MODELING OF THE DECISION-MAKING IN A PAIR INTERACTION Tatiana N. Savchenko

Part 1:Mathematical psychology

The term "mathematical psychology" sounded in the report IF Herbart in 1822 "On the possibility and the need to apply the psychology of mathematics" (The mathematical model of the emergence of representations in the mind)

In 1850, his disciple M.I. Drobish published a book "fundamental principle of the doctrine of mathematical psychology.(An attempt to create a mathematical psychology as a theoretical science, came to the concept of probability distribution)

Development of methods of data analysis, development of psychological theory of measurement.

Criticism of the ideas of Herbart

Vvedensky - "mathematical psychology - a dream, for which there is to take even the unsuccessful“ (1966)

.Vladislavlev - raised the question of the extent of feeling (70- 80)

Groth - created a descriptive mathematical models of mental processes, anticipated the idea of the graph as a mathematical object, foresaw the idea mulmnozhestva (1882)

Rossolimo - suggested that "psychological profiles" - psychometric scales (1910)

Chelpanov - the basis of elementary statistical processing (1912)

In 1963 the U.S. began a textbook on mathematical psychology, it also became a magazine on mathematical psychology.

It began the revival of mathematical psychology in the United States has

•Mathematical psychology is a branch of theoretical psychology, used to construct theories and models of mathematical tools and axiomatic-deductive method.

•The object of mathematical psychology are natural systems that have mental properties, meaningful psychological theories and mathematical models of such systems.

•The subject of mathematical psychology is the development and application of the formal apparatus for the adequate modeling systems that have mental properties, and method - mathematical modeling.

Mathpsychology

Psychophysics

Artificialintelligence

Synergetics

The main stages in the development of mathematical psychology:-transition from the individual models and the laws(60-70) to the axioms, theories (80)

-that the most significant regulatory models in the 70 years, then they were supplemented refined

-intensive development of the theory of psychological measurement (80)

-mat.modelirovaniya rapid growth in Russia    (90) (Krylov, Pospelov, Kurdyumov, Malenetsky, Zhuravlev, Sukhodolskiy, Sokolov, Tarasov, Druzhinin, Izmailov, Petrenko, Artemyev, Satarov, Averkin, Drynkov, Savchenko, Golovina)

Synergetic approach to the modeling of psychological systems

•Psihosinergetica as a possible new paradigm of psychological science

•From psychology to synergetics

•The most important mathematical models developed by the laboratory staff of the Institute of Mathematical Psychology Institute:

• -collective behavior (Krylov, Golovin, Ostryakov)reflexive behaviors (wing)

• -learning, representing automatic reinforcement denumerable (Drynkov)

• - purposeful behavior (Korenev, Pridvorov)

• - behavior in conflict situations (Savchenko)

• - Knowledge Dynamics (Golovina)

Mathematical methods of data analysis of empirical research:-MS-in pseudo-space (Krylov, Ostryakov)-CA-analysis based on the theory of concept development Vygotstkogo (Krylov Ostryakov)-MS-fuzzy sets in Zadeh (Golovin)-hierarchical CA with the assessment of division into classes (Savchenko, Drynkov)-Joint-use spacecraft CA and MS (Drynkov, Savchenko)-LSA with the assessment of division into classes (Savchenko, Drynkov)-CA on fuzzy sets (Savchenko)

The modern apparatus of mental simulation systems

•Fuzzy sets

•Multisets

•Synergetic approach

•Quality Integration

•Neural networks

Part 2:Mathematical game theory and conflict studies

Conflict Studies •One of the directions in the Conflict

Studies uses mathematical game theory as a description apparatus.

•This is due to the fact that game theory is an integral body of mathematics capable of predicting, and because matrix representation of possible outcomes is a convenient tool to describe various types of social interaction

•The concept of "game" used in game theory is similar to the concept of "interaction situation", and the concept of "game with non-opposite interests” is close to the concept of "interaction situation with uncertainty".

