MARKETING (UNR471)

Dr. Mohamed Sameh

GAME THEORY
GAME THEORY

• Contains a finite number of players.

• Each player has a number of actions called “strategies”.
• All the strategies and their effects are known to the players.
• No player knows his opponent’s strategy until he decides his own.
• The outcomes of strategies are put in a ‘pay-off matrix’.
• The players playing the game are assumed “Rationale”, i.e. always trying to choose ‘optimal
strategy’.
• The expected payoff is known as ‘Value of the game’.
• The game is said to be ‘fair’ if the value of the game is zero. Otherwise, it’s ‘unfair’.
STRATEGY

• Pure Strategy:
• Each player knows exactly what the other player is going to do.
• A deterministic situation.
• The objective is to maximize the actual gain.

• Mixed Strategy:
• Each player is guessing which activity is to be selected by the other.
• A probabilistic situation.
• The objective is to maximize the expected gain.
TWO-PERSON
ZERO-SUM GAME

• Most common form

• Two competitors, each will be rewarded
• Fixed reward total
• What one wins, the other loses
PAYOFF MATRIX

• Player A has ‘m’ strategies and player B has ‘n’ strategies.

• Rows (i) are the strategies to player A.
• Columns (j) are the strategies available to player B.
• Cell value Vij is the payment to player A if A chooses strategy i and B chooses
strategy j.
• In a zero-sum, two-person game, player B’s payoff is the negative of player A’s.
• Payoff matrices for player A and player B are ultimately zero.
MINIMAX CRITERION (PURE STRATEGY)
1. Identify the players.
2. Identify the possible strategies for each of them.
3. Identify the payoff for all strategy combinations.
4. Build the Pay- off matrix for one of the players.
5. Apply minimax criterion.
6. Each player should play in such a way as to minimize his maximum losses.
7. Player 1 should select the strategy whose minimum payoff is largest (max-mini ), whereas player
2 should choose the one whose maximum payoff to player 1 is the smallest (mini-max).
8. Determine the saddle point if it exists, where mini-max = max-mini (stable solution).
9. The Saddle point has always the minimum payoff in its row and the maximum in its column.
10. If there is no saddle point, the solution is unstable.
7
EXAMPLE
• Coke and Pepsi are competing against each other to have the highest beverage sales during summer. They are
both planning big advertising campaigns during July and August in two of the north coast villages, Marassi and
Amwaj. They have a choice of either spending a month in each village or spending the two months in one of them.
Since the necessary arrangements must be made in advance, neither company will learn the other’s campaign
schedule until after it has finalized its own. Therefore, each company has asked its advertising manager to assess
what the impact would be (in terms of sales won or lost) from the various combinations of strategies to use this
information to choose the best strategy. The Payoff for each combination (in terms of million bottles sales) for
Coke are tabulated.

Pepsi
Strat. 1 Strat. 2 Strat. 3
Spend one month in each village (strat. 1) 1 -1 1
Coke Spend two months in Marassi (strat. 2) -2 0 3
Spend two months in Amwaj (strat. 3) 3 1 2

8
SOLUTION
Pepsi
Strat. 1 Strat. 2 Strat. 3
Strat. 1 1 -1 1
Coke Strat. 2 -2 0 3
Strat. 3 3 1 2

Pepsi
Strat. 1 Strat. 2 Strat. 3 Minimum
Strat. 1 1 -1 1 -1
Coke Strat. 2 -2 0 3 -2
Strat. 3 3 1 2 1
Optimal strategy: Maximum 3 1 3
- For Coke : Strat.3
- For Pepsi: Strat.2 Mini-max value Max-mini value
Saddle point
- Value (V) = +1 for Coke
- V = -1 for Pepsi
DOMINATED STRATEGIES (PURE & MIXED)
1. Identify the players.
2. Identify the possible strategies for each of them.
3. Identify the payoff for all strategy combinations.
4. Build the Pay- off matrix for one of the players.
5. Apply dominated strategies to find the solution (pure) or thin out the
matrix(mixed).
6. A strategy is dominated by another if the second strategy has always an equal or
a better pay-off regardless of what the opponent does.
7. Usually used for big tables.

10
EXAMPLE
• Coke and Pepsi are competing against each other to have the highest beverage sales during summer. They are
both planning big advertising campaigns during July and August in two of the north coast villages, Marassi and
Amwaj. They have a choice of either spending a month in each village or spending the two months in one of them.
Since the necessary arrangements must be made in advance, neither company will learn the other’s campaign
schedule until after it has finalized its own. Therefore, each company has asked its advertising manager to assess
what the impact would be (in terms of sales won or lost) from the various combinations of strategies to use this
information to choose the best strategy. The Payoff for each combination (in terms of million bottles sales) for
Coke are tabulated.

