### Book chapter

## Degree of Manipulability of Known Social Choice Rules in the Case of Multiple Choice

The problem of the manipulability of known social choice rules in the case of multiple choice is considered. Several concepts of expanded preferences (preferences over the sets of alternatives) are elaborated. As a result of this analysis ordinal and nonordinal methods of preferences expanding are defined. The notions of the degree of manipulability are extended to the case under study. Using the results of theoretical investigation, 22 known social choice rules are studied via computational experiments to reveal their degree of manipulability.

### In book

We consider the problem of manipulability of social choice rules in the impartial anonymous and neutral culture model (IANC) and provide a new theoretical study of the IANC model, which allows us to analytically derive the difference between the Nitzan-Kelly index in the Impartial Culture (IC) and IANC models. We show in which cases this difference is almost zero, and in which the Nitzan-Kelly index for IANC is the same as for IC. However, in some cases this difference is large enough to cause changes in the relative manipulability of social choice rules. We provide an example of such cases.

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.

In the paper we analyze the reasons of Russian bank license withdrawal, formulated in orders of CB RF at the period 2005.2–2008.4. During this period, after establishing deposit insurance system in Russia, two main reasons were «money laundering» and «financial insolvency». We design binary choice logit models and multinomial logit models to model probability of license withdrawal one year ahead of the event. We use in model macro indicators to control for the varying economic environment and bank-specific financial indicators taken one year before the observation of the bank status. The models reveal factors important for the prediction of the license withdrawal, which are found to be different for the two reasons. Also we investigate if multinomial model outperform binary model in the bank license withdrawal forecast. We consider dynamics of impact of unaccounted factors, including human factor.

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).

Procedures aggregating individual preferences into a collective choice differ in their vulnerability to manipulations. To measure it, one may consider the share of preference profiles where manipulation is possible in the total number of profiles, which is called Nitzan-Kelly's index of manipulability. The problem of manipulability can be considered in different probability models. There are three models based on anonymity and neutrality: impartial culture model (IC), impartial anonymous culture model (IAC), and impartial anonymous and neutral culture model (IANC). In contrast to the first two models, the IANC model, which is based on anonymity and neutrality axioms, has not been widely studied. In addition, there were no attempts to derive the difference of probabilities (such as Nitzan-Kelly's index) in IC and IANC analytically. We solve this problem and show in which cases the upper bound of this difference is high enough, and in which cases it is almost zero. These results enable us to simplify the computation of indices.

*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.