Game theory is the formal study of conflict and cooperation between interdependent decision-makers. It has applications in economics, political science, military strategy, biology, and computer science. Key concepts in game theory include Nash equilibrium, where no player can benefit by changing their strategy alone, and mixed strategies, where players randomize between pure strategies. Zero-sum games model situations where one player's gains equal the other's losses. The example given presents a payoff matrix modeling the strategic choices of two firms. It has a saddle point equilibrium where each firm chooses a pure strategy that maximizes their minimum payoff.

1. A PRESESNTATION ON
Game theory
Presented by Vijay Patel
Coerce Teacher - Dr. Hulas Pathak

2. What is game theory ?
• Game theory is the branch of decision theory
concerned with interdependent decisions and it
is the formal study of conflict and cooperation.
Game theoretic concepts apply whenever the
actions of several agents are interdependent.
These agents may be individuals, groups, firms, or
any combination of these. The concepts of game
theory provide a language to formulate,
structure, analyze, and understand strategic
scenarios.

3. Application of Game theory
• The major applications of game theory are to -
economics, political science (on both the national and
international levels), tactical and strategic military
problems, evolutionary biology, and, most recently,
computer science.
• There are also important connections with account- ing,
statistics, the foundations of mathematics, social
psychology, and branches of philosophy such as
epistemology and ethics. Game theory is a sort of
umbrella or ‘‘unified field’’ theory for the rational side
of social science, where ‘‘social’’ is interpreted broadly,
to include human as well as non-human players
(computers, animals, plants).

4. History of Game theory
• The earliest example of a formal game-theoretic
analysis is the study of a duopoly by Antoine Cournot
in 1838. The mathematician Emile Borel suggested a
formal theory of games in 1921, which was furthered
by the mathematician John von Neumann in 1928 in a
“theory of parlor games.” Game theory was established
as a field in its own right after the 1944 publication of
the monumental volume Theory of Games and
Economic Behavior by von Neumann and the
economist Oskar Morgenstern. This book provided
much of the basic terminology and problem setup that
is still in use today.

5. • Problem. Let us view the problem from firm
A’s view point. Suppose that the firm has
three strategies, of selling its fruit in three
different package ,I,II,III, under consideration.
Suppose, also that its competitor ,firm Bis
considering four different market plases,
Amritsar, Panthankot, Ludhiana and Jabalpur (
shown by A,P,L,and J).
• consider now any pair of strategies open to
the two players. For example, firm A
employing strategy I, and firm B using strategy
J. such a pair of decision will determine A’s
market share. It will, let us say, result in a 9
per cent share of the market for this firms.

6. • This figer is called A’s pay- off. This information is
summarized in a pay –off matrix as given in figer 1.
this matrix show what a will receive as a result of
each possible combination of strategy chois by
himself and by his competitor.
B’ s strategy Row minimum
A P L J
A;s I 50 90 18 25 18
Stategy II 37 15 10 82 10
III 60 20 4 12 4
Column
Maximum 60 90 18 82 -

7. • We see from fig 1 that the number 10
indicates that if A choose strategy II and B
choose strategy L, A will receive a pay –off of
10. in this game what ever percentage is left,
that goes to B. this means, B’s pay –off from
this strategy combination is 90 percent.

8. Mixed strategies
• A game in strategic form does not always have a Nash
equilibrium in which each player deterministically
chooses one of his strategies. However, players may
instead randomly select from among these pure
strategies with certain probabilities. Randomizing one’s
own choice in this way is called a mixed strategy. Nash
showed in 1951 that any finite strategic-form game has
an equilibrium if mixed strategies are allowed. As
before, an equilibrium is defined by a (possibly mixed)
strategy for each player where no player can gain on
average by unilateral deviation. Average (that is,
expected) payoffs must be considered because the
outcome of the game may be random.

9. • The most obvious, and natural, interpretation uses objective
chances. What it means to play the strategy 〈x,1−x〉 is to grab
some chance device that goes into one state with chance x,
and another state with chance 1−x, see which state it goes
into, then play the s1 if it is in the first state, and s2 if it is in
the second state. Consider, for instance, the game Rock,
Paper, Scissors, here represented as
Game Paper Scissors Rock
rock 0, 0 -1, 1 1, -1
Paper 1, -1 0, 0 -1, 1
Scissors -1, 1 1, -1 0, 0
• For each player, the equilibrium strategy is to
play〈1/3,1/3,1/3〉. (Exercise: Verify this!) The chance
interpretation of this mixed strategy is that the player takes
some randomising device, say a die, that has a 1/3 chance of
coming up in one of three states. Perhaps the player rolls the
die and plays Rock if it lands 1 or 2, Paper if it lands 3 or 4,
Scissors if it lands 5 or 6.

10. Zero-sum games and computation
• The extreme case of players with fully opposed interests is
embodied in the class of two players
• zero-sum (or constant-sum) games. Familiar examples range from
rock-paper scissors to many parlor games like chess, go, or
checkers.
• A classic case of a zero-sum game, which was considered in the
early days of game theory by von Neumann, is the game of poker.
The extensive game in Figure 10, and its strategic form in Figure 11,
can be interpreted in terms of poker, where player I is dealt a strong
or weak hand which is unknown to player II. It is a constant-sum
game since for any outcome; the two payoffs add up to 16, so that
one player’s gain is the other player’s loss. When player I choose to
announce despite being in a weak position, he is colloquially said to
be “bluffing.” This bluff not only induces player II to possibly sell
out, but similarly allows for the possibility that player II stays in
when player I is strong, increasing the gain to player I.

11. B’ s strategy Row minimum
A P L J
A;s I 50 90 18 25 18
Stategy II 37 15 10 82 10
III 60 20 4 12 4
Column
Maximum 60 90 18 82 -
From table we can conclude that if a choose the ith strategy ( that is i-th row ) and player B
choose the j-th strategy (that is J-th column), then the element aij is assumed to represent
the pay off from player B to player A. thus, if aij is a positive number it represents payment
of A to B.
The problem posed in table is an example of what is called a strictly determined game
since it has a solution of pure strategies. The amount 18= a13) is the minimum amount in the
first row and the maximum amount in the third column. This is called an equilibrium value
or a saddle point. In this case, the value of the game is equal to the saddle value.