fdocuments.in_index-introduction-of-game-theory-introduction-of-game-theory-significance.ppt

Game theory

1. INDEXINDEX
•Introduction of game theoryIntroduction of game theory
• Significance of game theorySignificance of game theory
•Essential features of game theory Essential features of game theory
• AssumptionsAssumptions
• Elements of game theoryElements of game theory
• Limitations of game theoryLimitations of game theory
• Types of strategy Of game theoryTypes of strategy Of game theory
•Methods for game theoryMethods for game theory

2. INTRODUCTIONINTRODUCTION
•Game theory was developed by Game theory was developed by Prof. John Von Prof. John Von
Neumann Neumann andand Oscar Morgenstern in 1928 Oscar Morgenstern in 1928
game theory is a body of knowledge that deals game theory is a body of knowledge that deals
with making decisions when two or more with making decisions when two or more
rational and intelligent opponents are involved rational and intelligent opponents are involved
under situations of conflict and competition. The under situations of conflict and competition. The
approach of game theory is to seek to approach of game theory is to seek to
determine a rival’s most profitable counter-determine a rival’s most profitable counter-
strategy to one’s own best moves. It helps in strategy to one’s own best moves. It helps in
determining the best course of action for a firm determining the best course of action for a firm
in view of the expected counter moves from the in view of the expected counter moves from the
competitors.competitors.

3. What is Game Theory?What is Game Theory?
Game theory is a type of decision theory Game theory is a type of decision theory
which is based on reasoning in which the which is based on reasoning in which the
choice of action is determined after choice of action is determined after
considering the possible alternatives considering the possible alternatives
available to the opponents playing the available to the opponents playing the
same game. The aim is to choose the best same game. The aim is to choose the best
course of action, because every player course of action, because every player
has got an alternative course of actionhas got an alternative course of action

4. Significance of game theory
1. Game theory is a kind of decision theory which is based on 1. Game theory is a kind of decision theory which is based on
the choice of action. And choice of action is determining after the choice of action. And choice of action is determining after
considering the possible alternatives available to the considering the possible alternatives available to the
opponent.opponent.
2. It involves the player decision i.e. decision makers who have 2. It involves the player decision i.e. decision makers who have
different goals and objectives.different goals and objectives.
3. The game theory determine the rules of rational behavior of 3. The game theory determine the rules of rational behavior of
these players in which the outcomes are dependent on the these players in which the outcomes are dependent on the
actions of the interdependent players.actions of the interdependent players.
4.In a game theory there are number of possible outcomes, 4.In a game theory there are number of possible outcomes,
with different values to the decision makers.with different values to the decision makers.
5.They might have some control but do not have the complete 5.They might have some control but do not have the complete
control over others.control over others.

5. Essential features of game Essential features of game
theorytheory
1.1.There is Finite number of competitors.There is Finite number of competitors.
2.2.The list of Finite number of possible course The list of Finite number of possible course
of action is available to each players.of action is available to each players.
3.3.Each player has the Knowledge of Each player has the Knowledge of
alternatives.alternatives.
4.4. Each player makes a choice, i.e., the game Each player makes a choice, i.e., the game
is played.is played.
5.5.The play is associated with an outcome The play is associated with an outcome
known as gain.known as gain.
6.6.The possible gain or loss of each player The possible gain or loss of each player
depends upon the choice of his opponent.depends upon the choice of his opponent.

6. AssumptionsAssumptions
•(1) Each decision maker has available to him two or (1) Each decision maker has available to him two or
more more well-specifiedwell-specified choices choices or sequences of or sequences of
choices.choices.
•(2) Every possible combination of plays available to (2) Every possible combination of plays available to
the players leads to a the players leads to a well-defined end-statewell-defined end-state (win, (win,
loss, or draw) that terminates the game. loss, or draw) that terminates the game.
•(3) A (3) A specified payoffspecified payoff for each player is associated for each player is associated
with each end-state. with each end-state.

7. •((4) Each decision maker has 4) Each decision maker has perfect perfect
knowledgeknowledge of the game and of his of the game and of his
opposition. opposition.
•(5) All decision makers are (5) All decision makers are rationalrational; ;
that is, each player, given two that is, each player, given two
alternatives, will select the one that alternatives, will select the one that
yields him the greater payoff. yields him the greater payoff.

