopoly network with higher probability than its competitor.