The document discusses different classifications and concepts related to game theory, including: 1) Games can be zero-sum, where one player's gains equal another's losses, or non-zero-sum. They can involve 2 players or more. 2) Strategies can be pure, where players always choose the same action, or mixed, where players vary their actions randomly. 3) A payoff matrix outlines the potential payoffs for each combination of strategies between players in a 2-person, zero-sum game. The value of the game is when the maximin and minimax values are equal.

1. Yasir Hashmi 01

2. Game theory

3. CLASSIFICATION OF GAMES
• Zero Sum Games: Most instances involve repetitive
solution. The winner receives the entire amount of the
payoff which is contributed by the loser.
• Non-Zero Sum Games: The gains of one player differ
from the losses of the other. Some other parties in the
environment may share in the gain or losses.

4. CLASSIFICATION OF GAMES
• Two-Person Game: A game with 2 number of players.
• N-Person Game: A game with N number of players,
where >2.

5. CLASSIFICATION OF GAMES
• Pure-Strategy Game: A game in which the best
strategy for each player is to play one strategy
throughout the game.
• Mixed-Strategy Game: A game in which each player
employs different strategies at different times in the
game.

6. PAY OFF:
• It is the sum of gains and losses from the game that
are available to the players.
• If in a game sum of the gains to one player is exactly
equal to the sum of losses to another player, so that
the sum of the gains and losses equals zero then the
game is said to be a zero-sum game.

7. PAY OFF:
• There are also games in which the sum of the
player’s’ gains and losses does not equal zero, and
these games are denoted as non-zero-sum games.

8. STRATEGY:
• The strategy for a player is the set of alternative
courses of action that he will take for every payoff
(outcome) that might arise.

9. STRATEGY:
Strategy may be of two types:
(a) Pure strategy
If the players select the same strategy each time, then
it is referred as pure strategy. In this case each player
knows exactly what the opponent is going to do and
the objective of the players is to maximize gains or to
minimize losses.

10. STRATEGY:
(b) Mixed Strategy
When the players use a combination of strategies with
some fixed probabilities and each player kept guessing
as to which course of action is to be selected by the
other player at a particular occasion then this is known
as mixed strategy.

11. TWO PERSON ZERO SUM GAME:
A game which involves only two players, say player A
and player B, and where the gains made by one
equals the loss incurred by the other is called a two
person zero sum game.

12. TWO PERSON ZERO SUM GAME:
For example:
If two chess players agree that at the end of the game
the loser would pay 50Rs to the winner then it would
mean that the sum of the gains and losses equals zero.
So it is a two person zero sum game.

13. PAY OFF MATRIX:
• If Player A has m strategies represented as A1, A2, ---
, Am and player B has n strategies represented by
B1, B2,--- ,Bn.
• Then the total number of possible outcomes is m x n.
• Here it is assumed that each player knows not only
his own list of possible courses of action but also
those of his opponent.

14. PAY OFF MATRIX:
• It is assumed that player A is always a gainer whereas player B a loser. Let 푎푖푗 be the
payoff which player A gains from player B if player A chooses strategy 푖 and player B
chooses strategy 푗, then the pay off matrix is:
Player B`s strategies
푩ퟏ 푩ퟐ 푩ퟑ 푩풏
Player A`s
strategies
푨ퟏ 풂ퟏퟏ 풂ퟏퟐ 풂ퟏퟑ 풂ퟏ풏
푨ퟐ 풂ퟐퟏ 풂ퟐퟐ 풂ퟐퟑ 풂ퟐ풏
푨ퟑ 풂ퟑퟏ 풂ퟑퟐ 풂ퟑퟑ 풂ퟑ풏
푨풎 풂풎ퟏ 풂풎ퟐ 풂풎ퟑ 풂풎풏

15. PAY OFF MATRIX:
• By convention, the rows of the payoff Matrix denote
player A’s strategies and the columns denote player
B’s strategies.
• Since player A is assumed to be the gainer always so
he wishes to gain a payoff 푎푖푗 as large as possible
and B tries to minimize the same.

16. METHODS OF SOLVING 2-PERSON ZERO-SUM
GAMES:
1. In case of Pure Strategy game, maximizing player
arrives at optimal strategy on the basis of maximin
criterion and minimizing player’s strategy is based on
minimax criterion.
2. For no saddle point, we try to reduce the size of
game using dominance rules.

17. SOLUTION OF PURE STRATEGY GAMES:
Following methods are used for solving 2 person zero
sum games :
1. In a two person game if saddle point exists it is solved
using pure strategies but in case of no saddle point,
mixed strategies decide the results.
2. The game is solved when maximin value equals
minimax value. This value is reffered as the value of
game.

18. EXAMPLE:
Firm B Row
B1 B2 B3 minimum
Firm A A1 2 18 4 2
A2 16 10 8 8
Column maximum 16 18 8
As shown, The value of game is 8. The following steps are
followed:
1. Find maximin value:
a) Find minimum value in each row denoting minimum
possible game from each strategy of A.
b) Maximum value is the maximum of these minimum values.

19. EXAMPLE:
Firm B Row
B1 B2 B3 minimum
Firm A A1 2 18 4 2
A2 16 10 8 8
Column maximum 16 18 8
2. Find minimax value:
a) Find maximum value in each column denoting minimum
possible loss from each strategy of B.
b) Minimax value is minimum of these maximum values.

20. EXAMPLE:
Firm B Row
B1 B2 B3 minimum
Firm A A1 2 18 4 2
A2 16 10 8 8
Column maximum 16 18 8
3. Find saddle point:
a) At the right of each row, write the row minimum and
underline the largest of them.
b) At the bottom of each column, write the column maximum
and underline the smallest of them

21. EXAMPLE:
Firm B Row
B1 B2 B3 minimum
Firm A A1 2 18 4 2
A2 16 10 8 8
Column maximum 16 18 8
At the right If these two elements are equal, the
corresponding cell is the saddle point and the value is
value of the game.