Description
Ppt on dominance rule.
Presented by J Group
2
GROUP CONSIST OF
DOMINANCE THEORY
y In a game, sometimes a strategy available to a player might be found to be
preferable to some other strategy / strategies. Such a strategy is said to dominate the other one(s). y The rules of dominance are used to reduce the size of the payoff matrix. These rules help in deleting certain rows and/or columns of the payoff matrix, which are of lower priority to at least one of the remaining rows, and/or columns in terms of payoffs to both the players. y Rows / columns once deleted will never be used for determining the optimal strategy for both the players. y This concept of domination is very usefully employed in simplifying the two person zero sum games without saddle point. In general the following rules are used to reduce the size of payoff matrix.
Strategy Used in Dominance
y Pure Strategy : y If the player selects the same strategy each time, then
it is a pure strategy.
y In this case each player knows exactly what the other is
going to do, i.e. there is a deterministic situation and the objective of the players is to maximize gains or to minimize losses.
Strategy Used in Dominance
y Mixed Strategy : y When the players use a combination of strategies and
each player is always kept guessing as to which course of action is to be selected by the other, then it is known as a mixed strategy. y Thus, there is a probabilistic situation and the objective of the player is to maximize expected gains or to minimize losses. y Thus, mixed strategy is a selection among pure strategies with fixed possibilities.
PRINCIPLES OF DOMINANCE
y Rule 1: If all the elements in a row ( say ith row ) of a pay off matrix are
less than or equal to the corresponding elements of the other row ( say jth row ) then the player A will never choose the ith strategy then we say ith strategy is dominated by jth strategy and will delete the ith row. y Rule 2: If all the elements in a column ( say rth column ) of a payoff matrix are greater than or equal to the corresponding elements of the other column ( say sth column ) then the player B will never choose the rth strategy or in the other words the rth strategy is dominated by the sth strategy and we delete rth column . y Rule 3: A pure strategy may be dominated if it is inferior to average of two or more other pure strategies.
In an election campaign, the strategies adopted by the ruling and opposition party along with pay-offs (ruling party's % share in votes polled) are given below:
Ruling Party's Strategies
Campaign one day in each city Campaign two days in large towns Spend two days in large rural sectors
Campaign one day in each city
55
Campaign two days in large towns
40
Spend two days in large rural sectors
35
70
70
55
75
55
65
Assume a zero sum game. Find optimum strategies for both parties and expected payoff to ruling party.
Solution
y Let A1, A2 and A3 be the strategies of the ruling
party and B1, B2 and B3 be those of the opposition. Then Player B B1 B2 B3 A1 55 40 35 Player A A2 70 70 55 A3 75 55 65
Now with the given matrix, we have to find the saddle point. B1 B2 B3 A1 55 40 35 A2 70 70 55 A3 75 55 65
Row Minima Column Maxima
As maximin = 55 and minimax = 65, there is no saddle point.
Row 1 is dominated by Row 2 and Column 1 is dominated by Column 3
A1 A2 A3
B1 55 70 75
B2 40 70 55
B3 35 55 65
The reduced 2 x 2 matrix is as given below , B2 B3 A2 70 55 A3 55 65
The reduced 2 x 2 matrix is as given below , B2 B3 A2 70 55 A3 55 65
y For ruling party: Let the ruling party select strategy A2 with a
probability of p and therefore opposition party selects strategy A3 with a probability of Suppose the opposition selects strategy B2. Then the expected gain to ruling party for this game is given by :
Formulae Used
y X1 y X2 y Y1 y Y2
A22 A21 A11 - A12 A22 A21 A11 A12
X1 + X2 = 1
Y1 + Y2 = 1
A11 * A22 A21 * A12
yV
(Value of the Game) (A11 + A22) (A21 + A12)
y As the reduced pay-off matrix does not possess any
saddle point, the players will use mixed strategies. The optimum mixed strategy for player A is determined by : X1 A22 A21 65 - 55 10 2 X2 A11 - A12 70 55 15 3 X1
2 /5
X2
3 /5
y As the reduced pay-off matrix does not possess any
saddle point, the players will use mixed strategies. The optimum mixed strategy for player B is determined by : Y1 A22 A12 65 - 55 10 2 Y2 A11 - A21 70 55 15 3 X1
2 /5
X2
3 /5
The expected value of the game is given by :
A11 * A22 A21 * A12
yV
(Value of the Game) (A11 + A22) (A21 + A12) 70 * 65 - 55 * 55 4550 3025 1525 25
V (70 + 65) (55 + 55) 135 - 110
V 61
Hence the optimal mixed strategies are
PLAYER A Strategies :A1 0 A2 2/5 A3 3/5
Hence the optimal mixed strategies are
PLAYER B Strategies :A1 0 A2 2/5 A3 3/5
Value of the Game:-
y
V 61
Thank You
doc_604771650.ppt
Ppt on dominance rule.
