Book chapter
Сложность порядковых правил коллективного выбора
When a society needs to make a collective decision one could apply some aggregation method, particularly, voting. One of the main problems with voting is the manipulability of voting rules. We say a social choice rule is vulnerable to manipulation if there exists at least one voter who can achieve a better voting result by misrepresenting her preferences. The popular approach to comparing manipulability of social choice rules is defining a complexity class of the corresponding manipulation problem. This paper provides a survey into manipulation complexity literature considering variety of problems with different assumptions and constraints.
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.
The paper investigates problems arising from necessityto make joint group decision on the base of heterogeneous individual preferences. It is shown that the problem illustrated on a set of beautiful paradoxes is deeper than that of bad mechanisms of the collective choice, instead arising from incompatibility of democracy with the basic principles of any reasonable decision-making. Arrow's impossibility theorem, whose shortest proof is presented in the paper,deals with it. Also the paper suggests some compromising collective choice mechanisms which seem to be better than their widespread analogues.
In situations when a group of people has to make a decision based on the set of individual preferences, they use a certain aggregation method, in particular, voting. One of the main problems for any non-dictatorial social choice rule is the possibility for the voters to achieve a more preferable outcome of the voting by misrepresenting their preferences. Such actions on behalf of the voters are called manipulation, or strategic voting. One approach used to compare social choice rules in terms of how hard they are to manipulate is to find the complexity classes of manipulation problems for a given aggregation method. In this work, we present a survey of the studies of complexity classes of manipulation problems under various model assumptions and constraints.