ch 3 qm.pptxhttps://www.slideshare.net/slideshow/ppt-solarpptx/265379690

What is a Game?
• A game is a situation where the participants
payoffs depend not only on their decisions but
also on their rivals decisions.
• A game consists of:
– a set of players
– a set of strategies for each player
– the payoffs to each player for every possible list of
strategy choices by the players.

Game Theory
• Game theory
– refers to the general situation of conflict and
competition in which two or more competitors are
involved in decision-making activities in
anticipation of certain outcomes over a period of
time.
– deals with mathematical analysis of competitive problems
based on the principles of Minimax or Maximin
• Minimax and Maximin principles put forward by Von Neumann
which implies that each competitor acts to minimize his maximum
loss or maximize his minimum gain.

…continued
• Game theory
– was developed for the purpose of analyzing
competitive situations involving conflicting
interests.
– is one way to consider the impact of the strategies
of others on our strategies and outcomes.
– is the study of how optimal strategies are
formulated in situations of conflict and/or
competition.

In short the basic problem of playing a
game is to deal with the issue of
 How does our company select its
optimal strategy to get the largest pie in
a market/industry without the
knowledge of the competitors to earn the
best .

Basic Terminology in Game Theory
• Player - refers to participants in a game- it could be individuals, groups or
organizations
• Strategy is the decision rule by which a player determines his/her course
of action.
• Mixed strategy- happens when a player needs to use all or some of his
available strategies in some fixed proportion.
• Two-person zero-sum game & n-person zero-sum game – if their net
gain is zero in both cases
• Payoff- is the outcome of playing the game.
• A payoff matrix is a table showing the amount received by the player
named at the left hand side after all possible plays of the game where as the
payment is made by the player named at the top of the table.

…continued
• Nash Equilibrium: is a situation in which neither
player has an incentive to change strategy, given the
other player’s choice.
• Pure Strategy
– is the same strategies each player always follows regardless
of the other player’s strategy
– occurs when a player uses only one particular
strategy
– A dominant strategy = Pure strategy
• A saddle point is a situation where both players are
facing pure strategies.

Assumptions of Game Theory
• Each player has a finite number of possible strategies.
– The list may not be the same for each player
• Player A attempts to maximize gains and player B minimize
losses.
• The decision of both players are made individually prior to the
play with no communication between them.
• The decisions are made simultaneously and also announced
simultaneously .
– so that neither player has an advantage resulting from direct
knowledge of the player’s decision.
• Both players know not only possible pay-offs to themselves
but also of each other.

Classifications of Game Theory
• The game theory models can be classified into
several categories. Some important categories
are:
• Two-person & N-person games
– If the number of players is two, it is known as two-
person game.
– On the other hand, if the number of players is N, it
is known as N-person game.

…continued
• Zero sum & Non-zero sum game
– In a zero sum game, the sum of the points won equals
the sum of the points lost, i.e., one player wins at the
expense of the other.
– To the contrary, if the sum of gains or losses is not
equal to zero, it is either positive or negative, then it is
known as non-zero sum game.
• An example of non-zero sum game is the case of two
competing firms each with a choice regarding its advertising
campaign. In such a situation, both the firms may gain or
loose, though their gain or loss may not be equal.

…continued
• Games of Perfect and Imperfect information
– If the strategy of a player can be discovered by his
competitor, then it is known as a perfect information game.
– In case of imperfect information games no player has
complete information and tries to guess the real situation.
• Pure & Mixed strategy games
– If the players select the same strategy each time, then it is
referred to as pure strategy games.
– If a player decides to choose a course of action for each play
in accordance with some part probability distribution, it is
called mixed strategy game.

Pure Strategy
• Pure strategy.
– is a decision always to choose a particular one
course of action.
– In other words, if the best strategy for each player
is to play one particular strategy throughout the
game, it is called pure strategy.
– The simplest type of game is one where the best
strategies for both players are pure strategies.
– This is the case if and only if, the pay-off matrix
contains a saddle point.

Classical Example of Pure Strategy :
Prisoners’ Dilemma
• Two suspects are arrested for armed robbery. They are
immediately separated. If convicted, they will get a term of
10 years in prison. However, the evidence is not sufficient
to convict them of more than the crime of possessing
stolen goods, which carries a sentence of only 1 year.
• The suspects are told the following: If you confess and
your accomplice does not, you will go free. If you do not
confess and your accomplice does, you will get 10 years in
prison. If you both confess, you will both get 5 years in
prison.