8. The Two-Person, Zero-Sum The Two-Person, Zero-Sum
GameGame
•Two person zero sum game is the situation which Two person zero sum game is the situation which
involves two persons or players and gains made involves two persons or players and gains made
by one person is equals to the loss incurred by by one person is equals to the loss incurred by
the other.the other.
•For example there are two companies coca-cola For example there are two companies coca-cola
and Pepsi and are struggling for the larger share and Pepsi and are struggling for the larger share
in the market.in the market.
•Now any share of the market gained by the coca-Now any share of the market gained by the coca-
cola company must be lost share of Pepsi and cola company must be lost share of Pepsi and
therefore the sums of the gains and losses therefore the sums of the gains and losses
equals zero.equals zero.

9. n-persons gamen-persons game..
A game involving n persons is called a n person A game involving n persons is called a n person
game. In this two person game are most game game. In this two person game are most game
are most common. When there are more than are most common. When there are more than
two players in a game, obviously the complexity two players in a game, obviously the complexity
of the situation is increased.of the situation is increased.
Pay offsPay offs..
Outcomes of a game due to adopting the different Outcomes of a game due to adopting the different
courses of action by the competing players in the courses of action by the competing players in the
form of gains or losses for each of the players is form of gains or losses for each of the players is
known as pay offs.known as pay offs.

10. •Pay off matrixPay off matrix
In a game , the gains and losses, resulting from different In a game , the gains and losses, resulting from different
moves and counter moves, when represented in the form moves and counter moves, when represented in the form
of a matrix is known as pay off matrix or gain matrix . This of a matrix is known as pay off matrix or gain matrix . This
matrix shows how much payment is to be made or matrix shows how much payment is to be made or
received at the end of the game in case a particular received at the end of the game in case a particular
strategy is adopted by a player.strategy is adopted by a player.
Pay off matrix shows the gains and losses of one of the two Pay off matrix shows the gains and losses of one of the two
players, who is indicated on the left hand side of the players, who is indicated on the left hand side of the
matrix .matrix .
Negative entries in the matrix indicate losses.Negative entries in the matrix indicate losses.
This is generally prepared for the maximizing player.This is generally prepared for the maximizing player.
However’ the same matrix can be interpreted for the other However’ the same matrix can be interpreted for the other
players also.players also.

11. As in a zero sum game, the game of one player As in a zero sum game, the game of one player
represents the losses of the other player , and vice represents the losses of the other player , and vice
versa .versa .
Thus, the pay off matrix of Mr. A is the negative pay off Thus, the pay off matrix of Mr. A is the negative pay off
matrix for Mr. B.matrix for Mr. B.
The other player is known as the minimizing player. He is The other player is known as the minimizing player. He is
indicated on the top of the table.indicated on the top of the table.
•Decision of a game Decision of a game ::
In a game theory best strategy for each player is In a game theory best strategy for each player is
determined on the basis of some criteria . Since both determined on the basis of some criteria . Since both
the players are expected to be rational in their the players are expected to be rational in their
approach , this is known as the criteria of optimality . approach , this is known as the criteria of optimality .
The decision criteria in game theory is criteria of The decision criteria in game theory is criteria of
optimality i.e. maximin and minimax.optimality i.e. maximin and minimax.

12. Limitations of game theoryLimitations of game theory
•The assumption that the players have the knowledge about their own The assumption that the players have the knowledge about their own
pay-offs and pay-offs of others is rather unrealistic. He can only make pay-offs and pay-offs of others is rather unrealistic. He can only make
a guess of his own and his rivals’ strategies.a guess of his own and his rivals’ strategies.
•As the number of maximum and minimize show that the gaming As the number of maximum and minimize show that the gaming
strategies becomes increasingly complex and difficult. In practice, strategies becomes increasingly complex and difficult. In practice,
there are many firms in an oligopoly situation and game theory cannot there are many firms in an oligopoly situation and game theory cannot
be very helpful inbe very helpful in such situation.such situation.
• The assumptions of maximum and minimize show that the players are The assumptions of maximum and minimize show that the players are
risk-averse and have complete knowledge the strategies. These do not risk-averse and have complete knowledge the strategies. These do not
seen practical.seen practical.
• Rather than each player in an oligopoly situation working under Rather than each player in an oligopoly situation working under
uncertain conditions, the players will allow each other to share the uncertain conditions, the players will allow each other to share the
secrets of business in order to work out a collusion. Thus, the mixed secrets of business in order to work out a collusion. Thus, the mixed
strategies are also not very useful.strategies are also not very useful.

