MODELING OF THE DECISIONMAKING IN A PAIR INTERACTION
Tatiana N. Savchenko
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.
Math
psychology
Artificial
intelligence
Psychophysics
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
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:
•
  aij
B  bij
, 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.
• 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
i
f  ,    h1 m1  ,    h2 m2  ,  
• i = 1,2 – player number
• hji- choice by the j-th player of the i-th strategy
 f a ( ,  )  h1 m ( ,  )  h2 mb ( ,  )

 f b ( ,  )  g1 mb ( ,  )  g 2 mb ( ,  )
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)
max min f1  ,  


max max f1  ,  


min max f1  ,  


min min f1  ,  


For two games of strategy maximum guaranteed
payoff Dj.fon Neumann
• 1-th player chooses a criterion
max min f1  ,  
•
•

and 2th
–

max max f1  ,  


• Thus, our definition of the game solution is a
generalization of the concept of game solution
suggested by J.fon Neumann
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
A
2
1
1
1
B
1
1
1
2
Areas of the expectation of winning the game «family
dispute»
α=0 β=0
2,5
a
2
c
1,5
1
α=1 β=1
p
h
0,5
Ряд1
0
-1,5
-1
b,d
0
-0,5
-0,5
α=1 β=0
α=0 β=1
-1
-1,5
0,5
1
1,5
2
2,5
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
Thank you for your attention