Confess Don't Confess
Confess (5, 5) (0, 10)
Don't Confess (10, 0) (1, 1)
Individual B
Individual A
Payoff Matrix (negative values)

Confess Don't Confess
Confess (5, 5) (0, 10)
Don't Confess (10, 0) (1, 1)
Individual B
Individual A
Dominant Strategy
Both Individuals Confess
(Nash Equilibrium)

Examples of Pure Strategy- Two person Zero-sum
game
• Example: Illustrate the following game, where competitors A
and B are assumed to be equal in ability and intelligence. A
has a choice to strategy 1 or strategy 2, while B can select
strategy 3 or 4 . How much is the value of the game?
Competitor B
Competitor A
Strategy 3 Strategy 4
Strategy 1
+4 +6
Strategy 2
+3 +5

…continued
Solution:
• A maximize his gains where as B minimize his
losses since there is no negative flows .
• B loses the same amount as A gains which results
a zero-sum game
• The strategy that A uses is to ensure that his
average gain per play is at least equal to the
value of the game, where B ensures that his
average loss per play is no more than the game
value

…Solution
Competitor B
Competitor
A
Strategy 3 Strategy 4 Maximin
Strategy 1
+4 +6 +4
Strategy 2
+3 +5 +3
Minimax +4 +6

…Solution
• Since each player always follows the same strategy
regardless of the other player’s strategy
– thus the players are using a dominant strategy or Pure
strategy
• Since both players are facing pure strategies-
– thus there is a saddle point in this game.
• Using the rules of Minimax for the column player and
Maximin for the row player
– A saddle point occurs when the row player uses strategy 1
and the column player applies strategy 3.
– the value of the game is 4

…continued
• Example 3: consider the following game
– Determine the value of the game and
– The pure strategies for both player A and Player B if there is any
– Does it have a saddle point?
Player B
Plan P Plan Q
Player A
-3 3
-2 4
2 3

…Solution
• The negative values are a gain for player B where as the
positive values are a gain for player A
• When Minimax value = Maximin value – the corresponding
pure strategies are called optimal strategies and the game is
said to have a saddle point or equilibrium point.
• If minimax for the column ≠ maximin for the row – there is
no saddle point and the value of the game lies between them.

…Solution
Player B
Plan P Plan Q Maximin
Player A
Choice L -3 3 -3
Choice M -2 4 -2
Choice N 2 3 2
Minimax 2 4

Example 4: Consider the following game
Find
• The saddle point:
• Strategy of A and B:
• The Game value:
B
A
16 4 0 14 -2
10 8 6 10 12
2 6 4 8 14
8 10 2 2 0

…Solution:
• Use Maximin rule for the Player A and Minimax rule for
the player B
• The saddle point = row 2 and column 3
• Strategy A = playing row 2 and Strategy B = playing
column 3
• The game value = +6

Mixed Strategies
• Mixed strategy.
– is a decision to choose a course of action for each
play in accordance with some probability
distribution.
– In other words, if the optimal plan for each player
is to employ different strategies at different times,
we call it mixed strategy.

Mixed Strategy - The Rule of Dominance
• If no pure strategies exist, the next step is to eliminate
certain strategies (rows and/or columns) by dominance. The
resulting game can be solved by some mixed strategy
• The principle of dominance can be used to reduce the size
of games by eliminating strategies that would never be
played.
• A strategy can be eliminated if all its game’s outcomes are
the same or worse than the corresponding game outcomes
of another strategy

…continued
• Dominance rule for column(s): every value in the dominating
column must be greater than or equal to the corresponding
value of the dominated column(s).
– So, the dominant column must be deleted because players will never
play that strategy since it results a heaviest losses to one player and a
heaviest gains to the other player
• Rule of dominance for row(s): every value in the dominating
row must be less than or equal to the corresponding value of
the dominated row(s).
– So delete the dominant row in the same fashion

Example: Does this game have a saddle point?
Player Q
W B R
Player P
W 0 -2 7
B 2 3 6
R 3 -3 8

…Solution
• This payoff matrix has no saddle point and pure
strategy
– thus, the matrix should be reduced.
• Therefore, apply the rule of dominance:
– to reduce the matrix into a 2 x 2 or 2 x n or n x 2 and
– to identify the mixed strategies and the values of the
game using either the algebraic or graphical or
simplex algorithm.

… solution - rules of Dominance
• Hence, Player Q will not play strategy R since this
will result in heaviest losses to him and highest
gains to player P.
– Each elements in column R is greater than or
equal to the corresponding elements in the other
column(s).
– So, it has to be deleted since this column is
dominant and will never be played by player Q

...Solution
• For the same example, Each element in row
W is less than or equal to the corresponding
elements in the other row(s). So, it has to be
deleted. Thus, obtain:
W B
B 2 3
R 3 -3

Other rules of dominance
• A given strategy can be dominated if it is inferior/superior
to an average of two or more other strategies
– i.e., if a given column is greater than or equal to the
average of the other columns, that column is dominant
and will never be played. Hence, it has to be deleted
– If a given row is less than or equal to the average of the
other rows, that row is dominant and will never be
played. Hence, it has to be deleted so that you will
obtain a reduced matrix game

Game Theory-Using Algebraic technique
• In case where there is no saddle point and the
rule of dominance has been used to reduce the
game matrix, players will resort to mixed
strategies.
• In the case of mixed strategies we can
determine the strategies of players and value of
the game by using the following techniques:
• Algebraic or Graphical or simplex
algorithm

• Algebraic Method- the solution of the game is:
– A play’s (p, 1 - p)
• where: p = d – c/ (a + d) - (b + c)
– B play’s (q, 1 - q)
• where: q = d – b/(a + d) - (b + c)
– Value of the game, V =ad – bc/(a + d) - (b + c)