13. StrategyStrategy
•It is pre- determined rule by which each player decides his course It is pre- determined rule by which each player decides his course
of action from his list available to him. How one course of of action from his list available to him. How one course of
action is selected out of various courses available to him is action is selected out of various courses available to him is
known as strategy of the game. known as strategy of the game.
•TYPES OF STRATEGY :TYPES OF STRATEGY :
There are two types of strategy are employedThere are two types of strategy are employed
1.1.Pure strategyPure strategy : it is predetermined course of action to be : it is predetermined course of action to be
employed by the player. The player knew it in advance. It is employed by the player. The player knew it in advance. It is
usually represented by a number with which the cause of action usually represented by a number with which the cause of action
is associated. is associated.

14. •::Mixed strategy Mixed strategy
In mixed strategy the player decides his course of action in In mixed strategy the player decides his course of action in
accordance with some fixed probability distribution . accordance with some fixed probability distribution .
Probability are associated with each course of action and Probability are associated with each course of action and
the selection is done as per these probabilities.the selection is done as per these probabilities.
In mixes strategy the opponent cannot be sure of the In mixes strategy the opponent cannot be sure of the
course f action to be taken on any particular occasion.course f action to be taken on any particular occasion.
•Decision of a game :Decision of a game :
In a game theory, best strategy for each player is In a game theory, best strategy for each player is
determined on the basis of some rules. Since both the determined on the basis of some rules. Since both the
players are expected to be rational in their approaches players are expected to be rational in their approaches
this is known as criteria of optimality.this is known as criteria of optimality.

15. The Maximin –Minimax PrincipleThe Maximin –Minimax Principle
Maximin CriteriaMaximin Criteria : The maximizing player lists his : The maximizing player lists his
minimum gains from each strategy and selects the minimum gains from each strategy and selects the
strategy which gives the maximum out of these strategy which gives the maximum out of these
minimum gains.minimum gains.
Minimax CriteriaMinimax Criteria : The minimizing player lists his : The minimizing player lists his
maximum loss from each strategy and selects the maximum loss from each strategy and selects the
strategy which gives him the minimum loss out of these strategy which gives him the minimum loss out of these
maximum losses.maximum losses.

16. •Value of gameValue of game : : In game theory, the concept value of In game theory, the concept value of
game is considered as very important . The value of game game is considered as very important . The value of game
is maximum guaranteed gain to the maximizing player if is maximum guaranteed gain to the maximizing player if
both the players use there best strategy .both the players use there best strategy .
It refers to the average pay off per play of the game over the It refers to the average pay off per play of the game over the
period of time. period of time.
•SADDLE POINTSADDLE POINT : : the saddle point in a pay off matrix is the saddle point in a pay off matrix is
one which is the smallest value in its row and the largest in one which is the smallest value in its row and the largest in
its column its column
The saddle point is also known as equilibrium point in the The saddle point is also known as equilibrium point in the
theory of games.theory of games.
An element of a matrix that is simultaneously minimum of An element of a matrix that is simultaneously minimum of
the row in which it occurs and the maximum of the column the row in which it occurs and the maximum of the column
in which it occurs is a saddle point of thein which it occurs is a saddle point of the matrix game.matrix game.

17. In a game having a saddle point optimum strategy In a game having a saddle point optimum strategy
for a player X is always to play row containing for a player X is always to play row containing
saddle point and for a player Y to play the saddle point and for a player Y to play the
column that contains saddle point.column that contains saddle point.
The following are the steps required to find out The following are the steps required to find out
saddle point :saddle point :
1.1.Select the minimum value of each row & put Select the minimum value of each row & put
a circle around it.a circle around it.
2.2.Select the maximum value of each column Select the maximum value of each column
and put square around it. and put square around it.
3.3.The value with both circle and square is the The value with both circle and square is the
saddle point. saddle point.

