Game theory

1. KULDEEP MATHUR
M.B.A.
JIWAJI UNIVERSITY GWALIOR
GAME THEORY:

2. Game is defined as an activity between two or more
persons involving activities by each person
according to a set of rules , at the end of which each
person receives some benefit or satisfaction or
suffers loss.
The set of rules define the game .
Going through the set of rules once by the
participants defines a play.

3. Game theory attempts to study “decision making
in situations where two or more intelligent and
rational opponents are involved under conditions
of conflict and competition. ”

4. Terminology:
Strategy :
A strategy for a player is defined as a set of rules
or alternative courses of action available to him in
advance , by which player decides the course of
action that he should adopt.
TYPES:
 Pure strategy
 Mixed strategy
 Optimal strategy

5. Pure strategy:
pure strategy is a decision rule always to select a
particular course of action.
Mixed strategy:
Mixed strategy is a selection among pure strategies with
fixed probabilities.
Optimal strategy:
A course of action or play which puts the player in the
most preferred position , irrespective of the strategy of
his competitors is called an optimal strategy.

6. Number of players:
If a game involves n players , then it is called a n-
person game.
Payoff:
Outcomes of a game when different alternatives
are adopted by the competing players are called
the payoffs.

7. Payoff matrix:
The payoffs in terms of gains or losses when
player select their particular strategies can be
represented in the form of a matrix called the
payoff matrix.

8. Zero-sum games:
If the players make payments only to each other
i..e
“loss of one is the gain of others ” , then the
competitive game is called zero-sum game.

9. Two person zero-sum game:
A game of two person in which the gains of one
player are the losses of the other player is called
a two person zero-sum game.

10. Example:
Player A: gainer
Player B: loser
All the payoffs are assumed in terms of player A.
aij : the payoff which player A gains from the player
B chooses strategy Ai and player B chooses
strategy Bj
 Negative entry in the table means that the
payments are to be made by A to B.

11. Player A: Maximizing player
Player B: Minimizing player
Maximin Principle:
The player A decides to play that strategy which
corresponds to the maximum of the minimum gains
for his different courses of action.
Minimax Principle:
The player B would like to play that strategy which
corresponds to the minimum of the maximum losses
for his different courses of action.

12. Saddle point:
The maximin value = The minimax value
We get a saddle point.
The saddle point is the solution of the game .
Therefore ,
The strategies of A and B corresponding to saddle
point are the optimal strategies of A and B.

13. Value of the game:
It is the expected payoff of play when all the
players of the game follow their optimum
strategies.
The game is called fair if the value of the game is
zero and strictly determinable if it is non zero.

14. Problem:
Player A can choose his strategies from (A1,A2,A3) only .while
B can choose from the set (B1,B2) only. The rules of the game
state that the payments should be made in accordance with the
selection of strategies :
Strategies pair selected Payment to be made
(A1,B1) Player A pays Re 1 to player B
(A1,B2) Player B pays Rs 6 to player A
(A2,B1) Player B pays Rs 2 to player A
(A2,B2) Player B pays Rs 4 to player A
(A3,B1) Player A pays Rs 2 to player B
(A3,B2) Player A pays Rs 6 to player B
What strategies should A and B play in order to get the optimum
benefit of the play

15. Rectangular games without saddle
point:
The game has no saddle point
The concept of optimum strategies can b
extended to all matrix games by introducing a
probability with choice and mathematical
expectation with payoff.

16. Let player A chooses a particular activity i such that
1<i<m. then the set x=(xi, 1<i<m) of probability
constitute the strategy of A.
Similarly,
The set y=(yi, 1<i<n) of probability constitute the
strategy of B.
Thus the vector x=(x1,x2,…xm) of the non negative
numbers satisfying x1+x2+…+xm =1 is called mixed
strategy of the player A.
Similarly, the vector y=(y1,y2,…yn) of the non negative
numbers satisfying y1+y2+…+yn =1 is called mixed
strategy of the player B.

17. The mathematical expectation of the payoff
function E(x , y) in a game whose payoff matrix is
Vij is defined by
E(x , y)=∑∑ xi vij yi
where x and y are the mixed strategies of the
player A and player B.
Thus,
The player A chooses x so as to maximize his
minimum expectation and the player B should
chooses y so as to minimize the player A ‘s
greatest expectation.
The player A tries max min E(x,y)
The player B tries min max E(x,y)

18. Strategic saddle point:
If
min max E(x,y)=E(xo,yo)=max minE(x,y)
Then (xo,yo) is called the strategic saddle point of
the game where xo and yo define the optimum
strategies and v=E(xo,yo) is the value of the
game.

19. Principle of dominance:
The concept of dominance is especially useful for the
evaluation of two person zero-sum games where a
saddle point does not exist.
RULE :
 When all element in a row of a payoff matrix are less
than or equal to the corresponding elements in
another row, then the former row is dominated by the
latter and can be deleted from the matrix.

20.  When all element in a column of a payoff matrix
are greater than or equal to the corresponding
elements in another column, then the former row
is dominated by the latter and can be deleted
from the matrix.
 A pure strategy may be dominated if it is inferior
to average of two or more other pure strategies.

21. Graphical method for 2×n and
m×2games:
The graphic method consists of two graphs:
(i) The payoff available to player A versus his
strategies, and
(ii) The payoff available by player B versus his
strategies

22. Graphical method for 2×n:
Graph for the player A:
 The highest point on the lower boundary of these
lines will give maximum expected payoff among
the minimum expected payoffs on the lower
boundary and the optimal value of the probability
p1 and p2.
 Now, the two strategies of player B corresponding
to these lines passes through the maximin point
can be determined.
 It helps in reducing the size of the game (2×2).

23. Graphical method for m×2:
Graph for the player B:
 The lowest point on the upper boundary of these
lines will give minimum expected payoff among
the maximum expected payoffs on the upper
boundary and the optimal value of the probability
q1 and q2.
 Now, the two strategies of player A corresponding
to these lines passes through the minimax point
can be determined.
 It helps in reducing the size of the game (2×2).