Dividing Goods or Bads Under Additive Utilities

NRU Higher School of Economics , 2016. No. 147.

Bogomolnaia A., Moulin H., Sandomirskiy F., Yanovskaya E. B.

The Competitive Equilibrium with Equal Incomes is an especially appealing efficient and envy-free division of private goods when utilities are additive: it maximizes the Nash product of utilities and is single-valued and continuous in the marginal rates of substitution. The CEEI to divide bads captures similarly the critical points of the Nash product in the ecient frontier. But it is far from resolute, allowing routinely many divisions with sharply different welfare consequences. Even the much more permissive No Envy property is profoundly ambiguous in the division of bads: the set of ecient and envy-free allocations can have many connected components, and has no single-valued selection continuous in the marginal rates. The CEEI to divide goods is Resource Monotonic (RM): everyone (weakly) benets when the manna increases. But when we divide bads eciently, RM is incompatible with Fair Share Guarantee, a much weaker property than No Envy.

Keywords: fair division of goods, fair division of bads, competitive equilibrium with equal incomes, Nash product, envy-freeness, resource monotonicity, independence of lost bids

Publication based on the results of: Algorithmic aspects of fair division problems (2016)