18. TYPES OF PROBLEMTYPES OF PROBLEM
1.1.Games with pure strategies or Two person Zero sum game with Games with pure strategies or Two person Zero sum game with
Saddle point Or Two person Zero sum game with pure strategySaddle point Or Two person Zero sum game with pure strategy : :
In case of pure strategy , the maximizing player arrives at his optimal In case of pure strategy , the maximizing player arrives at his optimal
strategy on the basis of maximin criterion. The game is solved strategy on the basis of maximin criterion. The game is solved
when maximin value equals minimax value.when maximin value equals minimax value.
Example : Example :
4
18
20
12
6
10
Y1 Y2 Y3
FIRM Y
X1
X2
FIIRM X

19. 64
18
20
12
10
Y1 Y2 Y3
X1
X2
Firm Y
Firm X
The saddle point Exits and the value of game (v) is 10 and
the pure strategy for X is X2 and for Y is Y3

20. Games with mixed strategiesGames with mixed strategies
•All game problems where saddle point does not exists All game problems where saddle point does not exists
are taken as mixed strategy problems. Where row are taken as mixed strategy problems. Where row
minima is not equal to column maxima, then the different minima is not equal to column maxima, then the different
methods are used to solve the problems. Both players methods are used to solve the problems. Both players
will use different strategies with certain probability to will use different strategies with certain probability to
minimize.minimize.
•Methods :Methods :
1.1.ODDS method (2x2 game without saddle point)ODDS method (2x2 game without saddle point)
2.2.Dominance methodDominance method
3.3.Sub games method For (mx2) or (2xn) matrices .Sub games method For (mx2) or (2xn) matrices .
4.4.Equal gains method.Equal gains method.
5.5.Linear programming method- graphic methodLinear programming method- graphic method

21. ODDS METHODODDS METHOD
•Step 1Step 1 Find out the difference in the value of in cell (1,1) Find out the difference in the value of in cell (1,1)
and the value in cell (1,2) of the first row and place it in and the value in cell (1,2) of the first row and place it in
front of second row.front of second row.
•Step 2 Step 2 find out the difference in the value of cell (2,1) and find out the difference in the value of cell (2,1) and
(2,2) of the second row and place it in front of first row.(2,2) of the second row and place it in front of first row.
•Step 3Step 3 find out the difference in the value cell (1,1) and find out the difference in the value cell (1,1) and
(2,1) of the first column and place it below the second (2,1) of the first column and place it below the second
column.column.
•Step 4Step 4 . Similarly find the difference between the value of . Similarly find the difference between the value of
the cell (1,2) and the value in cell (2,2) of the second the cell (1,2) and the value in cell (2,2) of the second
column and place I below the first column.column and place I below the first column.

22. The above odds or differences are taken positive (ignoring the negative The above odds or differences are taken positive (ignoring the negative
sign) sign)
Mathematically :Mathematically :
a1
b1
a2
b2
X1
X2
X
Y1 Y2
strategy
(b1-b2)
(a1-a2)
(a2-b2) (a1-b1)
0dds
odds

23. The valve of game is determined with the help of following The valve of game is determined with the help of following
equation :equation :
Valve of game = a1(b1-b2) + b1(a1-a2) Valve of game = a1(b1-b2) + b1(a1-a2)
(b1-b2) + (a1-a2)(b1-b2) + (a1-a2)
Probabilities for x1 = b1-b2 X2 = a1-a2Probabilities for x1 = b1-b2 X2 = a1-a2
(b1-b2) = (a1-a2) (b1-b2) + (a1-a2)(b1-b2) = (a1-a2) (b1-b2) + (a1-a2)
Probabilities for Y1 = a2-b2 Y2= a1-b1Probabilities for Y1 = a2-b2 Y2= a1-b1
(a2-b2) + (a1-b1) (a2-b2) + (a1-b1)(a2-b2) + (a1-b1) (a2-b2) + (a1-b1)

24. Dominance MethodDominance Method
•Dominance method is also applicable to pure strategy and Dominance method is also applicable to pure strategy and
mixed strategy problems. In pure strategy the solution is mixed strategy problems. In pure strategy the solution is
obtained by itself while in mixed strategy it can be used for obtained by itself while in mixed strategy it can be used for
simplifying the problems.simplifying the problems.
•Principle of dominance : Principle of dominance : The principle of dominance states The principle of dominance states
that if the strategy of a player dominates over the other that if the strategy of a player dominates over the other
strategy is ignored because it will not effect the solution in strategy is ignored because it will not effect the solution in
any way.any way.
For the gainer point of view if a strategy gives more gain For the gainer point of view if a strategy gives more gain
than the another strategy , then first strategy dominates than the another strategy , then first strategy dominates
over the other and he second strategy can be ignored over the other and he second strategy can be ignored
altogether.altogether.
So determination of superior or inferior strategy is base d on So determination of superior or inferior strategy is base d on
the objective of player.the objective of player.

