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1. PRESENTATION
ON
GAME THEORY
PRESENTED TO PRESENTED BY
Dr. HULAS PATHAK OM PRAKASH SONVANEE
PROFESSOR M.Sc.(AG.) PRE. YEAR
Deptt. Of Argil. Economics Deptt. Of Argil. Economics

2. GAME THEORY
INTRODUCTION-
 Many practical problem require decision-making in a
competitive situation where there are two or more
opposing parties with the conflicting interests and
where the action of one depends upon the one taken
by the opponent.
 For example, candidates for election, advertising and
marketing campaigns, etc.
 In a competition situation the course of action
(alternatives for each competitor may be either finite
or infinite.
 Competitive situation will be called a ‘Game.

3. If it has the following properties :
 There are a finite number of competitors
(participants) called players.
 Each player has a finite number of strategies
(alternatives) available to him.
 A play of game takes place when each player
employs his strategy .
 Every game results in an outcome, e.g., loss or gain a
draw, usually called payoff, to some player.

4. TWO-PERSON ZERO –SUM GAME
 When there are two competitors playing a game , it
is called a two-person game .
 In case the number of competitors exceeds two ,say
n, then the game is termed as a ʻn-person game ʼ.
 Game having the zero-sum character that the
algebraic sum of gains and losses of all the players is
zero are called zero-sum games.

5. Example-
 Consider a two-person coin tossing game. Each
player tosses an unbiased coin simultaneously .
 Player B pays Rs. 7 to A, if {H,H} occurs and Rs. 4, if
{T, T} occurs;
 Otherwise player A pays Rs. 3 to B . This two-person
game is a zero-sum game, since the winnings of one
player are the losses for the other.
 Each player has choice from amongst two pure
strategies H and T. if we agree conventionally to
express the outcome of the game in term of the
payoff to one player only,

6.  say A, then the above information yield the following
payoff matrix in terms of the payoff to the player A.
 Clearly, the entries in B ʼ s payoff matrix will be just the
negative of the corresponding entries in A ʼs payoff
matrix so that the sum of payoff matrices for player A
and B is ultimately a null matrix.
 We generally display the payoff matrix of that player
who is indicated on the left side of the matrix.

7. For example-
A ʼs payoff matrix may be displayed as below:
Player B
H H
Player A H
T
7 -3
-3 4

8. GAMES WITHOUT SADDLE POINT- MIXED
STRATEGIES
 As determining the minimum of column maxima and
the maximum of row minima are two different
operation, there is no reason to expect that they
should always lead to unique payoff position –the
saddle point.
 In all such cases to solve games, both the players
must determine an optimal mixture of strategies to
find a saddle point.

9.  The optimal strategy mixture for player may be
determined are called mixed strategies because they
are probabilistic combination of available choices of
strategy.
The value of game obtained by the use of mixed
strategies represents which least player A can
expect to win and least which player can lose.
The expected payoff to a player in a game with
arbitrary payoff matrix (a ij) of order m × n is
defined as:

10. E ( p, q) = pi aij qj = pT Aqm
Where p and q denote the mixed strategies for
player A and B respectively.
Maxi mine-minimax criterion – consider an m× n
game (a ij) without any saddle point, i.e.,
strategies are mixed. let p 1,p2,… pm be the
probabilities with which player A will play his
moves A1, A2 ,… , Am

11. respectively ; and let q1, q2, …, qn be the
probilities with which player B will play his
moves B1,B2 ,… , Bn respectively. Obviously , pi
≥ 0 (i= 1,2,…,m), qj ≥ 0 (j=1,2,…, n), and
p1+p2+…+pm = 1; q1 + q2+…+qn = 1.
The expected payoff function for player A,
therefore, will be given by
E ( p, q) = pi aij qj

12.  Making use of maximin- minimax criterion, we have
 For player A.
V= max. min. E(p,q ) = max.p (min.j ( ) )
= max.p[min.j { } ]
 Here max. ( ) denoted the expected gain to
player A uses his ith strategy.

13.  The relationship v_ ≤ v- hold good general and when
pi and qj correspond to the optimal strategies the
relation holds in `equation’ sense and the expected
value for both the players become equal to the
optimum expected value of game.
 Definition- a pair of strategies (p,q)for which v_=v- =
v is called a saddle point of E(p,q).
 Theorem 17-2. For any 22 two-person zero-sum
game without any saddle point having the payoff
matrix for player A

14. The optimum mixed strategies
SA = and SB =
Are determined by
= ,
Where p1+p2= 1 and q1+q2 =1 .the value v of
the game to A is given by

15. V=
GENERAL SOLUTION OF m× n RECTENGULAR
GAMES
In a rectangular game with an m n payoff matrix,
if there dose not exist any saddle point and also
it is not possible to reduce the size of the game
to m2 or 2 payoff matrix by using the concept of
dominance ,the following method are generally
used to solve the game:

