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2The problem of correlation of the institutional order, the institutional environment and the institutional structure can be solved in the framework of complementarity - the system integrity, not conflicting to each other rules and incentives of activities. It was found that the complementarity of the institutional order, the institutional environment and the institutional structure provides the focal effect of subjects and objects of the regional services market.
In the summary, we note that the regional services market as the economic institution can be characterized by: the multilevelty (includes the rules of federal, regional and municipal levels); duality (derived from formal and informal rules, institutional dichotomy); fragmentation (inability to provide frontal blocking undesirable type of behavior, which manifests itself in the circumvention of the formal institutional framework).
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THE SUFFICIENT CONDITION FOR THE SOLVABILITY OF THE ELEMENTARY ZERO-SUM MATRIX GAMES (NX N)
ABSTRACT. THE AUTHOR INTRODUCES THE CONCEPT OF STABILIZING STRATEGIES IN THE MATRIX GAMES (NXN); DESCRIBES THE CONDITIONS OF THEIR EXISTENCE AND OPTIMALITY; GIVES THE EXAMPLES AND SUGGESTS THE NEW INTERPRETATION OF MATRIX GAMES SOLVING AS THE SEARCH OF STABILIZING STRATEGIES.
KEYWORDS: ANTAGONISTIC MATRIX GAMES (NXN), ELEMENTARY SOLVABILITY, VON NEUMANN THEOREM, STABILIZING STRATEGIES, STABLE WIN (LOSS) OF THE PLAYER.
OLEG KOLTUNIVSKIYI
CANDIDATE OF PHYSICAL AND MATHEMATICAL SCIENCES, HEAD OF THE CHAIR OF NATURAL SCIENCES, YUZHNO-SAKHALINSK INSTITUTE OF ECONOMICS, LAW AND COMPUTER SCIENCES
In this work, the payoff matrix of the game is the square matrix of n order.
The mixed strategies X and Y of the first and second players, as usual, will be called a row vector of size (1xn) and the column vector (nx1) correspondently, consisting of the probabilities of players pure strategies.
Definition 1. Strategies Х- or Y+ are called stabilized, if exist the numbers v- u v+, which fulfil the conditions X-AY = v- or XAY+= v+ for any strategy.
The meaning of stabilization: strategies X- or Y+ provide the constant win or loss for the first or second player, independently from the chosen strategy.
Lemma 1. If the meanings v- u v+ are equal, then they are equal to the cost of the game v:
v- = v = v+
It should be reminded, that the well-known von Neumann theorem about minimax said that there are the optimal balanced players strategies X* u Y* , such as
max min XAY= min max XAY= v=X*AY* X Y Y X
Carrying out the conditions of Lemma 1, the strategies X* u Y* can be strategies X- u Y+ correspondingly.
It is obvious, that
v- < v < v+
Defining the main results in two new matrixes from the matrix A the next way:
Subtract the first column from each previous one (elementwise). Number 1 should replace the elements of the last column. It will be the matrix B.
The analogous transformations should be held within the last and previous strings of the matrix A. It will be matrix C.
The next theorems are justly:
Theorem 1. If the algebraic adjuncts Bin (i=1,..., n) of the last column of the matrix B elements are all non-positive or non-negative and are not equal to 0, then the stabilizing strategy X of the first player and his stable win v exist and are determined by the equations:
dete det B
Theorem 2. If the algebraic adjuncts Cnj(j=1,... ,n) of the last column of the matrix B elements are all non-positive or non-negative and are not equal to 0, then the stabilizing strategy Y+ of the first player and his stable win v+ exist and are determined by the equations:
Theorem 3. If the matrixes BmC fulfil the conditions of the theorems 1 and 2 correspondingly, than
1) The determinants are equal
dete = v Bm H detc = vc„,
2) One of the optimal strategies X* m Y* in the antagonistic game with the playoff matrix A are described in the theorems 1 u 2 stabilized strategies of the players X- u Y+ correspondingly:
X*=X-, Y*=Y+
And the game cost v is equal to the determinants ratio
_deM_detA V ~ detB ~ detC "
Note 1. The theorems 1-3 are reversible in the condition that one or both players have the stabilizing strategies and the correspondingly algebraic adjuncts of the matrixes B and (or) C are all non-positive, or non-negative.
Note 2. From the theorem 3 follows that the same sign of the matrix elements adjuncts B and C involves the activity (usage) of each players strategy in the optimal solution (X*, Y*).
The reverse is not correct - in the common case the activity of each strategy in one (!) of the optimal solutions does not lead to the positivity or negativity of all the adjuncts. It is enough to see the matrixes with the same (!), when all the elements are the same.
The examples prove the written above.
Example 1. It is known [1, 69], that for the playoff square matrix A, satisfying the Minkovskiyi -Leontiev condition, all the players strategies will be active.
Example 2 shows that for the game with the playoff matrix A, satisfying the theorem 3 condition, not all players strategies should be active. (a - any number)
Example 3 shows that for the game with the dominating (strict) strategy for one player the combined strategy (activating the dominating one), making the constant win or loss of the player, can exist.
