# Publications

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).

Supposing that Player 1’s computational power is higher than that of Player 2, we give three examples of different kinds of public signal about the state of a two-person zero-sum game with symmetric incom- plete information on both sides (both players do not know the state of the game) where Player 1 due to his computational power learns the state of the game meanwhile it is impossible for Player 2. That is, the game with incomplete information on both sides becomes a game with incomplete information on the side of Player 2. Thus we demonstrate that information about the state of a game may appear not only due to a private signal but as a result of a public signal and asymmetric computational resources of players.

We prove a general possibility result for collective decision problems where individual allocations are one-dimensional, preferences are single-peaked (strictly convex), and feasible allocation pro les cover a closed convex set. Special cases include the celebrated median voter theorem ([10], [21]) and the division of a non disposable commodity by the uniform rationing rule ([48]). We construct a canonical peak-only rule equalizing in the leximin sense individual gains from an arbitrary benchmark allocation: it is ef cient, group-strategyproof, fair, and (for most problems) continuous. These properties leave room for many other rules, except for symmetric non disposable division problems.

We consider finite noncooperative (Formula presented.) person games with fixed numbers (Formula presented.), (Formula presented.), of pure strategies of Player (Formula presented.). We propose the following question: is it possible to extend the vector space of finite noncooperative (Formula presented.)-games in mixed strategies such that all games of a broader vector space of noncooperative (Formula presented.) person games on the product of unit (Formula presented.)-dimensional simplices have Nash equilibrium points? We get a necessary and sufficient condition for the negative answer. This condition consists of a relation between the numbers of pure strategies of the players. For two-person games the condition is that the numbers of pure strategies of the both players are equal.

We consider repeated zero-sum games with incomplete information on the side of Player 2 with the total payoff given by the non-normalized sum of stage gains. In the classical examples the value of such an N-stage game is of the order of N or of square root of N, as N tends to infinity. Our aim is to find what is causing another type of asymptotic behavior of the value observed for the discrete version of the financial market model introduced by De Meyer and Saley. For this game Domansky and independently De Meyer with Marino found that the value remains bounded, as N tends to infinity, and converges to the limit value. This game is almost-fair, i.e., if Player 1 forgets his private information the value becomes zero. We describe a class of almost-fair games having bounded values in terms of an easy-checkable property of the auxiliary non-revealing game. We call this property the piecewise property, and it says that there exists an optimal strategy of Player 2 that is piecewise-constant as a function of a prior distribution. Discrete market models have the piecewise property. We show that for non-piecewise almost-fair games with an additional non-degeneracy condition the value is of the order of square root of N.

Users share the cost of unreliable non-rival projects (items). For instance, industry partners pay today for R&D that may or may not deliver a cure to some viruses, agents pay for the edges of a network that will cover their connectivity needs, but the edges may fail, etc. Each user has a binary inelastic need that is served if and only if certain subsets of items are actually functioning. We ask how should the cost be divided when individual needs are heterogenous. We impose three powerful separability properties: Independence of Timing ensures that the cost shares computed ex ante are the expectation, over the random realization of the projects, of shares computed ex post. Cost Additivity together with Separability Across Projects ensure that the cost shares of an item depend only upon the service provided by that item for a given realization of all other items. Combining these with fair bounds on the liability of agents with more or less flexible needs, and of agents for whom an item is either indispensable or useless, we characterize two rules: the Ex Post Service rule is the expectation of the equal division of costs between the agents who end up served; the Needs Priority rule splits the cost first between those agents for whom an item is critical ex post, or if there are no such agents between those who end up being served.

The sum (resp. the sum of the squares) of the defects in the triangle in- equalities for the area one lattice parallelograms in the first quadrant has a surprisingly simple expression.

Namely, let f(a,b,c,d)=√a2 +b2+√c2 +d2− (a+c)2 +(b+d)2. Then, f(a,b,c,d)2 =2−π/2, (Ж)

f(a,b,c,d) = 2, (ж)

wherethesumrunsbyalla,b,c,d∈Z≥0 suchthatad−bc=1. This paper is devoted to the proof of these formulae. We also discuss possible

directions in study of this phenomenon.

Suppose that there exists a hypersurface with the Newton polytope Δ, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve *C* which passes through generic points p1,…,pn with prescribed multiplicities m1,…,mn. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve *C* is at least max(mi). Using tropical floor diagrams (a certain degeneration of p1,…,pn on a horizontal line) we prove that

area(Δ)≥12n∑i=1m2i−S, where S=12max(n∑i=1s2i|si≤mi,n∑i=1si≤ω(Δ)).

In the case m1=m2=⋯=m≤ω(Δ) this estimate becomes area(Δ)≥12(n−ω(Δ)m)m2. That rewrites as d≥(√n−12−12√n)m for the curves of degree *d*. We consider an arbitrary toric surface (i.e. arbitrary Δ) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not *a priori* clear what is *a collection of generic points* in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory.

In the bilateral assignment problem, source a holds the amount ra of resource of type a, while sink i must receive the total amount xi of the various resources. We look for assignment rules meeting the powerful separability property known as Consistency: “every subassignment of a fair assignment is fair”. They are essentially those rules selecting the feasible flow minimizing the sum ∑i,aW(yia), where W is smooth and strictly convex.