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15MATHEMATICAL SCIENCES
SOLUTION OF BIMATRIX GAMES IN PREFERRED MIXED STRATEGIES
Beltadze G.
Doctor of Physical and Mathematical Sciences, Ph.D., Professor emeritus, Department Artificial
Intelligence Georgian Technical University, 0175 Tbilisi, Georgia https://doi.org/10.5281/zenodo.6594502
Abstract
In the article the task of finding the most preferred mixed strategies in finite scalar m x n bimatrix T(A, B) game is studied. The equilibrium situation in the mixed strategies always exists in T(A, B) game according to the Nash theorem. The problem of finding an equilibrium in the T( A, B) game has a long history, but due to the complexity of the known algorithms and methods, that cause various problems, its study is being continued today. Our approach is different from these methods. In T(A, B) game each player primarily has his own interest in order to do so he can act different principles in addition to the principle of Nash equilibrium. Therefore we consider such a principle in mixed strategies. To apply such a principle, the player uses Adam Smith's principle of optimality, which does not take into account the interests of the partner and acts to achieve the best result so that it is the best for him.To achieve such a result, the player in the game T( A, B) ranks his pure strategies according to the advantages, for which he finds the weights of the strategies. The weight of the strategy corresponds to the probability of its choice. For such a ranking of the multiplicity of player strategies, we use Thomas Saaty's simple method of analytical hierarchy. We obtain the weight vector of the pure strategies as the preferred mixed strategy. Relevant examples are given.
Keywords: Bimatrix game, Nash Equilibrium, Preferred, Mixed strategy, Weight vector.
1. INTRODUCTION
Game theory is divided into two parts: one is a noncoalitive (as the same as noncooperative, strategic) game theory, and the second is a cooperative game theory [1;2]. Such division is based on the premise that the main unit of a noncooperative game's analysis is a rational individual participant, who tries clearly, with defined rules and possibilities to get maximal utility (payoff) from the game independently. If individuals use such actions, that can be named as "cooperation" in the ordinary sense of the word, then this is because, that such cooperative behavior is in the interest of all individuals: each avoids cooperative breach.
Strategic theory is strategicaly oriented. Therefore, according to this approach, the players' result depend on their abilities in the game. But the cooperative approach is concerned with the multitude of possible outcomes, not how they can be achieved. Noncooperative theory is a pecular microtheory that provides a detailed description of what is happening in the process of the game. Thus, cooperative theory is a macrotheory compared to noncooperative, the basis of which is the theory of noncooperative games. At the same time, game theory deals with the modelling of socioeconomic processes, and is oriented toward socioeconomic applications. Consequently, the methodological and metaphysical aspect is even more essential for game theory than for other branches of mathematics, and demands move careful consideration [3].
A strategic game has two forms of presence. One is a positional form, another one is normal form. We will discuss the game in its normal form. There is no dynamics in them; each player makes only one decision (makes one move), and all players make decisions
simultaneously and independently from each other, however none of them knows what decisions have made or will make their partners. Therefore, such game is static game, where the player's strategy and move are the same. The difference can only arise in dinamic (including positional) games. In general, a player's strategy in game theory is to plan his action throughout the game, taking into account all the information recieved.
Definition 1.1. A normal (or strategic) form of
a noncooperative game is called triple (model)
T=< N ,{St} ie N' {H} ie N >, (1)
where N = {l,...,n} is a set of players; S is
i e N player's set of pure strategies (moves) and H : S = ^ S ^ R is i e N player's function
ieN
of payoff (utility function). This function to every s = (s,. ., s„) set of the players strategies that is called the game's result, i.e. situation, i.e. profile, matches this player's payoff (utility) - Hi (s), i e N. If there are two players participate in the (1) game, i.e. the set of players' N = {l,2}, and their sets of strategies are finite and they respectively are S = {1,. .,m} and S2 = {1,...,n}, there exists at
least one situation (i, j) e S x S 2, where the payoffs sum of players' is not zero H (i, j) + H (i, j) ^ 0, we get two players strategy game, called m x n bimatrix game. Therefore, in bimatrix game the players' interest may be diametrically opposed, as well their interest may
partially coincide. Designate the players' functions of
payoff H = A = (a ), H = B = (b ) and from
(1) thus received bimatrix game let be T(A, B) (or
(A, B) bimatrix game), that will be written payoff's
seperately A and B matrices, or by one matrix that is composed by the pairs of players' payoffs:
f ain ^ [¿11 b12 ■ - ¿in ^
A = a21 a22 a2n , B = b21 b22 ■ ■ bln ; (2)
V^ml am2 a mn / Vml bm2 ■ - bmnJ
In (2) payoff's A and B matrices may be numeral or other elements of nature (for example vectorial). In the first case we have scalar bimatrix game. Both types of games are very topical with their theoretical and practical values. There are many scientific papers dedicated to solving their studies nowadays, but there are a lot of problems in such games and it is imposible to complete them. In this article we will only discuss the problems of solving the first types of games with some point of view.