25. •For deleting the ineffective rows and columns the For deleting the ineffective rows and columns the
following general rules are to be followed :following general rules are to be followed :
1.1.If all the elements of ith row of a pay off matrix are If all the elements of ith row of a pay off matrix are
less than or equal to(less than or equal to(<<) the corresponding each element ) the corresponding each element
of the other jth row then the player A will never choose of the other jth row then the player A will never choose
the ith strategy or ith row is dominated by he jth row . the ith strategy or ith row is dominated by he jth row .
Then delete ith row.Then delete ith row.
2.2.If all the elements of a column say jth column are If all the elements of a column say jth column are
greater than or equal to the corresponding elements of greater than or equal to the corresponding elements of
any other column say ith column then the ith column is any other column say ith column then the ith column is
dominated by jth column. Then delete ith column.dominated by jth column. Then delete ith column.

26. •A pure strategy of a player may also be dominated if it is A pure strategy of a player may also be dominated if it is
inferior to some convex combination of two or more pure inferior to some convex combination of two or more pure
strategies. As a particular case, if all the elements of a strategies. As a particular case, if all the elements of a
column are greater than or equal; to the average of two or column are greater than or equal; to the average of two or
more other columns then this column is dominated by the more other columns then this column is dominated by the
group of columns. Similarly if all the elements of row are group of columns. Similarly if all the elements of row are
less than or equal to the average of two or more rows less than or equal to the average of two or more rows
then this row is dominated by other group of row.then this row is dominated by other group of row.
•By eliminating some of the dominated rows a columns By eliminating some of the dominated rows a columns
and if the game is reduced to 2x2 form it can be easily and if the game is reduced to 2x2 form it can be easily
solved by odds method.solved by odds method.

27. Sub –games Method (in case 2xn or mxn matricesSub –games Method (in case 2xn or mxn matrices
•A game where one player has two alternatives while the other A game where one player has two alternatives while the other
player has more than two alternatives. In case of 2xn or mx2 player has more than two alternatives. In case of 2xn or mx2
matrices this can be solved by Sub games method. When matrices this can be solved by Sub games method. When
there is no saddle point or it can not be reduced by dominance there is no saddle point or it can not be reduced by dominance
method then in such situation sub games method is very method then in such situation sub games method is very
useful . This technique is discussed as blow.useful . This technique is discussed as blow.
•PROCEDUREPROCEDURE
1.1.Step 1 . Divide the mx2 or 2x n game matrix into as many 2x2 Step 1 . Divide the mx2 or 2x n game matrix into as many 2x2
sub games as possible.sub games as possible.
2.2.Step 2. taking each game one by one and finding out the Step 2. taking each game one by one and finding out the
saddle point of each game and then that sub game has pure saddle point of each game and then that sub game has pure
strategies.strategies.
3.3.Step 3. in case there is no saddle point and then that sub Step 3. in case there is no saddle point and then that sub
game should be solved by odds method.game should be solved by odds method.

28. Probability method or equal gains methodProbability method or equal gains method
•(solution of 2x2 matrix without saddle point)(solution of 2x2 matrix without saddle point)
•In case of game not having saddle point, each player In case of game not having saddle point, each player
has to use mixed strategies. As the players are has to use mixed strategies. As the players are
reported to be rationale in their approach, the reported to be rationale in their approach, the
selection of their combination of strategies will be selection of their combination of strategies will be
done in such a way that the net gain is not done in such a way that the net gain is not
influenced by the selection of any combination of influenced by the selection of any combination of
strategy by the opponent.strategy by the opponent.
•In this, player select each of the available strategies In this, player select each of the available strategies
for certain proportion of the time i.e. each player for certain proportion of the time i.e. each player
selects a strategy with some probability.selects a strategy with some probability.