• Let the game of two persons with non-zero sum (bimatrix game) is given by two matrices:

• , i = 1, 2, ..., т; j = 1, 2, ..., n

• • Let α = (α1, ..., αm) β = (β1, ..., βn) vectors mixed strategies,

respectively, the 1 st and 2 nd player.Let m1 = (α, β) and m2 = (α, β) the expectation of winning the 1 st and 2 nd players, if they use their mixed strategies α and β, respectively.Each player, choosing their own mixed strategy affects the winnings, which are the two players.

ija ijbB

• J. von Neumann proved that for a two-person non-zero sum games (with non-opposite interests) there are such mixed strategies that maximize the guaranteed gain of each player.

• It was shown by experiments with specified two-person non-zero sum games that players used different criteria which were often not optimum. Example: 1) maximize his winnings; 2) minimizing the winning partner; 3) maximization of winning partner; 4) maximizing the amount of gains for both partners; 5) maximizing the difference between a win and the winning partner etc.

• All these examples of criteria can be described in terms of maximizing or minimizing each player given a linear combination of wins with fixed coefficients of a linear combination

• •

• i = 1,2 – player number

• hji- choice by the j-th player of the i-th

strategy

,,, 2211 mhmhf ii

),(),(),(

),(),(),(

21

21

mgmgf

mhmhf

bbb

ba

Substituting the coefficients h1 h2 for the first player equal to 0, 1, -1, we obtain the function fa, equal to the expectation of either a win or win a partner, or the total payoff. Similarly, for the function g

Definition of solutions for the game of two persons with opposite interests

• Decision Games, in this case - a lot of mixed strategies of all players who meet the criteria selected by each player

• The criterion is defined as the maximization or minimization of a given linear combination of winnings

• Each player chooses his strategy independently

Examples of criteria used by any player (1)

,minmax 1f

,maxmax 1f

,maxmin 1f

,minmin 1f

For two games of strategy maximum guaranteed payoff Dj.fon Neumann

•1-th player chooses a criterion

•

• and 2th –

• Thus, our definition of the game solution is a generalization of the concept of game solution suggested by J.fon Neumann

,minmax 1f

,maxmax 1f

Proposed method

• Depending on the individual characteristics and situation each participant forms a criterion of his behavior in the proposed situation.

Knowledge about the criteria and the game matrix determines thу solution

If the sets α and β have no intersections, then the solution does not exist.

The structure of the paired experiment

-Learning from a legend game-Objective - to collect the maximum number of points-Selection procedure: together with a partner or individually (the cooperative strategy is chosen, only with the consent of both parties)-The negotiation process-Commit acts-Getting wins and the next move-The end of the game occurred after a stroke or in the case of the desire of one participant to interrupt the game.-The experiment was conducted in two groups: an informal familiar and unfamiliar

An example of an experiment the pair playing a "family dispute"

• Game a family dispute - this bimatrix game with matrices Wins

11

12

A21

11

B

Areas of the expectation of winning the game «family dispute»

-1,5

-1

-0,5

0

0,5

1

1,5

2

2,5

-1,5 -1 -0,5 0 0,5 1 1,5 2 2,5

Ряд1

α=1 β=0

α=1 β=1

α=0 β=1

α=0 β=0

a

c

ph

b,d

Analysis procedure. Results

-Formally, the criteria determined by the results of bargaining (the frequencies of selection strategies for participants)-Moments of change - as a result of bargaining (most often proposed pair of strategies)-The content analysis of bargaining to determine additional goals, which put in front of him participating, the strategy of their behavior, as well as moments of change strategies and objectives of Conduct-Analysis of the protocols also allows for a set of solutions to provide one, corresponding to the chosen strategy.

-A theoretical study of the expectation of winning depends on a variety of mixed strategies-Possible criteria for decisions in a game of two persons with non-zero sum-For a number of strategies to obtain necessary and sufficient conditions for existence of the solution of the game-The experiment game of two persons with non-zero sum games for example "family dispute" and "prisoner dilemma"-A number of criteria used by the players-The inverse problem: the results of bargaining strategies are determined and, accordingly, possible criteria for selection of solutions and existence of solutions games-A study of the negotiating process when choosing a joint decision

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