16. Linear programming method, and
Iterative method for approximate solution.
Linear programming method- to illustrate the
connection between a game problem and a
linear programming problem, let us consider an
mn payoff matrix (aij) for player A. let
Sa = and Sn = where
be the mixed strategies for the two players respectively.
Then, the expected gains gj ( j=1,…,n) of player A against
Bs pure strategies will be

17. g1 = a11 p1 +a21 p2 +…+am1 pm
g2 = a12 p1 +a22 p2+…+am2 pm
gn = a1np1 + a2np2+ …+amnpm
And the expected losses ∫I ( i=1, …, m) of player B
against A’ s pure strategies will be
∫I = q11 q1 +a12 q2 +…+a1m qn
∫2 = a21 q1 +a22 q2+…+a2n qn

18. the objective of player A is to select pi(i= 1,2,…,m) such
that he can maximize his minimum expected gains; and
the player B desires to select qj (j= 1,2,…,n) that will
minimize his expected losses.
Thus, if we let u= min j ;
 The problem of two players could be written as:
Player A
Maximize= minimize subject to the constraints:
≥ µ and ∑pi= 1, pi ≥ 0 (i= 1,2,…,m)

19. Player B
maximize v = minimize = subject to the
constrints:
≤ v and ∑qj= 1, qj ≥ 0 (j= 1,2,…,n)
 Assuming that µ ˃ 0 and v ˃ 0, we introduce new variables
defined by
 Pi’ = pi/µ and qj; = qj/ v (i= 1,2,…,m; j= 1,2,…,n)
Then, the pair of linear programming problems can be re-written as:

20. Player A
minimize Po= p1’ + p2’ + … +pm’ subject to constraints:
a1j p1’ + a 2j p 2’ + … + a mj p m’ ≥ 1
p i’ ≥ 0 (i= 1,2,…,m; j= 1,2,…,n)
Player B
maximize q o = q1’ + q2’ + … +qm’ subject to constraints:
q i1 q1’ + q i2 q 2’ + … + a in q n’ ≤ 1

21. Sample problem -
 Solve the following game by linear programming technique:
Player B
Player A
Solution. Since some of the entries in the payoff matrix are negative,
we add a suitable constant to such of the entries to ensure then all
position .thus, adding a constant c=4 to each element, we get the
following revised payoff matrix :
Player B
 Player A

22. let the strategies of the players be
S A = S B = ,
Where p1+ p2+p3 =1 and q1+q2+q3 =1.
The linear programing formula for the ′ two players’ are:
For player A
maximize v= minimize = x1+x2+x3 subject to constants:
5x1+7x2+10x3 ≥ 1, 3x1+9x2+6x3 ≥ 1, 7x1+x2+2x3 ≥1 ; and x j ≥ o
, (j=1,2,3)

23. For player B
maximize v= minimize y1+y2+y3 subject to
constants:
5y1+3y2+7y3 ≤ 1, 7y1+9y2+y3 ≤ 1, 10y1+6y2+2y3 ≤ 1 ; and
y j ≥ o
(j=1,2,3)
,
where xj = pj /u, (j=1,2,3) yj /v, ( j =1,2,3); u= minimum
expected gain to A and v=minimum expected loss to B.
Let us solve the problem for player B. By introduction slack
variables s 1 ≥0, s2 ≥0 and s3 ≥0; the iterative simplex
tables are:

24. Initial iteration,introduction y3 and drop y4
Cb Yb Xb Y1 Y2 Y3 Y4 Y5 Y6
0 Y4 1 5 3 ⑦ 1 0 0
0 Y5 1 7 9 1 0 1 0
0 Y6 1 10 6 2 0 0 1
0 -1 -1 -1 0 0 0

25. First iteration, introduction y2 and y5
Cb yb xb y1 y2 y3 y4 y5 y6
1 Y3 1/7 5/7 3/7 1 1/7 0 0
0 Y5 6/7 44/7 60/7 0 -1/7 1 0
0 Y6 5/8 60/8 36/7 0 -2/7 0 1
1/v 1/7 -2/7 -4/7 0 1/7 0 0

26. Strategies available to them. However, in real
world situations, it is only but rarely that each
of the persons would have complete
knowledge about all the Strategies available
to his competitor, as also of the exact payoff
values associated with various combinations.
Business managers do not always adopted
conservative approach in their decision and
prefer to take risks.

27. • Further in a given situation, there are likely to
be more than two participants and the sum of
gains and losses of the opponents may not be
equal to zero. several extensions of the two
person zero games are possible. like the non-
zero sum games, n-person games ,games with
uncertain payoffs, games