# Malafeyev Oleg Alexeyevich

D.Sc., Professor, the Head Modelling in Social and Economical Systems Room 335, , tel. (812) 428-42-47E-mail: malafeyevoa@mail.ru |

## Other

## Scientific publications

- Малафеев О.А. Дифференциальные игры с бесконечным числом игроков // Исследование операций. Гавана, 1986. том 7 № 2.

## General Information

Present Position: Professor and Chair (Socioeconomical systems Modeling Department, Faculty of Applied Mathematics and Control Processes, Sankt-Petersburg State University)

Address: Faculty of Applied Mathematics and Control Processes, Sankt-Petersburg State University , Universitetskij Prospect, 35, Petergof, Russia, 198504

Email: malafeyevoa@mail.ru

## Academic qualifications

Full Doctor in Mathematics and Physics from the Leningrad State University Graduation date: June 1989 , Ph.D in Mathematics from Leningrad State University Graduation date: June 1971 University Diploma in Mathematics (M.Sc-level) from Leningrad State University, Faculty of Mathematics and Mechanics Graduation date: June 1967

## Employment history

October 1967 – October 1970 Postgraduate at Leningrad State University

November 1970 - December 1971 Junior Researcher at Leningrad State University

January 1972 – April 1990 Associate Professor (Docent) at Leningrad State University

March 1990 Professor at the Department of Mathematical Statistics , Reliability theory and Queueing theory

June 1991- Professor and Chair (Socio-economical Systems Modeling Department, Faculty of Applied Mathematics and Control Processes, Sankt-Petersburg State University)

1977-1978 Visiting Professor at the University Oriente , Santiago de Cuba,

1980 Visiting Professor at the State University of Ossetia ,Ordjonikidze (Tskhinval)

1985 Visiting Professor at the State University of Jakutia, Jakutsk

1987 Visiting Professor at the Silesian Mathematical Institute in Katovice, Poland

1994 Visiting Professor at the University of Stockholm, Sweden

Professor at Saint-Petersburg State University of Architecture and Civil Engineering 2002-2004 (part-time position)

Chairman of the State Attestation Commission in Mathematics of the Pushkin Leningrad State University 1995-1997

Head of the Sub-program -“Mathematical Modeling of Socioeconomical Systems ” – of All-Russia Scientific Research Program “Peoples of Russia” 1990-1994

Head of The Research Department of Synergistic and Competitive Systems at Saint-Petersburg State University 1991-1994

Head of The Scientific Research Program “Rastitelnost” at The Applied Problems Department of The Academy of Sciences of USSR 1967-1971

Head of The Scientific Research Program “Janr” at The Applied Problems Department of The Academy of Sciences of USSR 1972-1976

Head of The Scientific Research Program “Nord” at The Applied Problems Department of The Academy of Sciences of USSR 1982-1987

## Teaching and administration

(a) (1970- ) (Leningrad ) Saint-Petersburg State University :

I developed and delivered the courses on High Algebra , Combinatorial Topology, Algebraic Topology, Generalized Dynamical Systems Theory, Optimization Theory, Control Processes of Conflict theory, Mathematical Modeling of Socioeconomical Systems, Operation Research and Game Theory, Dynamic Programming Theory, Differential Games Theory, Mathematical Economics, Many-agents Systems of Competition and Cooperation; also supervised many dozens students in Applied Mathematics, and supervised PhD Dissertations on Mathematical Modeling of Banks Activity, on Qualitative methods in Two-person Zero-sum Differential Games , on Numerical Methods in Two-person Zero-sum Differential Games.

## Research

My research interests lie in Operations Research , General Game theory , Differential Game theory , Control Theory, Mathematical Economics, Optimization theory with applications to economics, business, ﬁnance and some other areas.