• Example: Two person zero-sum Game without
saddle point
• EX: In a game of matching coins, player A wins
br2 if there are two heads, wins nothing if there are
two tails and loses br1 when there are one head and
one tail.
– Determine the payoff matrix of each player,
– Best strategies for each player and
– The value of game

Solution: Matrix form of the game
Player B
H T
Player A
H 2 -1
T -1 0
• This game doesn’t have a saddle point and hence
there will never be a pure strategy to be taken

• Example 2 - Reduce the following game by dominance and
find the game value, the optimal strategy that both player A
and Player B chooses to take
PLAYER B
I II III IV
PLAYER
A
I 3 2 4 0
II 3 4 2 4
III 4 2 4 0
IV 0 4 0 8

…continued …Solution
• Row I is dominating- each elements in row one is less than
or equal to the corresponding value of row III
PLAYER B
I II III IV
PLAYER
A
I
II 3 4 2 4
III 4 2 4 0
IV 0 4 0 8

• Column I is dominant- each element in column I is greater
than or equal to the corresponding value of column III
PLAYER B
I II III IV
PLAYER
A
I
II 4 2 4
III 2 4 0
IV 4 0 8

• column II is dominant- each element in column II is greater
than or equal to the corresponding average value of column
III and IV
PLAYER B
I II III IV
PLAYE
R A
I
II 2 4
III 4 0
IV 0 8

• Row II is dominating- each element in row II
is less or equal the corresponding average
value of row III and IV
PLAYER B
I II III IV
PLAYER A
I
II
III 4 0
IV 0 8

Player Y
Y1 Y2
Payer X X1 4 3
X2 2 20
X3 1 1
• Example 3 -Solve the following game using algebraic
method and find the game value and the optimal strategies
for both players

…..Solution
Player Y
Y1 Y2
Player X
X1 4 3
X2 2 20
• Let the proportion of time that player X will play X1 is X and
X2 is 1-X and the proportion of time that player Y will play Y1
is y and Y2 is 1-y.

In class exercise
• Two firms A and B have for years been selling
computing product which forms a part of both
firms’ total sales. The marketing executive of
firm A raised the question: “what should be the
firms’ strategies in terms of advertising for the
product in question?” The market research
team of firm A developed the following data
for varying degree of advertising:

Cont...
• No advertising, medium advertising and large
advertising for both firms will result in equal market
share
• Firm A with no advertising: 40 percent of the market
with medium advertising by firm B and 28 percent of
the market with large advertising by firm B
• Firm A using medium advertising: 70 percent of the
market with no advertising by firm B and 45 percent
of the market with large advertising by firm B.

Cont...
• Firm A using large advertising: 75 percent of the market with
no advertising by firm B and 47.5 percent of the market with
medium advertising by firm B.
• Based on the forgoing information, answer the marketing
executives question
• What advertising policy should firm A pursue when
consideration is given to the above factors selling price, $4 per
unit, variable cost of product $2.5 per unit; annual volume of
30000units for firm A. Cost of annual medium advertising
$5000 and cost of annual of large advertising $15000. What
contribution, before other fixed costs, is available ?

Game Theory-Using Graphic method
Player B
Y1 Y2 Y3
Player A
X1 3 4 1
X2 -2 -3 3
• This method is applicable to only those games
in which one of the players has two strategies
only. Example !: Solve the following game
using graphic method

• Example 2: Consider the following payoff
matrix
Player A
Player B
B1
q
B2
1-q
A1 -2 4
A2 8 3
A3 9 0

• Consider the following payoff matrix
Player A
Expected
Pay-off
q=1 q=0
A1 -6q+4 -2 4
A2 5q+3 8 3
A3 9q 9 0

• Maximin = 4, Minimax = 3
• First, we draw two parallel lines 1 unit distance apart and mark a scale
on each.
– The two parallel lines represent strategies of player B in this
example.
• If player A selects strategy A1, player B can win –2 (i.e., loose 2 units)
or 4 units depending on B’s selection of strategies.
– The value -2 is plotted along the vertical axis under strategy B1 and
the value 4 is plotted along the vertical axis under strategy B2.
• A straight line joining the two points is then drawn. Similarly, we can
plot strategies A2 and A3 also.
• The problem is graphed in the following figure.

• The minimax point is situated just at the
intersection point between strategy A1 and
strategy A2. Thus, at this juncture the pay-off
matrix is reduced into 2 by 2 so that we can
use the algebraic technique.

• The reduced pay of matrix – after we
eliminated the strategy A3 is given below:
Player A
Player B
B1
q
B2
1-q
A1 -2 4
A2 8 3

In class exercise: solve the following
problem using graphical method
Player B
Y1 Y2 Y3 Y4
Player A
X1 19 6 7 5
X2 7 3 14 6
X3 12 8 18 4
X4 8 7 13 -1

Solution
• Follow the given below steps:
– Check the saddle point if there is, if so stop there
– Use the rule of dominance as much as to reduce
the matrix game
– Finally use graphic method to solve if you couldn’t
reduce it to 2x2 game matrix
– In this example we can apply the dominance rule

The End of the Fourth Chapter
Play Fair Game!

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