The advantage of capitalism vs. socialism depends on the criterion

The talk given by the second author, L. A. Shepp, at the Linnik symposium, St.Petersburg, April 2005. We consider two distinct government tax policies towards companies: the republican policy gives tax breaks to the richer companies, while the democratic policy would perhaps give breaks to the weaker companies in hopes to keep them alive and so reduce unemployment. Which policy is better? We show that this depends on the optimization criterion, at least for the case of two companies, which is all that we can handle, in the stated mathematical formulation of the question.