INDEX Introduction of game theory Introduction of game theory Significance of game theory...

36
INDEX INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game Essential features of game theory theory Assumptions Assumptions Elements of game theory Elements of game theory Limitations of game theory Limitations of game theory Types of strategy Of game theory Types of strategy Of game theory Methods for game theory Methods for game theory

description

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

Transcript of INDEX Introduction of game theory Introduction of game theory Significance of game theory...

Page 1: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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

Page 2: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

INTRODUCTIONINTRODUCTION•Game theory was developed by Game theory was developed by Prof. John Prof. John

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

Page 3: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

What is Game What is Game Theory?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

Page 4: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

Significance of game theory1. Game theory is a kind of decision theory which is 1. Game theory is a kind of decision theory which is based on the choice of action. And choice of action is based on the choice of action. And choice of action is determining after considering the possible determining after considering the possible alternatives available to the opponent.alternatives available to the opponent.

2. It involves the player decision i.e. decision makers 2. It involves the player decision i.e. decision makers who have different goals and objectives.who have different goals and objectives.

3. The game theory determine the rules of rational 3. The game theory determine the rules of rational behavior of these players in which the outcomes are behavior of these players in which the outcomes are dependent on the actions of the interdependent dependent on the actions of the interdependent players.players.

4.In a game theory there are number of possible 4.In a game theory there are number of possible outcomes, with different values to the decision outcomes, with different values to the decision makers.makers.

5.They might have some control but do not have the 5.They might have some control but do not have the complete control over others.complete control over others.

Page 5: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

Essential features of Essential features of game theorygame theory

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.

Page 6: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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.

Page 7: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

•((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.

Page 8: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

The Two-Person, The Two-Person, Zero-Sum GameZero-Sum Game

• 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.

Page 9: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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.

Page 10: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

• Pay off matrixPay off matrixIn 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.

Page 11: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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.

Page 12: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

Limitations of game Limitations of game theorytheory• 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.

Page 13: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

StrategyStrategy• It is pre- determined rule by which each player It is pre- determined rule by which each player

decides his course of action from his list available to decides his course of action from his list available to him. How one course of action is selected out of him. How one course of action is selected out of various courses available to him is known as strategy various courses available to him is known as strategy of the game. of the game.

• TYPES OF STRATEGY :TYPES OF STRATEGY : There are two types of strategy are employedThere are two types of strategy are employed1.1.Pure strategyPure strategy : it is predetermined course of action : it is predetermined course of action

to be employed by the player. The player knew it in to be employed by the player. The player knew it in advance. It is usually represented by a number with advance. It is usually represented by a number with which the cause of action is associated. which the cause of action is associated.

Page 14: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

•::Mixed strategy Mixed strategy In mixed strategy the player decides his In mixed strategy the player decides his course of action in accordance with some course of action in accordance with some fixed probability distribution . fixed probability distribution .

Probability are associated with each course of Probability are associated with each course of action and the selection is done as per these action and the selection is done as per these probabilities.probabilities.

In mixes strategy the opponent cannot be In mixes strategy the opponent cannot be sure of the course f action to be taken on sure of the course f action to be taken on any particular occasion.any particular occasion.

•Decision of a game :Decision of a game :In a game theory, best strategy for each In a game theory, best strategy for each player is determined on the basis of some player is determined on the basis of some rules. Since both the players are expected to rules. Since both the players are expected to be rational in their approaches this is known be rational in their approaches this is known as criteria of optimality.as criteria of optimality.

Page 15: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

The Maximin –Minimax The Maximin –Minimax PrinciplePrinciple

Maximin CriteriaMaximin Criteria : The maximizing player : The maximizing player lists his minimum gains from each lists his minimum gains from each strategy and selects the strategy which strategy and selects the strategy which gives the maximum out of these minimum gives the maximum out of these minimum gains.gains.

Minimax CriteriaMinimax Criteria : The minimizing player : The minimizing player lists his maximum loss from each strategy lists his maximum loss from each strategy and selects the strategy which gives him and selects the strategy which gives him the minimum loss out of these maximum the minimum loss out of these maximum losses.losses.

Page 16: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

• 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.

Page 17: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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.

Page 18: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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 :

418

20

12

6

10

Y1 Y2 Y3FIRM Y

X1

X2FIIRM X

Page 19: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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

Page 20: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

Games with mixed Games with mixed strategiesstrategies

• 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 method3.3. 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

Page 21: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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.