Consider m x n bimatrix game T(A, B) with A
and B matrices of scalar payoff. It will have following form
1 2 n
l (aiv b11) ^ b12) KP bm)
(A, B) = 2 (a2^ b21) (a22, b22) Kp b2n)
m (amV bmi) ^ bm 2) (a , b ) mn mn
(3).
In the strategic (1) game and therefore in (3) bimatrix game in the role of solution Nash equilibrium (or shortly equilibrium) situation is considered. Define it in (3) game in pure and in mixed strategies.
Definition 1.2. The situation (i , j ) in (3) game is called an equilibrium (or solution in the pure strategies), if the following inequalities are fulfilled
a , , > a , Vi = 1,..., m ; b,, > b, ,Vj = 1,...,n. .(4)
i j v i j i j J
The equilibrium situation in the (3) game may not exist in pure strategies and the equilibrium situation in the mixed strategies always exists according to the Nash theorem. Define an equilibrium situation in mixed strategies.
Like matrix games, in the given (3) T(A, B) game let's note the first and the second player's mixed strategies note respectively P = (p,..., pm )T and
0 = (qi,.., qn )T.
In (P, Q) situation the player's mixed payoffs (average payoffs, expected payoffs, expected utilities) respectively are equal
v(A) = A(P, Q) = YL a^yj = PTAQ,
i=i j=i m n
v(B) = B(P,Q) = LLbjXiy] = PTBQ. (5)
i=1 j=1
Definition 1.3. In r(A, B) game, the situation
(P* , Q* ) is called an equilibrium in mixed strategies (or solution in the mixed strategies), if for
VP, VQ strategies the following inequalities are fulfilled:
P*TAQ* > PTAQP*TBQ* > P*TBQ. (6)
Also, Nash theorem asserts that in any bimatrix game there exists at least one equilibrium situation in mixed strategies, does not give the ways of finding the equilibrium situation in mixed strategies. This is a separate problem for bimatrix games, that is solved by different algorithms - Vorob'ev [4], Kuhne [5] and Mangasarian [6]. For n player's case there are algorithms for Lemke - Howson [7] and Rosenmuller [8] for noncoalition (1) games. All of these algorithms are quite complex and can not be used by students. It is relatively easy to solve a bimatrix 2 x 2 game, for it a graphical method is used. This requires elmentary actions. 2 x 2 bimatrix games simulate many simple social - political situations. In particular, their usage have been studied in a teaching organization [9,10}. A lot of needs however, require the resolution of more dimensional games, for which algorithm K. Lemke [11] is formed. In addition to the listed algorithms, different methods and algorithms are used to find Nash equilibrium in bimatrix games [12;13].
All of the listed algorithm need to solve bimatrix games contain a very complex mathematical apparatus and is very difficult to use. So we tried to discuss different approaches to solve this problem.
Firstly, note that in the strategic game each player's task is to make a prediction other players' behavior. The player discusses which strategy not to use. So we have to find some way of compearing of two strategies. Obviously, none of the players will choose a strategy, if another strategy brings him more payoff. The simpliest and the most natural principle to compare strategies with, is the principle of dominance. It is explored in the article [14]. In the same article we olso studied other methods of preferred pure strategies.
Note, that in T(A, B) game each player primarily has his own interest in order to do so he can act different principles in addition to Nash equilibrium principle. Therefore let's discuss some of this kind of principles in preferred mixed strategies.
2. FINDING the PREFERRED MIXED STRATEGY by HIERARCHICAL ANALYSIS METHOD
Consider m x n bimatrix game T(A, B) with a A and B matrices of payoffs
A =
a2l a22
a ,
V ml
m2
B =
f bu bu ... bln Ï b21 b22 ... b2n
mn J
V bm1
b
b
. (7)
mn J
Assume that the goal of the 1st player in a given game is to win as many as possible regardless of the interests of the 2nd player. This means that he does not use the principle of Nash equilibrium and therefore uses the principle of Adam Smith's optimality, which does not take into account the interests of the partner and according to which "to act for the best result so that it is best for him". To do this, of course, he will not use his dominated pure strategies and will free the profit matrix from such [14]. The player will then try to rank the number of his strategies according to the advantages. The order of such advantages can be determined through certain weights of strategies. Match the weight of the strategy to the probability of its choice. This is how we get the weight vector of clear strategies, which we consider to be the player's preferred mixed strategy. At the same time it will be a fully mixed strategy.
