presentation lecture of game theory .pdf

3. BASIC TERM USED IN GAME THOERY
(1) PLAYER:
The competitor is referred to as player A player may be individual, a
group of individual, or an organization.

(1) PLAYER:
(2) TWO-PERSON GAME/ N-PERSON GAME:
 If a game involves only two players (competitors), then it is called a
two-person game.
 If numbers of player are more than two, then game is referred to as
n-person game.

(1) PLAYER:
(2) TWO-PERSON GAME/ N-PERSON GAME:
 If a game involves only two players (competitors), then it is called a
two-person game.
 If numbers of player are more than two, then game is referred to as
n-person game.
(3) ZERO SUM GAME/NON-ZERO SUM GAME:
In a game, if sum of the gain to one player is exactly equal to the
sum of losses to another player, so that the sum of the gains and losses
equals to zero, then the game is said to be a zero-sum game. Otherwise
it is said to be non zero-sum game.

(4) STRATEGY:
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.

(4) STRATEGY:
arise.
1. PURE STRATEGY
 A player always chooses the same strategy-same row or column.

(4) STRATEGY:
arise.
1. PURE STRATEGY
2. MIXED STRATEGY
 A player chooses the strategy with some fix probabilities.

(4) STRATEGY:
arise.
1. PURE STRATEGY
2. MIXED STRATEGY
 A player chooses the strategy with some fix probabilities.
3. OPTIMAL STRATEGY
 A strategy that maximizes a player’s expected pay-off.

(5) PAYOFF MATRIX:
The payoff (a quantitative measure of satisfaction that a player gets
at the end of the play) in terms of gains and losses, when player select
their particular strategies, can be represented in the form of a matrix,
called the payoff matrix.
In this pay-off matrix, positive pay-off is the gain to maximizing
player (X) and loss to minimizing player (Y). E.g., if X chooses strategy
X2 and Y chooses strategy Y1, then X’s gain is 32 and Y's loss is 32.
24 36 8
32 20 16
PLAYER
X
PLAYER Y
Y1 Y2 Y3
X1
X2

5. RULES FOR GAME THEORY
RULE 1: Look for pure Strategy (Saddle point)

5.RULES FOR GAME THEORY
RULE 2: Reduce game by Dominance
If no pure strategies exist, the next step is to eliminate certain strategies
(row/column) by law of Dominance.

5.RULES FOR GAME THEORY
RULE 2: Reduce game by Dominance
If no pure strategies exist, the next step is to eliminate certain strategies
(row/column) by law of Dominance.
RULE 3: Solve for mixed Strategy
A mixed strategy game can be solved by different solution method, such
as
1. Algebraic Method
2. Arithmetic Method
3. Matrix Method
4. Graphical method
5. Linear Programming Method

6. PURE STRATEGIES (MINIMAX AND MAXIMIN PRINCIPLE)
(1) Maximin principle:
Maximize the player’s minimum gains. That means select the
strategy that gives the maximum gains among the row minimum value.

6.PURE STRATEGIES (MINIMAX AND MAXIMIN PRINCIPLE)
(2) Minimax Principle:
Minimize the player’s maximum gains. That means, select the
strategy that gives the minimum loss among the column maximum
values.

6.PURE STRATEGIES (MINIMAX AND MAXIMIN PRINCIPLE)
(2) Minimax Principle:
Minimize the player’s maximum gains. That means, select the
strategy that gives the minimum loss among the column maximum
values.
(3) Saddle Point:
If the maximin value equals the minimax value, the game is said to
have a saddle (equilibrium) point and the corresponding strategies are
called "Optimal Strategies".

(4) Value of game:
This is the expected payoff at the end of the game, when each
player uses his optimal strategy.

(4) Value of game:
This is the expected payoff at the end of the game, when each
player uses his optimal strategy.
7. RULE TO FINDOUT SADDLE POINT
 Select the minimum (lowest) element in each row of the payoff
matrix and write them under ‘Row Minimum’ heading. Then, select
the largest element among these element and enclose it in a
rectangle [ ].
 Select the maximum (largest) element in each column of the pay off
matrix and write them under 'Column Maximum' heading. Then,
select the lowest element among these elements and enclose it in a
circle ( ).
 Find out the element(s) that is same in the circle as the well as
rectangle and mark the position of such element(s) in the matrix.
This element represents the value of the game and is called the
Saddle(or equilibrium) point.

Example: Find Solution of game theory problem using saddle
point
Player APlayer B
B1 B2
A1 4 6
A2 3 5
Solution:
 We apply the maximin (minimax) principle to analyze the game,
Player APlayer B
B1 B2
Row
Minimum
A1 [(4)] 6 [4] Maximin
A2 3 5 3
Column
Maximum
(4)
Minimax
6

Select minimum from the maximum of columns
Column MiniMax = (4)

Select maximum from the minimum of rows
Row MaxiMin = [4]

Row MaxiMin = [4]
Here, Column MiniMax = Row MaxiMin = 4
∴ This game has a saddle point and value of the game is 4.

Row MaxiMin = [4]
Here, Column MiniMax = Row MaxiMin = 4
∴ This game has a saddle point and value of the game is 4.
The optimal strategies for both players are,
The player A will always adopt strategy A1.
The player B will always adopt strategy B1.

presentation lecture of game theory .pdf

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