Page 22: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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

X2X

Y1 Y2strategy

(b1-b2)

(a1-a2)

(a2-b2) (a1-b1)0dds

odds

Page 23: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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)

Page 24: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

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.

Page 25: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

• For deleting the ineffective rows and For deleting the ineffective rows and columns the following general rules are columns the following general rules are to be followed :to be followed :

1.1.If all the elements of ith row of a pay off matrix If all the elements of ith row of a pay off matrix are less than or equal to(are less than or equal to(<<) the corresponding ) the corresponding each element of the other jth row then the each element of the other jth row then the player A will never choose the ith strategy or player A will never choose the ith strategy or ith row is dominated by he jth row . Then ith row is dominated by he jth row . Then delete ith row.delete ith row.

2.2.If all the elements of a column say jth column If all the elements of a column say jth column are greater than or equal to the corresponding are greater than or equal to the corresponding elements of any other column say ith column elements of any other column say ith column then the ith column is dominated by jth then the ith column is dominated by jth column. Then delete ith column.column. Then delete ith column.

Page 26: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

•A pure strategy of a player may also be A pure strategy of a player may also be dominated if it is inferior to some convex dominated if it is inferior to some convex combination of two or more pure strategies. As a combination of two or more pure strategies. As a particular case, if all the elements of a column particular case, if all the elements of a column are greater than or equal; to the average of two are greater than or equal; to the average of two or more other columns then this column is or more other columns then this column is dominated by the group of columns. Similarly if dominated by the group of columns. Similarly if all the elements of row are less than or equal to all the elements of row are less than or equal to the average of two or more rows then this row is the average of two or more rows then this row is dominated by other group of row.dominated by other group of row.

•By eliminating some of the dominated rows a By eliminating some of the dominated rows a columns and if the game is reduced to 2x2 form columns and if the game is reduced to 2x2 form it can be easily solved by odds method.it can be easily solved by odds method.

Page 27: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

Sub –games Method (in case Sub –games Method (in case 2xn or mxn matrices2xn 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.

• PROCEDUREPROCEDURE1.1. 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.

Page 28: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

Probability method or Probability method or equal gains methodequal 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.

Page 29: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

LPP-Graphic Method- for LPP-Graphic Method- for (2xm) and (nx2)(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.

Page 30: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

• 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.

Page 31: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

• Example : solve the following :Example : solve the following :

-5

8

5

-4

0

-1

-1

6

8

-5A

1

2

1 2 3 4 5B

Page 32: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

•Since Minimize = Since Minimize = Maximinze Maximinze

•Thus, players will use Thus, players will use the mixed strategy.the mixed strategy.

•Since we do not have Since we do not have any saddle point Let p1 any saddle point Let p1 the probability of Mr. the probability of Mr. selecting strategy 1 & selecting strategy 1 & hence (1-p1) be the hence (1-p1) be the probability of Mr. a probability of Mr. a selecting strategy 2.selecting strategy 2.

Page 33: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

If B select If B select strategystrategy

Expected pay off of AExpected pay off of A

1122334455

-5(p1) + 8(1-P1)= --5(p1) + 8(1-P1)= -13p1+813p1+85(p1)-4 (1-p1) = 9p1 -45(p1)-4 (1-p1) = 9p1 -40(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+68(p1) + -5(1-p1)= 13p1-58(p1) + -5(1-p1)= 13p1-5

Page 34: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

8

7

6

5

4

3

210

1-p1

-1

-2

-3

-4

-5

Lower envelop

maximin

p1

P

Q

RB3

B4

B1

B2

B5

13p1-5

9p1-4

-7p1+6

P1-1

-13p1+8

8

7

6

5

4

3

21

-1-2

-3

-4

-5

Page 35: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

•Since R is the maximize point Since R is the maximize point and here B1’,B3 interest. and here B1’,B3 interest. These strategies will be These strategies will be selected & the resultant selected & the resultant matrix is produced.matrix is produced.

-5

8

0

-1

1 3

1

2

A

B

odds

odds

9

5

1 13

Page 36: INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

• V = a1(b1-b2) + b1(a1-a2) = (-5x9) + (8X5) = -V = a1(b1-b2) + b1(a1-a2) = (-5x9) + (8X5) = -55

(b1-b2) + (a1-a2) 9+5 (b1-b2) + (a1-a2) 9+5 1414 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 1/1422 003 3/143 3/14

2 5/142 5/14 4 04 05 05 0