arXiv subjects

Fedor Sandomirskiy

Publications and source records attributed to Fedor Sandomirskiy.

26 records · Page 2Linked to original sources

arXiv2020

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

We consider fair allocation of indivisible items under additive utilities. When the utilities can be negative, the existence and complexity of an allocation that satisfies Pareto optimality and proportionality up to one item (PROP1) is an open problem. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. We point out that the result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stronger concepts.

cs.GT↗

arXiv2020

Representative Committees of Peers

A population of voters must elect representatives among themselves to decide on a sequence of possibly unforeseen binary issues. Voters care only about the final decision, not the elected representatives. The disutility of a voter is proportional to the fraction of issues, where his preferences disagree with the decision. While an issue-by-issue vote by all voters would maximize social welfare, we are interested in how well the preferences of the population can be approximated by a small committee. We show that a k-sortition (a random committee of k voters with the majority vote within the committee) leads to an outcome within the factor 1+O(1/k) of the optimal social cost for any number of voters n, any number of issues $m$, and any preference profile. For a small number of issues m, the social cost can be made even closer to optimal by delegation procedures that weigh committee members according to their number of followers. However, for large m, we demonstrate that the k-sortition is the worst-case optimal rule within a broad family of committee-based rules that take into account metric information about the preference profile of the whole population.

cs.GT↗

arXiv2017

Dividing goods or bads under additive utilities

We compare the Egalitarian Equivalent and the Competitive Equilibrium with Equal Incomes rules to divide a bundle of goods (heirlooms) or a bundle of bads (chores). For goods the Competitive division fares better, as it is Resource Monotonic, and makes it harder to strategically misreport preferences. But for bads, the Competitive rule, unlike the Egalitarian one, is multivalued, harder to compute, and admits no continuous selection. We also provide an axiomatic characterization of the Competitive rule based on the simple formulation of Maskin Monotonicity under additive utilities.

cs.GT↗

arXiv2017

Competitive division of a mixed manna

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.

cs.GT↗

arXiv2016

Dividing goods and bads under additive utilities

When utilities are additive, we uncovered in our previous paper (Bogomolnaia et al. "Dividing Goods or Bads under Additive Utilities") many similarities but also surprising differences in the behavior of the familiar Competitive rule (with equal incomes), when we divide (private) goods or bads. The rule picks in both cases the critical points of the product of utilities (or disutilities) on the efficiency frontier, but there is only one such point if we share goods, while there can be exponentially many in the case of bads. We extend this analysis to the fair division of mixed items: each item can be viewed by some participants as a good and by others as a bad, with corresponding positive or negative marginal utilities. We find that the division of mixed items boils down, normatively as well as computationally, to a variant of an all goods problem, or of an all bads problem: in particular the task of dividing the non disposable items must be either good news for everyone, or bad news for everyone. If at least one feasible utility profile is positive, the Competitive rule picks the unique maximum of the product of (positive) utilities. If no feasible utility profile is positive, this rule picks all critical points of the product of disutilities on the efficient frontier.

cs.GT↗

arXiv2016

On repeated zero-sum games with incomplete information and asymptotically bounded values

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 $V_N$ of such an $N$-stage game is of the order of $N$ or $\sqrt{N}$ as $N\to \infty$. Our aim is to find what is causing another type of asymptotic behavior of the value $V_N$ 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 $V_N$ remains bounded as $N\to\infty$ 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 $p$. Discrete market models have the piecewise property. We show that for non-piecewise almost-fair games with an additional non-degeneracy condition $V_N$ is of the order of $\sqrt{N}$.

cs.GT↗

arXiv2013

Repeated games of incomplete information with large sets of states

The famous theorem of R.Aumann and M.Maschler states that the sequence of values of an N-stage zero-sum game G_N with incomplete information on one side converges as N tends to infinity, and the error term is bounded by a constant divided by square root of N if the set of states K is finite. The paper deals with the case of infinite K. It turns out that for countably-supported prior distribution p with heavy tails the error term can decrease arbitrarily slowly. The slowest possible speed of the decreasing for a given p is determined in terms of entropy-like family of functionals. Our approach is based on the well-known connection between the behavior of the maximal variation of measure-valued martingales and asymptotic properties of repeated games with incomplete information.

cs.GT↗

arXiv2013

An exact renormalization formula for the Maryland model

We discuss the difference Schrödinger equation $ψ_{k+1}+ψ_{k-1}+λ\cot(πωk+θ)ψ_k=Eψ_k$, $k\in\mathbb{Z}$, where $λ$, $ω$, $θ$ and $E$ are parameters. We obtain explicit renormalization formulas relating its solutions for large $|k|$ to solutions of the equation with new parameters $λ$, $ω$, $θ$ and $E$ for bounded $|k|$. These formulas are similar to the renormalization formulas from the theory of Gaussian exponential sums.

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