29. LPP-Graphic Method- for (2xm) and (nx2)LPP-Graphic Method- for (2xm) and (nx2)
•Graphic method is applicable to only those games in Graphic method is applicable to only those games in
which one of the players has two strategies only. which one of the players has two strategies only.
Through sub game method provides simple approach Through sub game method provides simple approach
, but in case ‘n’ or ‘m’ is large then a graphic method , but in case ‘n’ or ‘m’ is large then a graphic method
is relatively fast and easy.is relatively fast and easy.
•The following are the steps involved in this method .The following are the steps involved in this method .
•Step 1Step 1 : the game matrix of 2xm or nx2 sub matrices. : the game matrix of 2xm or nx2 sub matrices.
•Step 2.Step 2. next taking the probabilities of the two next taking the probabilities of the two
alternatives of the first player say A as p1 and (1-p1) alternatives of the first player say A as p1 and (1-p1)
then the ne gain of A from the different alternatives then the ne gain of A from the different alternatives
strategies of B is expressed with equations.strategies of B is expressed with equations.

30. •Step 3.Step 3. the boundaries of the two alternatives the boundaries of the two alternatives
strategies of the first player are shown by the two strategies of the first player are shown by the two
parallel line shown on the graph.parallel line shown on the graph.
•Step 4Step 4 . The gain equation of different sub games are . The gain equation of different sub games are
then plotted on the graphthen plotted on the graph
•Step 5 .Step 5 . In case of maximizing player A, the point is In case of maximizing player A, the point is
identified where minimum expected gain is identified where minimum expected gain is
maximized. This will be the highest point out the maximized. This will be the highest point out the
inter section of the gains lines in the lower envelopinter section of the gains lines in the lower envelop
•In case of minimizing player B, the point where In case of minimizing player B, the point where
maximum loss is minimized is justified, this will be maximum loss is minimized is justified, this will be
the lowest point out at the intersection of the the lowest point out at the intersection of the
equation in the intersection of the equations in the equation in the intersection of the equations in the
upper envelop.upper envelop.

31. •Example : solve the following :Example : solve the following :
-5
8
5
-4
0
-1
-1
6
8
-5
A
1
2
1 2 3 4 5
B

32. •Since Minimize = Maximinze Since Minimize = Maximinze
•Thus, players will use the mixed Thus, players will use the mixed
strategy.strategy.
•Since we do not have any saddle Since we do not have any saddle
point Let p1 the probability of Mr. point Let p1 the probability of Mr.
selecting strategy 1 & hence (1-p1) selecting strategy 1 & hence (1-p1)
be the probability of Mr. a selecting be the probability of Mr. a selecting
strategy 2.strategy 2.

33. If B select If B select
strategystrategy
Expected pay off of AExpected pay off of A
11
22
33
44
55
-5(p1) + 8(1-P1)= --5(p1) + 8(1-P1)= -
13p1+813p1+8
5(p1)-4 (1-p1) = 9p1 -45(p1)-4 (1-p1) = 9p1 -4
0(p1) + - 1(1-p1) +p1-10(p1) + - 1(1-p1) +p1-1
-1(p1) +6(1-p1) = -7p1+6-1(p1) +6(1-p1) = -7p1+6
8(p1) + -5(1-p1)= 13p1-58(p1) + -5(1-p1)= 13p1-5

34. 8
7
6
5
4
3
2
1
0
1-p1
-1
-2
-3
-4
-5
Lower envelop
maximin
p1
P
Q
R
B3
B4
B1
B2
B5
13p1-5
9p1-4
-7p1+6
P1-1
-13p1+8
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5

35. •Since R is the maximize point and Since R is the maximize point and
here B1’,B3 interest. These here B1’,B3 interest. These
strategies will be selected & the strategies will be selected & the
resultant matrix is produced.resultant matrix is produced.
-5
8
0
-1
1 3
1
2
A
B
odds
odds
9
5
1 13

36. • V = a1(b1-b2) + b1(a1-a2) = (-5x9) + (8X5) = -5V = a1(b1-b2) + b1(a1-a2) = (-5x9) + (8X5) = -5
(b1-b2) + (a1-a2) 9+5 14(b1-b2) + (a1-a2) 9+5 14
AA BB
Probability of selecting strategies Probability of selecting strategies
no.no.
Probability of selecting Probability of selecting
strategies no.strategies no.
1 9/141 9/14 1 1/141 1/14
22 00
3 3/143 3/14
2 5/142 5/144 04 0
5 05 0