**Main achievements and research interests areas**

As early as at years 1967-68 [ ]an axiomatic continuous formalization for conflict control systems in metric spaces was given by him that included the ones described by ordinary differential equations with constraints on phase coordinates and by partial differential equations etc. In this frame on the base of given formalization the existence of an equilibrium points was proved for general nonlinear competitive dynamical systems (two-person non-zero sum differential games) with independently operating and completely informed of the current process trajectory participants (players) with arbitrary continuous payoff function. It was proved that when one agent’s actions restrict potential abilities of another one an evolution of the system along equilibrium trajectory one can not guarantee without changing the process’s definition. However it was proved, that if an informational symmetry of the system is changed for the benefit of one participant informing him of current controls of his adversary in advance of arbitrarily small time interval then the existence of an equilibrium evolution of the system appears anew albeit with some loses for a player discriminated. One can easily see that players interests in socio-econimical processes are not absolutely opposite – at some periods agents purposes may coincide but sometimes these ones are almost antagonistic. So just after studying two person zero-sum dynamic processes a question of equilibrium existence for dynamical processes with many players have arisen. Rather natural approach to construct equilibrium profile exploiting punishment idea was realized in[] . The most unpleasant feature of this approach is that almost any trajectory of the process (which can guarantee maxmin of payoff) can be chosen as an equilibrium one. However it is desirable to prove existence of absolute equilibrium with a property of perfectness when every subgame has the same equilibrium with an appropriate initial point (in another words the equilibrium is invariant with regards to all sub-processes of the total process ). The method of lower and upper approximative multi-step games with complete information used previously for the case of two-person zero-sum games can not be applied here because agents payoffs are different in different equilibrium points. The problem was solved at 1970-th and the proof of absolute equilibrium existence for n-person non-cooperative dynamical games with prescribed finite time horizon, with complete information about the history of the process up to the current time of the play, continuous payoff functions of the players, defined on the trajectories of the process and separate dynamics was given. The result was reported at The Leningrad State University seminar (Prof. L.Petrosijan-head);Leningrad Institute of Socioeconomical Problems seminar,(Prof. N.Vorobijev – head) and published in []. It was proved also that the equilibrium exists when payoff functions are additive and players make their operative decisions using the current state of the process information only. At early nineties these results were extended to the case of conflict control processes with jumps , defined by differential equations with coefficients – measures []. Properties of competitive equilibrium for many-person conflict processes with infinite horizon were studied and characterized by theorems of turnpike type in [] . Above described axiomatic system of n-person conflict control in metric spaces was extended to the case of infinite number of players with a help of formalism and methods of multivalued measurable functions and in the frame of this axiomatic existence of competitive equilibrium was proved for the case of the infinite number of agents with an appropriate adaptation of the competitive equilibrium concept[]. Sufficient conditions for equilibrium in conflict control processes with many participants are deduced in [] which helps to construct equilibrium trajectories of the processes in some cases.

For two-person zero-sum differential games is proved under minimal conditions of strong continuity type supposed for payoff function that value function are weak solutions (in a sense defined in [])of HJ-equation of this game []. Solving this equation with a help of difference schemes makes it possible to get value function and equilibrium trajectory. Similar results are derived for non-zero sum differential games with finite and infinite continuum number of players where two particular cases of pure and mixed strategies are separately considered.

Stability properties of competitive Nash equilibrium with respect to changes of conflict dynamical systems parameters are studied in []. Necessity of study such kinds stability appears everywhere in natural and social sciences, technics, when solving simulating and constructing problems, and in numerical analysis areas.

With a help of general and differential topology formalism, approach and technics results on local and global robustness (structure stability with regard to model parameters)of competitive equilibria for the models of conflict with continuous and smooth profit functions that strengthen and extend known results by Wu Wen-tsun and Harsanyi.It is proved that the space of non-cooperative n-person game of given format can be subdivided by means of smooth hypersurfaces into a finite number of linearly connected components so that each game from these components has a finite number of equilibria smoothly depending on games from the component. Nash , Wald-Ky Fan, Gliksberg results on equilibrium existence are extended to the case of n-person non-cooperative normal form games with arbitrary strategy sets and almost periodical payoff functions.

Constructed and investigated: the dynamic model of the optimal distribution resources in the context of the conflict, the dynamic analogies of Cournot oligopoly models, model of smoothing of cyclical dynamics based on cooperative interaction economy, model of the development of the industry , consisting of competitive interacting firms, the model of evolutionary economics, ompetitive model of spatial capital allocation, spatial models of competitive pricing, model of capital market securities distribution, dynamic competitive models for option prices in the securities market, the model of optimal control securities portfolio, the optimization bank's business model under competition, the capital’s dynamics model for the securities market, a two-stage model of voting, where a large number of electors (experts) select the only alternative just in one step, thermodynamic model for the dynamics of the funds, a capital diffusion model with space competition between technologies and a number of other models.

These results were summarized in a more than 20 books, 2 of them were marked by**St. Petersburg State University 1 degree Prize ' “For Scientific Work” in 2012' (Book “Understanding Game Theory”)****St. Petersburg State University 1 degree Prize ' “For Scientific Work” in 2001' (Book “Conflict Processes of Control”)**

Some of results were obtained in the course of 3 projects granted by Russian foundation for basic research, 1 project granted by Russian Foundation for Humanities and 1 project granted by High Education Ministry