To rank the multiplicity of player strategies in this way, use the Thomas Saaty's analytical hierarchy method [15].
Let's start the game T(A, B) by solving a set of strategies in the game for a set of 1 player strategies ^ = {1,..., m}. Use the method of analytical hierarchy to pairwise compare
(i, j),i = 1,...,m; j = 1,...,n situations on the set of A matrix elements. Here are the qualitative advantages: 1) equally superior, 2) weakly superior, 3) strongly superior, 4) very strongly superior, and 5) absolutely superior. In the role of the corresponding quantitative scale, consider a transit scale with an advantage a = 2 coefficient. Then the mentioned quality scale corresponds to the corresponding quantitative scale: 1) 1, 2) 2, 3) 4, 4) 8, 5) 16.
As it is known, using the method of analytical hierarchy is quite time consuming. We therefore use a simple method of calculating the weight vector [16], according to which: the player determines the elements of the first line of the pairwise comparison (or estimation) SA matrix, for which he compares the situation
(1,1) with the situations
(1,1), (1,2),..., (1, n);(2,1),...,(2, n);...,(m,1),...,(m, n), whose number is m • n. In comparison with other situations (i, j),i = 1,..., m; j = 1,..., n of the situation
(1,1) we mean how much profit is superior an to other a^, i = 1,..., m; j = 1,..., n gains on the mentioned scale. One of the difficulties in using this method is how much the point a considered as the starting point of
y
the calculation of profits (benefits) compared to the numbers taken on the number axis is superior to each other separately.
Axiom. The weight of the player strategy (probability of selection) is equal to the sum of the weights of the situations corresponding to this strategy:
P, = J P(i, j X i = ^^
m,
j=1
P is the weight of the 1st player's pure strategy, p (i, j) is the weight of the situation (i, j).
Let's start by evaluating the situation (1,1) with advantage over other situations (1, j), j = 1,..., n; (2, j), j = 1,..., n; ...; (m, j), j = 1,..., n. Let's denote these rating accordingly as follows:
a11 = 1 a12,...,a1n ; a21,...,a2n ; ...; am\,...,amn . (8)
Evaluations a,,, j = 1,...,n are called situation
ij > .J
assessments of the 1st player's i (i = 1,..., m) strategy. Place these estimates in the first row of the rating matrix ^.
(8) calculate the following divisions in the order given for each i = 1,..., m :
a = amn/aj > 0, j = 1,...,n, (9)
where a° = , a°„ = 1. We place these val-
11 mn > mn *
ues even in the last column of the matrix ^. Fractions (9) are the division of the last amn evaluation of the first line of the evaluation ^ matrix with the evaluations of the same line au = 1, a12,...,aln; ...;
am1,..., amn :
aa
a
11
12
a
2
(1,1) (1,2)
(1,«); = (21
(2, n); (m,1)
(m, n)
(1,1) (1,2)... (1, n); (2,1)
(2, n);
1 = O
o
(m,1)...(m, n)
O™, ...o
o
mn 0
o
o
0;
1n ; 0 21
o
o
0
2 n; 0
m1
1 = o0
(10)
All numbers in the last column of the rating matrix SA are positive. Draw a vector from them P:
P = (a0 a0 • a0 a0 • • a0 a0 )
P (a11 ,...,a1n ; a21,...,a2n v"; am1,..., amn ) .
Let us normalize this vector and let's denote it
PN = (pi,..., pin; p221,..., P22n;...; pmi.....p! ).
The components of the weighted vector (p'a,...,p]n),i = 1,..., m create the weights of the situations corresponding to the first player's i strategy. According to the axiom, for each i find the sums
n
** \ 1 i
p* = L p, i = 1,..., m, that represent the weight of
j=1
the pure i strategy or the probability of choosing it.
* * *
Conclusion 1. p*,p*,. .,pm weight will rank for the advantages of pure strategies i = 1,..., m -more weight corresponds to the preferred pure strategy.
_ * / * * * \T
Thus the vector P = (p*, p2,..., p* ) obtained from the vector PN represents the preferred mixed strategy of the 1st player in the bimatrix game (7) (or the preferred strategy according to A. Smith). The preferred mixed strategy of the preferred mixed strategy of the 2nd player in the given game is denote by
Q* = ^ q*).