Pepsi
Strat. 1 Strat. 2 Strat. 3
Spend one month in each village (strat. 1) 2 3 5
Coke Spend two months in Marassi (strat. 2) 2 1 4
Spend two months in Amwaj (strat. 3) 0 -1 -3

11
SOLUTION
Pepsi Dominating
Strat. 1 Strat. 2 Strat. 3 strategy

Strat. 1 2 3 5 Dominated
Coke Strat. 2 2 1 4 strategy
Strat. 3 0 -1 -3

Pepsi
Strat. 1 Strat. 2 Strat. 3
Strat. 1 2 3 5 Dominated
Coke
Strat. 2 2 1 4 strategy

Dominating strategy
SOLUTION
Pepsi Dominating
strategy
Strat. 1 Strat. 2
Strat. 1 2 3 Dominated
Coke strategy
Strat. 2 2 1

Pepsi
Strat. 1 Strat. 2
Coke Strat. 1 2 3 Dominated
strategy
Dominating
strategy
Optimal strategy:
Pepsi - For Coke : Strat.1
Strat. 1 - For Pepsi: Strat.1
Coke - V = +2 for Coke
Strat. 1 2
- V = -2 for Pepsi
MIXED STRATEGIES
1. Identify the players.
2. Identify the possible strategies for each of them.
3. Identify the payoff for all strategy combinations.
4. Build the Pay- off matrix for one of the players.
5. Apply minimax criterion.
6. No saddle point is found.
7. Apply dominance to thin out the table.
8. Use one of the following:
 ODDS METHOD (2x2 game)
 Sub Games Method. – For (mx2) or (2xn) Matrices
 Equal Gains Method.
 Linear Programming Method-Graphic solution (For (mx2) or (2xn) Matrices).

14
ODDS METHOD (MIXED STRATEGIES)
1. Find out the difference in the value of in cell (1, 1) and the value in the cell (1,2) of the first row and place it in
front of second row.
2. 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.
3. Find out the differences in the value of cell (1, 1) and (2, 1) of the first column and place it below the second
column.
4. Find the difference between the value of the cell (1, 2) and the value in cell (2, 2) of the second column and
place it below the first column.
5. The above odds or differences are taken as positive.
Player 2
Strat.1 (Y1) Strat.2 (Y2)
Strat.1 (X1) a1 a2 |b1-b2|
Player 1
Strat.2 (X2) b1 b2 |a1-a2|
|a2-b2| |a1-b1|
15
 ODDS METHOD
1. Calculate the following:

16
EXAMPLE
• Coke and Pepsi are competing against each other to have the highest beverage sales during summer.
They are both planning big advertising campaigns during July and August in two of the north coast villages,
Marassi and Amwaj. They have a choice of either spending a month in each village or spending the two
months in one of them. Since the necessary arrangements must be made in advance, neither company will
learn the other’s campaign schedule until after it has finalized its own. Therefore, each company has asked
its advertising manager to assess what the impact would be (in terms of sales won or lost) from the
various combinations of strategies to use this information to choose the best strategy. The Payoff for each
combination (in terms of million bottles sales) for Coke are tabulated.

Pepsi
Strat. 1 Strat. 2 Strat. 3
Spend one month in each village (strat. 1) 1 5 5
Coke Spend two months in Marassi (strat. 2) 4 2 4
Spend two months in Amwaj (strat. 3) 0 -1 -3

17
SOLUTION
Pepsi
Strat. 1 Strat. 2 Strat. 3
Strat. 1 1 5 5
Coke Strat. 2 4 2 4
Strat. 3 0 -1 -3

Pepsi
Strat. 1 Strat. 2 Strat. 3 Minimum
Strat. 1 1 5 5 1
Coke Strat. 2 4 2 4 2
Strat. 3 0 -1 -3 -3
Maximum 4 5 5
Max-mini value
Mini-max value
No Saddle point
SOLUTION

Pepsi Dominating
Strat. 1 Strat. 2 Strat. 3 strategy

Strat. 1 1 5 5 Dominated
Coke Strat. 2 4 2 4 strategy
Strat. 3 0 -1 -3

Pepsi
Strat. 1 Strat. 2 Strat. 3
Strat. 1 1 5 5 Dominated
Coke
Strat. 2 4 2 4 strategy

