A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others.
If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is welfarist (determined utility-wise by the feasible set of profiles), single-valued and easy to compute.
We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive division is still welfarist and related to the product of utilities or disutilities. If the zero utility profile (before any manna) is Pareto dominated, the competitive profile is unique and still maximizes the product of utilities. If the zero profile is unfeasible, the competitive profiles are the critical points of the product of disutilities on the efficiency frontier, and multiplicity is pervasive. In particular, the task of dividing a mixed manna is either good news for everyone, or bad news for everyone.
We refine our results in the practically important case of linear preferences, where the axiomatic comparison between the division of goods and that of bads is especially sharp. When we divide goods and the manna improves, everyone weakly benefits under the competitive rule; but no reasonable rule to divide bads can be similarly Resource Monotonic. Also, the much larger set of Non Envious and Efficient divisions of bads can be disconnected so that it will admit no continuous selection.
I consider the problem of allocating N indivisible objects among N agents according to their preferences when transfers are absent and an outside option may exist. I study the tradeoff between fairness and efficiency in the class of strategy-proof mechanisms. The main finding is that for strategy-proof mechanisms the following efficiency and fairness criteria are mutually incompatible: (1) ex-post efficiency and envy-freeness, (2) ordinal efficiency and weak envy-freeness, and (3) ordinal efficiency and equal division lower bound. Result 1 is the first impossibility result for this setting that uses ex-post efficiency ; results 2 and 3 are more practical than similar results in the literature. In addition, for N=3, I give two characterizations of the celebrated random serial dictatorship mechanism: it is the unique strategy-proof, ex-post efficient mechanism that (4) provides agents that have the same ordinal preferences with assignments not dominated by each other (weak envy-freeness among equals), or (5) provides agents that have the same cardinal preferences with assignments of equal expected utility (symmetry). These results strengthen the characterization by Bogomolnaia and Moulin (2001); result 5 implies the impossibility result by Zhou (1990).
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.