We will discuss examples below and compare preferred mixed strategies with equilibrium mixed strategies in terms of profitability and sustainability: We will look at examples below and compare it to equilibrium mixed strategies in terms of gain and
sustainability: Will a player win more by using a preferred mixed strategy if his partner can not predict and use a balanced mixed strategy? In addition to the above, a variety of content questions can also be asked here. We will try to answer some of them. We note, however, that the introduction of a predominant mixed strategy into a strategy game is a problematic issue, and it requires a deep analysis.
3. EXAMPLES
Example 1. Find the preferred mixed strategies in r(A , B) bimatrix game
f 2 3 > f 4 6 ^
A = , B =
v 1 V 6 0,
Solution. In the present game there is no equilibrium situation in pure strategies. Here is the only equilibrium situation in mixed strategies
(X *, Y *) = ((0,75;0,25)T ,(0,5;0,5)T ) and there
are player wins v(A) = 2,5; v(B) = 4,5.
To find the preferred mixed strategy P = (pj, p *) of the 1st player, compare the number an = 2 with the other elements of the matrix A on the above-mentioned quantitative scale of advantages. To do this, express such advantages in the following table of advantages (table 1). According to this table, the advantage 2 ^ 1 is weak and it is rated by 2. Because 3 ^ 2 it is weakly valued at 2, so 2 ^ 3 it is valued
at
. Also, 4 ^ 2 strongly or by 4, , so 2 ^ 4 it
is evaluated by Advantages.
'4.
Table 1
2 ^ 1 3 4
situation (i, j) (2,1) (1,2) (2,2)
rating 2 1/ /2 1/ /4
1 n ; o 21
Find the rating S^ matrix:
(1,1) (1,2) (2,1) (2,2)
(1,1) 1 1 2 2 1 4 1 2
(1,2) 1
(2,1) 1 1 8 1
(2,2)
(11)
From here P = (1
15
^X'1) i
4'/2'/8
its components and will be the normalized weight
vector PN = get
P* = (2/ 3/Y
P V5VV .
Now find the preferred mixed strategy Q* of the 2nd player, for which we consider the winning matrix
15'%5;X5'/15i From here we
1st player preferred mixed strategy
BT =
( 4
v6 0y
Consider the following table of benefits here and compile a rating matrix:
4 ^ 0 6
situation (i, j) (2,2) (1,2),(2,1)
rating 4 0,5
(1,1) Sb = (1,2); (2,1) (2,2)
(1,1) (1,2); (2,1) (2,2)
1
0,5 1
0,5
4 8 8 1
From here Q = (4,8;8,1), By which normalization we will get the vector
QN =(y21,y21y21,y21), from which the 2nd
player will be the preferred mixed strategy
Q* = (4/ 3/Y
Suppose the 1st player uses a superior mixed while the 2nd uses a
strategy P* = ,
balanced mixed Y = ( ^,/'2
strategy. Then the
payoffs of the players in the situation (P *, Y *) will
be V!(P*,Y*) = 3,1; v2 (P*,Y*) = 3,8. Hence, the winnings of the 1st increased (were 2,5 ) compared to the equilibrium situation of Nash, and the winnings of the 2nd decreased (were 4,5).
What is the maximum strategy of the 1st player? There is a saddle point (1,1) in the matrix payoff A
is the sum of
and the corresponding payoff 2. In a bimatrix game, payoff 1 by Nash equilibrium is greater than, that, as we have seen, the payoff obtained by the preferred mixed strategy are greater than all the others.
The advantage of players in a situation of mixed
strategies (P*,Q*) is the payoffs of the players
M P *, Q *) = P *T AQ * = 2,34, v2 (P*, Q*) = P*TBQ * = 4.
Thus, if both players use the preferred mixed strategy, then both winnings are less than they were in the equilibrium situation, ie. it is better for both of them to use balancing strategies
Note 1. It is easy to check that the situation
(P* ,Q* ) does not satisfy the Nash equilibrium conditions.
Now assume that the 2nd player uses the preferred mixed strategy and the 1st the balanced one. Then
vx(X *, Q *) = 2,32, v2( X *, Q*) = 4,57, e.g. compared to the equilibrium situation, the gain of 1 decreased slightly and the gain of 2 increased slightly.
Based on the answers received, it is the best for the player to use a preferred mixed strategy if his partner is unable to guess this decision.
As we have seen so far, there is some difficulty in compiling a table of such priorities, the rest being easily calculated by the other established method.
Example 2. A bimatrix game of "Prisoners' Dilemma" is given
A =
-1 -10^
v 0 -8 y
B =
-1 0 ^
v-10 - 8y
Here Nash equilibrium is a situation (2,2), in which the players payoffs is equal to (-8), while in mixed strategies there is no equilibrium situation. Find the players' preferred mixed strategies.