Dominating strategy
SOLUTION
Pepsi
Strat. 1 Strat. 2
Strat. 1 1 5
Coke
Strat. 2 4 2

Pepsi
Strat.1 (Y1) Strat.2 (Y2)
Strat.1 (X1) 1 5 |b1-b2|= 2
Coke
Strat.2 (X2) 4 2 |a1-a2|= 4
|a2-b2|=3 |a1-b1|=3
1∗2 +(4∗4)
Expected Value of the game = =3
(2+4)
For coke; probability for strat. 1 = 1/3 , strat. 2 = 2/3
For Pepsi; probability for strat. 1 = 1/2 , strat. 2 = 1/2
GRAPHICAL SOLUTION (MIXED STRATEGIES)
1. Find which player has only two strategies.
2. Assign X as the probability of using strat. 1 and (1-X) as the probability of using strat. 2.
3. Plot the expected payoff as a function of X for each of the opponent’s pure strategies.
𝑚𝑚
Expected payoff = ∑𝑖𝑖=1 ∑𝑛𝑛𝑗𝑗=1 𝑃𝑃𝑖𝑖𝑖𝑖 𝑥𝑥𝑖𝑖 𝑦𝑦𝑗𝑗

Hint: put X = 0 , and X = 1 as your graph limits.

4. This graph can then be used to identify the point that maximizes the minimum expected payoff.
5. The probability x can be found from the lowest intersection of the drawn lines.
6. The expected value can be then calculated from the equation of any of the lines.
7. The probabilities of the other player strategies can be calculated from the expected payoff
equation.
21
EXAMPLE
• Coke and Pepsi are competing against each other to have the highest beverage sales during summer. They are both
planning big advertising campaigns during July and August in two of the north coast villages, Marassi and Amwaj. They
have a choice of either spending a month in each village or spending the two months in one of them. Since the
necessary arrangements must be made in advance, neither company will learn the other’s campaign schedule until
after it has finalized its own. Therefore, each company has asked its advertising manager to assess what the impact
would be (in terms of sales won or lost) from the various combinations of strategies to use this information to
choose the best strategy. The Payoff for each combination (in terms of million bottles sales) for Coke are tabulated.

Pepsi
Strat. 1 Strat. 2 Strat. 3
Spend one month in each village (strat. 1) 1 5 5
Coke Spend two months in Marassi (strat. 2) 4 2 4
Spend two months in Amwaj (strat. 3) 0 -1 -3

22
SOLUTION
Pepsi
Strat. 1 Strat. 2 Strat. 3
Strat. 1 1 5 5
Coke Strat. 2 4 2 4
Strat. 3 0 -1 -3

Pepsi
Strat. 1 Strat. 2 Strat. 3 Minimum
Strat. 1 1 5 5 1
Coke Strat. 2 4 2 4 2
Strat. 3 0 -1 -3 -3
Maximum 4 5 5
Max-mini value
Mini-max value
No Saddle point
SOLUTION
Pepsi Dominating
Strat. 1 Strat. 2 Strat. 3 strategy

Strat. 1 1 5 5 Dominated
Coke Strat. 2 4 2 4 strategy
Strat. 3 0 -1 -3

Pepsi
Strat. 1 Strat. 2 Strat. 3
Strat. 1 1 5 5 Dominated
Coke
Strat. 2 4 2 4 strategy

Dominating strategy
SOLUTION
Pepsi
Strat. 1 Strat. 2
Strat. 1 1 5
Coke
Strat. 2 4 2
Pepsi
Strat.1 (Y1) Strat.2 (Y2)
Strat.1 (X) 1 5
Coke
Strat.2 (1-X) 4 2

First plot line equation = 1 * X + 4 * (1-X) = 4 – 3X

Second plot line equation = 5 * X + 2 *(1-X) = 2+ 3X
SOLUTION
From graph; point of intersection occurs at (1/3, 3)
The probability of using strat.1 for Coke = 1/3
The probability of using strat.1I for Coke = 2/3
𝑚𝑚
Expected Payoff = ∑𝑖𝑖=1 ∑𝑛𝑛𝑗𝑗=1 𝑃𝑃𝑖𝑖𝑖𝑖 𝑥𝑥𝑖𝑖 𝑦𝑦𝑗𝑗 =3.
For Pepsi; expected payoff =(4-3X)* Y1+ (2+3X)* Y2
=3
Put: x =0 4Y1 +2Y2 = 3 & x= 1  Y1+5 Y2 = 3
Solving for both equations;
The probability of using strat.1 for Pepsi = 1/2
The probability of using strat.1 for Pepsi = 1/2