As part of the research, we:
- Made the first generalizations of threshold procedures in the case of equally important criteria;
- Construct a basic behavior model of the stock exchange as a reaction to incoming signals regarding the state of the economy, developed new models that suggest that players learn from their successes, from the results of testing with real data, we obtained numerical parameters of the models;
- Through computer simulation we evaluated various procedures of collective choice in terms of the degree of manipulability;
- Developed axiomatics of the proportional representation system in terms of rational choice, and studied existing electoral systems that use the single transferable vote rule;
- Developed an accurate and efficient method (as well as its modification free from rounding errors) of solving bilinear programming problems to compare preferred alternatives with interval information about relevance of the criteria and the value of scale evaluations, carried out an algorithmic implementation of the methods and computational testing, studied the hierarchy analysis method and demonstrated its inaccuracy from the point of view of mathematical theory of measurement;
- Developed and implemented in software a new generation of automatic classification algorithms based on formalizing the notion of intuitively right classification;
- Construct a formal model of the fair division problem for several players;
- Developed a set of programs to calculate several solutions to the social choice problem, based on the preferences of the majority and related ways of composing rankings, examined and specified set-theoretical relationships of these solutions;
- Constructed axiomatics of the Banzhaf index and Shapley-Shubik indices into power indices, depending on agents preferences to coalesce;
- Developed a game-theoretic model of the university selection process by applicants and the process of stage-by-stage enrollment according to the procedure approved in 2010; predicted student behavior in choosing a group of higher education institutions to apply to in the framework of one group of training areas, modeled the application and enrollment process with different distributions based on USE results; predicted student shortfall in medium-quality universities provided that these institutions comply with admission rules; determined Pareto inefficiency and instability (in terms of generalized pair-matchings) of applicant distribution across higher education institutions with different initial distributions of USE results ;
- Modified a system of indicators used in the dynamic analysis of the behavior patterns of Russian commercial banks; introduced new variables reflecting the bank's role in the interbank market and the use of state support for the banking sector; carried out a cluster analysis of banks in terms of selected indicators over time, evaluated the efficiency of Russian commercial banks in 2008 and the relationship between the efficiency of banks and their ownership structure;
- Formed a database of indicators of commercial bank functioning and indicators of socio-economic development of Russian regions over the period of 2006-2009, determined homogeneous groups of Russian banks, evaluated the efficiency of Russian banks, demonstrated the influence of a bank’s size, asset and liability structure, capital participation of foreign investors and access to state funds on the cost efficiency of peer banks; identified the most common business models of large and medium-sized Russian banks that are specific of the pre and post-crisis periods;
- Obtained asymptotic formulas for option price under any ratios of the Black-Scholes model parameters for large and small time periods, made a comparison of computation time for the correct solution of the model, expressed as an improper integral and obtained asymptotic formulas;
- For elimination games, the principle of single deviation was proved, all stationary equilibriums in pure strategies for three-player dual games with simultaneous shooting were found.
- Obtained for the first time comparative statics for equilibrium in the probability model of voting and demonstrated that candidate positions depend on how the value of a landslide victory differs from the value of the victory with a small margin;
- Developed mathematical basis and started experiment calculations of ontological analysis of the texts in Russian and English: (a) analysis of the content of academic program of mathematical and informatics courses at HSE (in Russian), (b) analysis of annotations and key words of the articles published in journals and magazines of the World Association of Computing Machinery (in English);
- Carried out a number of experiments to verify the efficiency of proposed criteria to terminate the cluster division process; carried out initial testing of the algorithm of the query set ascension across the taxonometry tree and result visualization, analyzed the results of the work in the course program, managed to represent some topics of the sets of their associations with more general topics on taxonomy, discovered unbalance in taxonomy;
- Offered methods of building tree and n-clusters (as a result of propagation of BOX bi-clusterization on ternary relations) and tested their experimental verification, demonstrating the stability of the methods to noise;
- Analyzed the change of the influence in IMF as a result of the 2008 reform and demonstrated that even though the change in relative influence corresponds to the aims set forth in the IFM reform provisions, in absolute terms this influence redistribution is negligible.
To conclude, these tasks have been solved with methods and models of social choice theory, multi-criteria decision theory, game theory, linear programming theory, and econometrics.
Despite many impressive achievements in decision making theory over the last 30 years, the problems considered in the context of this research are far from being thoroughly covered, and work on solving them is an attempt to learn more about possible ways to make a decision, which is a fundamental cognitive task.a