Presented by J Group
2
GROUP CONSIST OF
DOMINANCE THEORY
y In a game, sometimes a strategy available to a player might be found to be
preferable to some other strategy / strategies. Such a strategy is said to dominate the other one(s). y The rules of dominance are used to reduce the size of the payoff matrix. These rules help in deleting certain rows and/or columns of the payoff matrix, which are of lower priority to at least one of the remaining rows, and/or columns in terms of payoffs to both the players. y Rows / columns once deleted will never be used for determining the optimal strategy for both the players. y This concept of domination is very usefully employed in simplifying the two person zero sum games without saddle point. In general the following rules are used to reduce the size of payoff matrix.
Strategy Used in Dominance
y Pure Strategy : y If the player selects the same strategy each time, then
it is a pure strategy.
y In this case each player knows exactly what the other is
going to do, i.e. there is a deterministic situation and the objective of the players is to maximize gains or to minimize losses.
Strategy Used in Dominance
y Mixed Strategy : y When the players use a combination of strategies and
each player is always kept guessing as to which course of action is to be selected by the other, then it is known as a mixed strategy. y Thus, there is a probabilistic situation and the objective of the player is to maximize expected gains or to minimize losses. y Thus, mixed strategy is a selection among pure strategies with fixed possibilities.
PRINCIPLES OF DOMINANCE
y Rule 1: If all the elements in a row ( say ith row ) of a pay off matrix are
less than or equal to the corresponding elements of the other row ( say jth row ) then the player A will never choose the ith strategy then we say ith strategy is dominated by jth strategy and will delete the ith row. y Rule 2: If all the elements in a column ( say rth column ) of a payoff matrix are greater than or equal to the corresponding elements of the other column ( say sth column ) then the player B will never choose the rth strategy or in the other words the rth strategy is dominated by the sth strategy and we delete rth column . y Rule 3: A pure strategy may be dominated if it is inferior to average of two or more other pure strategies.
In an election campaign, the strategies adopted by the ruling and opposition party along with pay-offs (ruling party's % share in votes polled) are given below:
Ruling Party's Strategies
Campaign one day in each city Campaign two days in large towns Spend two days in large rural sectors
Campaign one day in each city
55
Campaign two days in large towns
40
Spend two days in large rural sectors
35
70
70
55
75
55
65
Assume a zero sum game. Find optimum strategies for both parties and expected payoff to ruling party.
Solution
y Let A1, A2 and A3 be the strategies of the ruling
party and B1, B2 and B3 be those of the opposition. Then Player B B1 B2 B3 A1 55 40 35 Player A A2 70 70 55 A3 75 55 65
Now with the given matrix, we have to find the saddle point. B1 B2 B3 A1 55 40 35 A2 70 70 55 A3 75 55 65
Row Minima Column Maxima
As maximin = 55 and minimax = 65, there is no saddle point.
Row 1 is dominated by Row 2 and Column 1 is dominated by Column 3
A1 A2 A3
B1 55 70 75
B2 40 70 55
B3 35 55 65
The reduced 2 x 2 matrix is as given below , B2 B3 A2 70 55 A3 55 65
The reduced 2 x 2 matrix is as given below , B2 B3 A2 70 55 A3 55 65
y For ruling party: Let the ruling party select strategy A2 with a
probability of p and therefore opposition party selects strategy A3 with a probability of Suppose the opposition selects strategy B2. Then the expected gain to ruling party for this game is given by :
Formulae Used
y X1 y X2 y Y1 y Y2
A22 A21 A11 - A12 A22 A21 A11 A12
X1 + X2 = 1
Y1 + Y2 = 1
A11 * A22 A21 * A12
yV
(Value of the Game) (A11 + A22) (A21 + A12)
y As the reduced pay-off matrix does not possess any
saddle point, the players will use mixed strategies. The optimum mixed strategy for player A is determined by : X1 A22 A21 65 - 55 10 2 X2 A11 - A12 70 55 15 3 X1
2 /5
X2
3 /5
y As the reduced pay-off matrix does not possess any
saddle point, the players will use mixed strategies. The optimum mixed strategy for player B is determined by : Y1 A22 A12 65 - 55 10 2 Y2 A11 - A21 70 55 15 3 X1
2 /5
X2
3 /5
The expected value of the game is given by :
A11 * A22 A21 * A12
yV
(Value of the Game) (A11 + A22) (A21 + A12) 70 * 65 - 55 * 55 4550 3025 1525 25
V (70 + 65) (55 + 55) 135 - 110
V 61
Hence the optimal mixed strategies are
PLAYER A Strategies :A1 0 A2 2/5 A3 3/5
Hence the optimal mixed strategies are
PLAYER B Strategies :A1 0 A2 2/5 A3 3/5
Value of the Game:-
y
V 61
Thank You
doc_604771650.ppt