This document provides an overview of game theory concepts including the Prisoner's Dilemma, Nash Equilibrium, payoff matrices, pure and mixed strategies, and zero-sum games. It discusses terminology used in game theory like the number of players, strategies, and payoffs. Examples are provided to illustrate concepts like saddle points, dominance rules, and solving 2x2, 2x3, and 4x2 games using graphical and algebraic methods.

1. GAME THEORY
“Trust None. For oaths are straws, men’s faiths are wafer-cakes.” -
William Shakespeare (Henry V)
Prisoners’ Dilemma
Sanya and Cinque: Two Players (Two robbers of Hibernia Savings
Bank)
Payoff Matrix
Bill
Confess Not Confess
Al
Confess 3, 3 1, 10
Not Confess 10, 1 2, 2

2. NASH EQULIBRIUM
Nash Equilibrium (Game theory) - A stable state
of a system that involves several interacting
participants in which no participant can gain by
a change of strategy as long as all other
participants remain unchanged.

3. Game Theory is basically a strategic interaction between mutually
aware players.
It is based on the fact that “You are self-interested and selfish”
So is everyone else....
Terminology
a) No. of players: Two-person game and n-person game
b) Sum of gains and losses: zero-sum game and non-zero sum game
c) Strategy: It is set of rules which a person should adopt at each play
Pure-strategy - If a player knows what course of action he is going to
adopt based on the knowledge about opponent’s course of action,
then he always selects a particular course of action (known with
certainty)
Mixed-strategy - When both the players are guessing as to which
course of action is to be selected on a particular occasion with certain
probability then it is a mixed-strategy game

4. d) Two-person, Zero-sum games: A game with two players where
the loss of one player is equal to gain of the other with net gain
being zero.
e) Payoff Matrix:Row designations are Player A’s strategies and
Column designations are Player B’s strategies.
Player B’s Strategies
B1 B2 ............ Bn
Player A’s
Strategies
A1 a11 a12 ............ a1n
A2 a21 a22 ............ a2n
.
. .
.
.
.
............
..
..
Am am1 am2 ............ amn

5. Pure strategies
Maximin and Minimax Strategies
Payoff matrix for A Firm B’s Strategy
B
1 2 -2
Firm A’s
A 2 -3 -4 Strategy
0 -1 2
Player B
Solve the following:
Player A
1 2
1 20 -6
2 8 2
3 -4 1
B1 B2 B3 B4
A1 1 7 3 4
A2 5 6 4 5
A3 7 2 0 3

6. saddle point
If Maximin = Minimax = Value of the game, then we have a
Saddle Point. Saddle Point of a payoff matrix is the position of an
element which is minimum in its row and maximum in its
column.
Player B
Player A
Player B
Player A
8 7 15 12
9 14 8 10
10 12 14 13
-7 -4
7 -3
8 -2

7. 2) Player B
Player A
3) Firm B
Firm A
B1 B2 B3 B4 B5
A1 -1 0 0 5 3
A2 3 2 3 2 2
A3 -4 -3 0 2 6
A4 5 3 -4 2 6
B1 B2 B3 B4 B5
A1 3 -1 4 6 7
A2 -1 8 2 4 12
A3 16 8 6 14 12
A4 1 11 -4 2 1

8. 4) Assume that two firms are competing for the market share for a
particular product. Each firm is considering what promotional
strategy to employ for the coming period. Assume that the
following payoff matrix describes the increase in market share of
Firm A and the decrease in market share for Firm B. Determine the
optimal strategies for each firm.
a) Which firm would be winner, in terms of market share?
b) What might the two firms do to maximize their profits or
minimize their losses?
Firm B
No
Promotion
Moderate
Promotion
More
Promotion
Firm A
No
Promotion
5 0 -10
Moderate
Promotion
10 6 2
More
Promotion
20 15 10

9. Mixed strategies
Consider this problem
20 8 -6
12 10 2
3 5 6
1/31/3 2/32/3
1/21/2
1/21/2

10. Algebraic Method (2 x 2) games
5) Solve the following 2 x 2 games without saddle points
B B
A A
6) Two players A & B without showing each other, put a coin on a
table, with head or tail up. A wins Rs.8 when both coins show
head and Re.1 when both are tails. B wins Rs.3 when the coins do
not match. Given the choice of being a matching player (A) and
non-matching player (B), which one would you choose and what
would be your strategy?
5 1
3 4
6 -3
-3 0

11. The Rules of Dominance: If a row or column is dominated by
another row or column in terms of pay-offs, then the dominated
row or column can be deleted to reduce a ‘n x m’ matrix to a ‘2 x 2’
matrix.
Player B
For ex.:
Player A
Player B
Player A
I II III
I -4 6 3
II -3 -3 4
III 2 -3 4
B1 B2 B3 B4
A1 3 2 4 0
A2 3 4 2 4
A3 4 2 4 0
A4 0 4 0 8

12. 7) In an election campaign, the strategies adopted by the ruling
and opposition party along with payoffs (ruling party’s % share in
votes polled) are given below. Assume a zero-sum game. Find
optimum strategies for both parties and expected payoff to ruling
party.
Opposition Party’s Strategies
Campaigning one
day in each city
Campaigning two
days in each city
Campaigning two
days in large rural
area
Ruling
Party’s
Strategies
Campaigning one
day in each city
55 40 35
Campaigning two
days in each city
70 70 55
Campaigning two
days in large
rural area
75 55 65

13. 8) A steel company is negotiating with its union for revision of
wages to its employees. The management, with the help of a
mediator, has prepared a payoff matrix shown below. Plus sign
indicates a wage increase, while a negative sign indicates a wage
decrease. Union has also constructed a table which is comparable
to that developed by management. What strategies are best for the
management and union and what is the value of the game?
Union Strategies
U1 U2 U3 U4
Steel Co.
Strategies
C1 2.5 2.7 3.5 -0.2
C2 2.00 1.60 0.80 0.80
C3 1.40 1.20 1.50 1.30
C4 3.00 1.40 1.90 0

14. Arithmetic Method
9) Solve the following game:
10) Two breakfast food mfgers, ABC and XYZ are competing for an increased
market share. The pay-off matrix, shown in the following table, shows the
increase in market share for ABC and decrease in market share for XYZ.
Simplify the problem by rule of dominance and find optimum strategies and
value of game.
Player B
B1 B2 B3
Player
A
A1 1 7 2
A2 6 2 7
A3 5 2 6
Give coupons Decrease Price
Maintain
Present Strategy
Increase
advertising
Give coupons 2 -2 4 1
Decrease Price 6 1 12 3
Maintain Present
Strategy
-3 2 0 6
Increase
advertising
2 -3 7 1

15. graphical method
Graphical method is used to convert a m x 2 and 2 x n matrices to
a 2 x 2 matrix.
For a 2 x n matrix problem, we have to find a maximin point and
for a m x2 matrix problem we have to find a minimax point.
2 x n matrix problem B’s Strategies
11) Solve the 2 x 3 game graphically
A’s strategies
12) Solve the 4 x 2 game graphically
m x 2 matrix problem
A
I II III
I 1 3 11
II 8 5 2
I II
I 2 4
II 2 3
III 3 2
IV -2 6

16. 13) A soft drink company calculated the market share of two
products against its major competitor having three products and
found out the impact of additional advertisement in any one of its
products against the other.
What is the best strategy for
the company as well as the
competitor? what is the
payoff obtained by the
company and the competitor
in the long run? Use graphical method to obtain the solution.
Competitor B
B1 B2 B3
Company
A
A1 6 7 15
A2 20 12 10