Consider the table of advantages of the 1st player as follows:
-1 ^ -10 -8 0
situation (i, j) (1,2) (2,2) (2,1)
rating 16 8 0,5
We get it P" = (%, 3/5 J-If
-1 0 ^
BT =
v-10 -8y
we discuss the table of advantages for the matrix with the same scale we used for the A matrix elements:
-1 ^ -10 -8 0
situation (i, j ) (2,1) (2,2) (1,2)
rating 16 8 0,5
Then the 2nd player's preferred mixed strategy will be the same as the 1st one Q = iT
1
The expected average winnings of the players in the received situation (P , Q ) is equal
V (P*, Q ) = v2 (P*,Q* ) = "5,9, to which is greater than the winnings obtained in the equilibrium situation. The situation (P , Q ) is therefore preferable to the Nash equilibrium situation for both players, indicating the advantage of using them.
Conclusion 2. The player's preferred mixed strategy is approaching to the Nash equilibrium mixed strategy.
Example 3. Find the preferred mixed strategies in a bimatrix game
f 6 0 2 > f 5 0 61
A = 0 4 3 ' B = 0 3 0
V 7 0 0 J V 1 2 3 J
11/ 4
/23'/ 23
J )
payoffs v(A) « 2,4, v(B) = 1,8.
Consider the table of 1 st player benefits as follows
6 ^ 0 2 3 4 7
situacion (i- j) (1,2),(2,1), (3,2),(3,3) (1,3) (2,3) (2,2), (3,1)
rating 16 8 4 2 0,5
We will find out through it P = (16,1,2;1,8,4;32,1,1) and the 1st player will be
the
preferred
mixed
strategy
Solution. Here we have one equilibrium situation in pure strategies (2,2) with payoffs (4,3) and one equilibrium situation in mixed strategies
For the 2nd player's payoff matrix '5 0 0 3
BT =
1 1 2
6 0 3
J
Suppose it is a table of advantages
5 ^ 0 1 2 3 6
Sit. (i- j ) (1,2),(2,1), (3,2) (1,3) (2,3) (2,2), (3,3) (3,1)
rating 16 8 4 2 0,5
From here
Q =
= (1/ 1/ 1/ • 1
2'/32'/16'/32'
i/ a V 1
1 /32'V
the payers' payoffs are v(A) = -0,5,
( p - Q )=((59'49/
(6/ 5
14'/14
and there players' payoffs are MP*,Q*) - -0,49, V2(P*,Q*) - 0,1.
The situation has a face in balanced mixed strategies
( * r ')T, (y^AoJ )
v( B) =
= 1
and QQ = j) In the situation
of the preferred mixed strategies, the players payoffs V(P*,Q*) - 1,65, v2(P*,Q*) - 2,7. Comparing the profit values, it can be seen that compared to the equilibrium situation in mixed strategies, the profit of 1 in this case decreased (was 2,4), and the profit of the 2nd increased (was 1,8). In both situations the player's payoffs are significantly less than in the equilibrium situation in pure strategies. So it is not advisable to use mixed strategies in a given game in order to make a high payoff.
Example 4. Solve a bimetric game in preferred mixed strategies
(-10 0 ^ ( 1 -10 ^ A = , B = .
I 0 "1 " V 10 1 " V
Solution. Without solving we will write the situation in the given game into the preferred mixed strategies
"3-
Here, too, the winnings of the players in both situations are close to each other.
Conclusion 3. If we consider that the situation of the preferred mixed strategies is a certain approximation to the equilibrium situation, then we can say that finding the preferred mixed strategies is incomparably easier to say compared to the Lemke algorithm.
4. CONCLUSION
The players preferred mixed strategies existence in a bimatrix game are explored. To determine them, the ranking of players' pure strategies with advantages over the winnings using a quantitative scale using a simple method of analytical hierarchy is considered. We find weights of clear strategies that correspond to the probabilities of their choice. The weighted vector of pure strategies thus obtained is the preferred mixed strategy. Examples are discussed that compare preferred mixed strategies with equilibrium mixed strategies in terms of profitability and sustainability. The situation in the preferred mixed strategies generally does not meet the Nash equilibrium conditions and such a situation represents an approximation of the Nash equilibrium situation. Finding the preferred mixed strategies in bimatrix games is very simple easier than any algorithm for solving it. Such an introduction of a predominant mixed strategy into a strategy game is a problematic issue that requires its continuation in the future.
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