Definition, Terms Used, Types of Strategies,Odd method, Saddle Point, Dominance method, Graphical Method.

1. Jagdeep Singh Jagdev

2. Definition of game theory
 The branch of mathematics concerned with the
analysis of strategies for dealing with competitive
situations where the outcome of a participant’s choice
of action depends critically on the actions of other
participants. Game theory has been applied to
contexts in war, business, and biology

3. A few examples of competitive and
conflicting decision environment
 Pricing of products, where sale of any product is
determined not only by its price but also by the price
set by competitors for a similar product
 The success of any TV channel programme largely
depends on what the competitors presence in the same
time slot and the programme they are telecasting
 The success of an advertising/marketing campaign
depends on various types of services offered to the
customers.

4. Terms Used in Game theory
Number of Players
• How many players are there?
• If a game involves only two players (competitors),
then it is called a two-person game. However, if the
number of players is more, the game is referred to as n-
person game
Strategies: The strategy for a player is the list of all
possible actions (moves or courses of action) that he will
take for every payoff (outcome) that might arise. It is
assumed that the rules governing the choices are known
in advance to the players.

5.  Payoffs are the consequences for each player for every
possible profile of strategy choices for all players.
 Zero-sum (or constant-sum) game : one player's
winnings are the others' losses, so the net gain is zero
across all players
 Optimal Strategy: The particular strategy by which a
player optimises his gains or losses without knowing
the competitor's strategies.
 Value of game: The expected outcome per play when
players follow their optimal strategy.

6. Assumptions of the Game theory
 Each player has available to him a finite number of
possible strategies (courses of action). The list may not
be same for each player.
 Player A attempts to maximize gains and player B
minimise losses.
 The decision of both players are made individually
prior to the play with no communication between
them.

7.  The decisions are made simultaneously and also
announced simultaneously so that neither player has
an advantage resulting from direct knowledge of the
other player’s decision.
 Both the players know not only possible payoffs to
themselves but also of each other.

8. Note
 By convention, the payoff table for the player whose
strategies are represented by rows (say player A) is
constructed.

9. Types of Strategies
• Pure Strategy
 It is the decision rule which is always used by the
player to select the particular strategy. Thus, each
player knows in advance of all strategies out of which
he always selects only one particular strategy
regardless of the other player’s strategy, and the
objective of the player is to maximize profit or
minimize losses.

10.  Mixed strategy
 Courses of action that are to be selected on a
particular occasion with some fixed probability are
called mixed strategies.

11. Pure Strategy
 Maximin – minimax principle
 Maximin Criterion: The player who is maximizing his
outcome or payoff finds out his minimum gains from
each strategy (course of action) and selects the
maximum value out of these minimum gains.
 Minimax Criterion: In this criterion the minimizing
player determines the maximum loss from each
strategy and then selects with minimum loss out of the
maximum loss list

12. Example 1
 For the game with payoff matrix:
 Determine the best strategies for players A and B. Also
determine the value of game. Is this game (i) fair? (ii)
strictly determinable?

13. Example 1
 Player A adopts A1 strategy.
 Player B adopts B3 strategy.
 Value of game V = -2 Not fair but strictly
determinable.

14. Saddle Point or Equilibrium Point
In a payoff matrix the value, which is
the smallest in its row and the largest
in the column, is called the saddle
point.

15. Example 2
 A company management and the labour union are
negotiating a new three year settlement. Each of these
has 4 strategies:
 (i) Hard and aggressive bargaining
 (ii) Reasoning and negotiating approach
 (iii) Legalistic strategy
 (iv)Conciliatory approach

16.  The cost to the company are given for every pair of
strategy choice
 What strategy will the two sides adopt? Also
determine the value of the game.

17.  The company will adopt strategy III And union will
always adopt strategy I. Value of game V = 12

18. Mixed Strategies
 A method of playing a matrix game in which the player
attaches a probability weight to each of the possible
options, the probability weights being nonnegative
numbers whose sum is unity, and then operates a
chance device that chooses among the options with
probabilities equal to the corresponding weights.

19. 1. Odds Method (2X2 matrix)
 If payoff matrix for player A is given by
 The following formulae are used to find the value of
game and optimal strategies:

20.  Value V ={a1 ( b1 – b2 ) + b1 ( a1 – a2 )}/{( b1 -b2 )+( a1 –
a2 )}
 Prob. of A1 = ( b1 – b2 )/ {( b1 -b2 )+( a1 – a2 )} Prob. of
A2 = ( a1 – a2 ) / {( b1 -b2 )+( a1 – a2 )} Prob. of B1 = ( a2
– b2 ) / {( a2 – b2 )+( a1 – b1 )} Prob. of B2 = ( a1 – b1 ) /
{( a2 – b2 )+( a1 – b1 )}

21. Example 2
 Two players A and B are involved in a game of
matching coins. When there are both heads, player A
wins 100 points and wins 0 when there are two tails.
When there is one head and one tail, B wins 50 points.
Determine the payoff matrix, the best strategy for both
players A and B. Find the value of game to A.

22.  Value V = {100 (50) + (-50)(150)} / {(-50) + (150)} = -12.5
 Prob. of A selecting strategy H = 50/200 = 1/4 Prob. of
A selecting strategy T = 150/200 = 3/4 Prob. of B
selecting strategy H = 50/200 = 1/4 Prob. of B selecting
strategy T = 150/200 = 3/4

23. Dominance Method
 Rule 1. If all the elements in a row (say ith row) of a
payoff 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 or
in other words the ith strategy is dominated by the jth
strategy

24.  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 other words the rth strategy is
dominated by the sth strategy.
 • Rule 3. A pure strategy may be dominated if it is
inferior to average of two or more other pure stategies.

25. Example 3
 Reduce the following game by dominance method and
find the game value:

31. Graphical Method
The graphical method is useful
for the game where the payoff
matrix is of the size m X 2 or 2 X
n

32. Consider the following pay-off
matrix
Player A
Player B
B1 B2
A1 -2 4
A1 8 3
A1 9 0

33. Solution.
The game does not have a saddle
point as shown in the following
table
Player A
Player B Minimu
m
Probabili
ty
B1 B2
A1 -2 4 -2 q1
A2 8 3 3 q2
A3 9 0 0 q3
Maximum 9 4
Probability p1 p1

34.  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.
 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.


36.  The lowest point V in the shaded region indicates the
value of game. From the above figure, the value of the
game is 3.4 units. Likewise, we can draw a graph for
player B.
 The point of optimal solution (i.e., maximin point)
occurs at the intersection of two lines:
 E1 = -2p1 + 4p2 and
E2 = 8p1 + 3p2

37.  Comparing the above two equations, we have
 -2p1 + 4p2 = 8p1 + 3p2
Substituting p2 = 1 - p1
-2p1 + 4(1 - p1) = 8p1 + 3(1 - p1)
p1 = 1/11
p2 = 10/11
 Substituting the values of p1 and p2 in equation E1
 V = -2 (1/11) + 4 (10/11) = 3.4 units

38. Graphic Method (mX2 or 2Xn)

40. Good management is the art of
making problems so interesting
and their solutions so constructive
that everyone wants to get to work
and deal with them. -Paul Hawken