The issue of estimating the voter power in weighted voting systems is considered. We show that the numerical values of the weighted power index previously introduced by the author can be determined by a simple minimization scheme. The efficiency of the method is demonstrated by estimating the influentiality of U.S. Congressmen.

A new approach to the solution of one-step games is constructed, without using the concept of mixed strategy. The notion of a “set” solution of a bimatrix game is defined. It is shown that this solution always exists and may be found by a finite procedure. Examples are given illustrating the form of the “set” solution and the structure of the set of best responses for various levels of information availability to the players regarding the opponent’s behavior.

A new quantitative definition of the “power index” is proposed for a voter in weighted voting systems (WV-schemes), where the voters a priori do not have equal rights. The proposed “power index” is generated in a self-consistent manner from information about the number of times that a voter enters any winning coalition in the WV-scheme. Explicit formulas are derived for computing the “power index”. The discussion is illustrated with prototype examples.

A new approach to the construction of utility under uncertainty is proposed. The approach relies on a binary relation on the set of outcomes of a probabilistic process. Kingman’s subadditive ergodic theorem is applied to prove the existence of a utility function. The differences between the proposed approach and the theories of von Neumann-Morgenstern and Savage are considered. Resolution of some classical paradoxes is demonstrated.

Methods of symbolic dynamics are applied to construct a description of two-person supergames, i.e., games involving infinite repetition of the same basis matrix game. The behavior of players choosing their moves based only on information about the m previous moves follows a stationary joint probability distribution. Equations for this distribution are derived and it is shown that a mixed extension of the basis game corresponds to its representation as a supergame with m = 0, i.e., with player behavior that ignores the effect of the game history on the choice of player strategies.

The article examines the solution of bargaining games on the basis of the previously proposed paradigm of stationary strategies in supergames. The value of axiomatic bargaining theory for choosing the unique solution in agreement games is considered. The “splitting the dollar” game for two and three players is discussed as an application.

The multirelation method originally developed for group choice problems is applied to study the formation of new links in a social network graph. A natural generalization of the “nearest neighbor” method is proposed for predicting the appearance of links between points that have no common neighbors in the original graph.

The dynamic approach developed for the description of supergames is applied to problems in the theory of cooperation. New methods are proposed to describe game programs both in the standard prisoner’s dilemma format and in the nonstandard format allowing nonsynchronous moves. A unique Pareto-efficient outcome is derived for the case when the players’ behavior is driven by the critical points they choose in the game.

The self-consistent choice theory is applied to solve the problem of ranking Internet pages given the link matrix. The damping factor introduced in the Google search engine distorts the initial information. A binary weak-order relation is constructed for ranking Internet pages and sites without the damping factor.

We construct an ordered family of web pages that allow for the canonical structure of the link matrix. The multirelation method is applied to obtain a restriction of this family and to investigate the kind of orderings that arise. A new interpretation of the link matrix is proposed, reducing the number of equivalence classes in its canonical form. The discussion is illustrated with some prototype examples.

We consider the behavior of four choice rules—plurality voting, approval voting, Borda count, and self-consistent choice—when applied to choose the best option from a three-element set. It is assumed that the two main options are preferred by a large majority of the voters, while the third option gets a very small number of votes and influences the election outcome only when the two main options receive a close number of votes. When used to rate the main options, Borda count and self-consistent choice contain terms that allow both for the “strength of preferences” of the voters and the rating of the main candidates by voters who vote for the third option. In this way, it becomes possible to determine more reliably the winner when plurality voting or approval voting produce close results.

We examine the properties of a new method for constructing the power index in voting systems where voters have unequal influence. A system of equations derived in this study produces the sought index by optimal approximation of the original voting system with another system that allows for the weights of the voters. The new index is applied to analyze the known paradoxes of committee voting theory.

We analyze the plurality runoff rule often used in national elections and prove that the corresponding choice function is Borda-rational. The compromise level attained in real voting systems is defined. The corresponding measures are calculated for the case of plurality runoff with three main candidates. We investigate the influence of the axiom of independence of irrelevant alternatives on the rationality type of choice functions realized in multicriterion choice systems.