Example 4 shows that for the game with the probability of consistent exclusion of the dominated players strategies the stabilizing strategies for each player in the initial game (!) exist.
1 2
2 1 1
0 2 1
. X"=X = (1/2;1/2;0), Y"T = (1/2; 0,1/2), v= 3/2
In the next examples the playoff matrix А has the saddle point. The stabilizing players strategies can be from zero to two.
Example 5. Two stabilizing strategies. (see also ex. 2)
Example 6. Only one player has the stabilizing strategy.
Example 7. Players do not have stabilizing strategies.
The author's attention to this problem was attracted by the problem 280 [2, a 184], when the elementary methods were viewed at the game theory lessons in some Sakhalin universities.
The other elementary algebraic methods of solving the antagonist matrix games ^n) are described in [3,o 39-52].
The author does not refer to the main definitions, notions and theorems, obviously known to each specialist.
The author suggests the following way of antagonist games solvation: non-active players strategies can be always chosen the way, that the playoff matrix A of free size (mхn) will be reduced to the square matrix А* of the size (n*хn*) with the properties:
1) The games costs are equal : v = v
2) The optimal players strategies in the game with the matrix A* are stabilizing, when uses the similar strategy and reaches the goal, at the same time he does not pay attention player behavior. Isn't it the antagonism?
Finally, we should note, that in the game (2x2) without the saddle point the stabilizing players behavior is at the same time minimax and maximin (!). So, the usually determined as "the best behavior in the worst situation" is at the same time "the worst behavior in the best situation"!
nen each player ntion to another
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1. Carlin S. Mathematical methods and theory in games, programming and economics. - New York: Wiley, 1964.
2. Kalikhman IR Collection of problems in mathematical programming. - M.: Vyssh.shkola, 1975.
3. Kolobashkin LV Fundamentals of the theory of games: a tutorial. - M.: BINOM. Knowledge Laboratory, 2011.
MATHEMATICAL MODELING IN PSYCHOLOGICAL RESEARCHES
ABSTRACT. THE AUTHOR CONSIDERS THE NATURE OF MATHEMATICAL MODELING AND ITS SIGNIFICANCE IN PSYCHOLOGICAL RESEARCHES. THE AUTHOR DISTINGUISHES THE TYPES OF MATHEMATICAL MODELS: DETERMINISTIC, STOCHASTIC MODELS AND SYNERGETIC MODELS. THE SYSTEM APPROACH IS PROPOSED AS AN INSTRUMENT OF IMPLEMENTATION OF MATHEMATICAL MODELLING IN PSYCHOLOGICAL RESEARCH.
KEYWORDS: MATHEMATICAL MODELING, PSYCHOLOGICAL RESEARCH, MATHEMATICAL MODELS, SYSTEM APPROACH
ALEKSANDRA ZELKO
CANDIDATE OF PEDAGOGIC SCIENCES, ASSOCIATE PROFESSOR AT THE CHAIR OF PEDAGOGICS AND EDUCATIONAL TECHNOLOGIES, IMMANUEL KANT BALTIC FEDERAL UNIVERSITY
Mathematical modeling is widely used in science and practice, because it allows to capture any structural changes in the system and to reflect them in the quantitative form. Modeling is necessary for analyzing the system effectiveness, forecasting and designing the system development.
However, in the process of psychological research, it was not found an adequate use for modeling, despite the fact that the essence of the process of cognition is inextricably linked with modeling: the study is based on the construction of the studying object image, settling its basic properties and relationships. One of the reasons of such position of modeling lies in the idea of methods of psychological research classification. In the classification proposed by B.G. Ananiev [1], modeling methods are empirical (empirical methods of obtaining scientific data and education facts) together with the methods of observation, experimental and biographical.
It should be noted, that B.G. Ananiev's classification is based on the need for a working methods classification, which "would be consistent with the order of operations in scientific research, determining the integrity of the cycle of modern psychological research" [1, c. 205].
We believe that consistently pursuing the idea of methods of psychological research classification, it is advisable to allocate the modeling methods (mathematical, cybernetic, simulation) in the separate class, because modeling takes the special place in the system of psychological research methods. We recognize that the subject of psychology is the processes of subjective reflection of objective reality, which are necessary for the regulation of behavior and activity. Actually, in observation, experiment and other empirical methods of psychological analysis, the object of the research is the behavior of subject's mental activity and its products. In this context, the adequate method of psychological research must a priori be the hypotheses about the mechanisms of subjective reflection and subsequent testing of these hypotheses. The most accurate method of bringing up and testing the hypotheses is to formulate the hypotheses about the mechanism of the studying phenomenon in the form of the model, and then the model observation with the behavior, recorded in the experiment with the real subject of mental activity. So, the phase of application the organizational, empirical and data processing methods corresponds to the certain researches, preceding the modeling stage.
V.Y. Krylov developed the general scheme of theoretical and experimental research, including the construction of the mathematical model of the studying phenomenon [5]. The highlight of the general scheme is to build the standard, and then the narrative (descriptive) model of the studying phenomenon. The comparison of V.Y. Krylov's scheme and the content of the cognitive procedures
