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## Manipulability of consular election rules

The Gibbard–Satterthwaite theorem is a cornerstone of social choice theory, stating that an onto social choice function cannot be both strategy-proof and non-dictatorial if the number of alternatives is at least three. The Duggan–Schwartz theorem proves an analogue in the case of set-valued elections: if the function is onto with respect to singletons, and can be manipulated by neither an optimist nor a pessimist, it must have a weak dictator. However, the assumption that the function is onto with respect to singletons makes the Duggan–Schwartz theorem inapplicable to elections which necessarily select multiple winners. In this paper we make a start on this problem by considering rules which always elect exactly two winners (such as the consulship of ancient Rome). We establish that if such a *consular election rule* cannot be expressed as the union of two disjoint social choice functions, then strategy-proofness implies the existence of a dictator. Although we suspect that a similar result holds for *k*-winner rules for k>2k>2, there appear to be many obstacles to proving it, which we discuss in detail.

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.

This article describesseveral impossibility results in social choice theory and demonstrates their importance for democratic theory. Since 1950s social scientists paid a great attention to the investigation of collective decision-making. This interest led to the formation of a new field of study within economics and political science, social choice theory. The main resultsof this strand of research are various impossibilitytheorems which illustrateinconsistencies indifferentvoting rules. Arrow`s impossibility theorem is usually considered to bethe most important result of this kind: however, many other impossibility theorems were proved during the last fifty years, among them the Gibbard-Satterthwaite theorem, Amartya Sen's liberal paradox and discursive dilemma. These paradoxical findingsreveal serious inner defects of democratic decision-making and therefore challenge the democratic idea itself, which is presumably the central project of modern political thought. Therefore, they are of great interest for democratic theorists.

Originally published in 1951, Social Choice and Individual Valuesintroduced “Arrow’s Impossibility Theorem” and founded the field of social choice theory in economics and political science. This new edition, including a new foreword by Nobel laureate Eric Maskin, reintroduces Arrow’s seminal book to a new generation of students and researchers.

"Far beyond a classic, this small book unleashed the ongoing explosion of interest in social choice and voting theory. A half-century later, the book remains full of profound insight: its central message, ‘Arrow’s Theorem,’ has changed the way we think.”—Donald G. Saari, author of Decisions and Elections: Explaining the Unexpected

Kenneth J. Arrow is professor of economics emeritus, Stanford University, and a Nobel laureate. Eric S. Maskin is Albert O. Hirschman Professor, School of Social Science, Institute of Advanced Study, Princeton, NJ, and a Nobel laureate.

*When a society needs to take a collective decision one could apply some aggregation method, particularly, voting. One of the main problems with voting is manipulation. We say a voting rule is vulnerable to manipulation if there exists at least one voter who can achieve a better voting result by misrepresenting his or her preferences. The popular approach to comparing manipulability of voting rules is defining complexity class of the corresponding manipulation problem. This paper provides a survey into manipulation complexity literature considering variety of problems with different assumptions and restrictions.*

Aleskerov et al. [1] and [2] estimated the degree of manipulability for the case of multi-valued choice (without using any tie-breaking rule) and for Impartial Culture (IC). In our paper, we address the similar question for the multi-valued choice and for Impartial Anonymous Culture (IAC). We use Nitzan-Kelly's (NK) index to estimate the degree of manipulability, which is calculated as the share of all manipulable voting situations, and calculate indices for 3 alternatives and up to 10000 voters. We have found that for the case of 3 alternatives Nanson's procedure shows the best results. Hare's procedure shows close, but a bit higher results. The worst aggregation procedure in terms of manipulability is Plurality rule. Additionally, it turned out that NK indices for IAC are smaller than NK indices for IC.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.