### Article

## Modeling optimal social choice: matrix-vector representation of various solution concepts based on majority rule

Various Condorcet consistent social choice functions based on majority rule (tournament solutions) are considered in the general case, when ties are allowed: the core, the weak and strong top cycle sets, versions of the uncovered and minimal weakly stable sets, the uncaptured set, the untrapped set, classes of k-stable alternatives and k-stable sets. The main focus of the paper is to construct a unified matrix-vector representation of a tournament solution in order to get a convenient algorithm for its calculation. New versions of some solutions are also proposed.

A set of related majority rule-based social choice correspondences are considered: the union of minimal *Р*-dominating sets MPD (Duggan 2011, Subochev 2016) the union of weakly stable sets MWS (Aleskerov & Kurbanov 1999), the union of minimal *P*-externally stable sets MPES (Wuffl et al. 1989, Subochev 2008) and the union of minimal *R*-externally stable sets MRES (Aleskerov & Subochev 2009, 2013). These tournament solutions have not attracted much attention so far. However, the analysis of their properties suggests that MPES and MRES can be useful as instruments of choice, for instance when it is necessary to aggregate rankings. Their implementation is also possible under certain conditions.

The results presented are the following.

1) In a general case of a topological space of alternatives, a sufficient and necessary condition has been provided for an alternative to belong to a minimal *P*-dominating set. This characteristic condition is related to some version of the covering relation. It has been established that the union of minimal *P*-dominating sets and the uncovered set are logically nested neither in a general case, nor in finite tournaments. The characterization obtained provides a sufficient condition of nonemptiness of MPES and MRES in a general case of a topological space of alternatives.

2) It has been found that MPES and MRES both satisfy the following axioms:

a) monotonicity with respect to changes in social preferences (*P*-monotonicity),

b) the generalized Nash independence of irrelevant alternatives,

c) the idempotence,

d) the Aizerman-Aleskerov property,

e) the independence of social preferences for irrelevant alternatives (the independence of losers),

but they do not satisfy the extension axiom (Sen’s property g). It has also been demonstrated that MPD satisfies neither of these axioms, and MWS satisfies *P*-monotonicity only.

3) It has been found that MPES and MRES both satisfy Sanver monotonicity (a.k.a. cover monotonicity). Thus, despite they are not Maskin monotonic, these social choice correspondences can be implemented in a nonstandard setting, where actors have (extended) preferences for sets of alternatives. It has also been demonstrated that MPD and MWS do not satisfy Sanver monotonicity.

We use data on economic, management and political science journals to produce quantitative estimates of (in)consistency of evaluations based on seven popular bibliometric indicators (impact factor, 5-year impact factor, immediacy index, article influence score, h-index, SNIP and SJR). We propose a new approach to aggregating journal rankings: since rank aggregation is a multicriteria decision problem, ordinal ranking methods from social choice theory may solve it. We apply either a direct ranking method based on majority rule (the Copeland rule, the Markovian method) or a sorting procedure based on a tournament solution, such as the uncovered set and the minimal externally stable set. We demonstrate that aggregate rankings reduce the number of contradictions and represent the set of single-indicator-based rankings better than any of the seven rankings themselves.

A game with {\em{restricted cooperation}} is a triple (N,v,\Omega), where N is a finite set of players, \Omega is a non-empty collection oft feasible coalitions , and v a characteristic function defined on \Omega. U.Faigle (1989) obtained necessary and sufficient conditions for the non-emptiness of the core for games with restricted cooperation. Unlike the classical TU games the cores for games with restricted cooperation may be unbounded. Recently Grabisch and Sudh\"olter (2012) studied the core for games whose collections of feasible coalitions has a hierarchical structure generated by a partial order relation of players.For this class of games they proposed a new concept -- the bounded core -- whose definition can be extended to the general class of games with restricted cooperation as the union of all bounded faces of the core. For this class of games the bounded core can be empty even the core is not empty. An axiomatization of the bounded core for the whole class of games with restricted cooperation is given with the help of axioms efficiency, boundedness, bilateral consistency, a weakening of converse consistency, and ordinality. Another axiomatization of the core is given for the subclass of games with non-empty cores that are bounded. The characterizing axioms are non-emptiness, covariance, boundedness, consistency, the reconfirmation property, superadditivity, and continuity.

We use data on economic, management and political science journals to produce quan- titative estimates of (in)consistency of the evaluations based on six popular bibliometric indicators (impact factor, 5-year impact factor, immediacy index, article influence score, SNIP and SJR). We advocate a new approach to the aggregation of journal rankings. Since the rank aggregation is a multicriteria decision problem, ranking methods from social choice theory may solve it. We apply either a direct ranking method based on the majority rule (the Copeland rule, the Markovian method) or a sorting procedure based on a tournament solution, such as the uncovered set and the minimal externally stable set. We demonstrate that the aggregate rankings reduce the number of contradictions and represent the set of the single-indicator-based rankings better than any of the six rankings themselves.

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.