When platforms compete for consumers, two types of consumer heterogeneity will matter: consumers value the presence of other consumers on a platform differently, and consumers contribute to the value of the platform differently. The optimal discriminatory pricing policy for platforms will depend...

We use a simple graphical approach to represent Social Welfare Functions that satisfy Independence of Irrelevant Alternatives and Anonymity. This approach allows us to provide simple and illustrative proofs of May's Theorem, of variants of classic impossibility results, and of a recent result on...