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­­­Axiomatic aspects of fair division problems

Priority areas of development: mathematics
The project has been carried out as part of the HSE Program of Fundamental Studies.

Goal of research:

To use a combination of modern approaches (axiomatic approach, algorithmic analysis, strategic analysis, computer simulation) for a comprehensive study of game-theoretic problems and economic mechanism design problems.


The methodology involved a combination of algorithmic and axiomatic approaches to the analysis of the problems. Under the methodology, the methods of strategic and cooperative game theory, mechanism design, microeconomics (general equilibrium theory), probability theory, information theory, stochastic optimization, differential geometry, convex geometry and convex optimization, tropical geometry and finite automata theory were used.

Empirical base of research:

Data on requests of quotations that are publicly available at URL: ftp: // free: free@ftp.zakupki.gov.ru/, and data provided by the authors of the site SPLIDDIT.ORG (http://www.spliddit.org/) were used.

Results of research:

The competitive mechanisms of the fair allocation of the mixture of goods and bads were considered. For example, when a partnership is dissolved, it is necessary to divide not only the total assets, but also the general obligations (debts or credits), or when assigning tasks (bads) between employees, managers can add incentives (goods). An analogue of the Eisenberg-Gale theorem was proved in the minimal assumptions about the utility of agents.

Axiomatic characterization of the competitive mechanism in mixed problems by applying partial non-manipulability, efficiency, symmetry and solidarity, as well as axiomatization by applying Suppes-Sen’s relation of dominance in the utility space were obtained.

An envy-free algorithm for rent division in the case when the capacity of the room is more than one was proposed.

A probabilistic model of agent preferences in problems of fair allocation with additive utility functions was constructed, the model was calibrated according to data from the site of Spliddit.org.

In a discrete problem of guaranteed search on graphs, mixed and connected mixed search, their monotonicity, connectivity, and the difference in their search numbers were investigated.

For antagonistic games with incomplete information for both parties, a public encrypted signal and various computing resources of the players, a class of random binary signals with a fixed length of outgoing rows was considered. We found the size of the automata available to the player, which allowed us to know the state of the game for any signal from the considered class.

A new approach to finding a solution for a cooperative differential game in conditions, when players do not have complete information about the game (motion equation, payoff function) over the entire time interval of the game, was proposed.

The supervised machine learning algorithms, that detect suspicious behavior in public procurement auctions, was constructed. It was found that the share of suspicious auctions was about 7-16%.

The mechanism of decision-making in the presence of a conflict of interests and incompleteness of information in case of the doctor-patient relationship was investigated.

In the sandpile model of self-organized criticality, the earlier experimentally predicted existence of solitons, triads, and nodes was proved.


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