Yasir Hashmi 01

Game theory

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

PAY OFF MATRIX:

• It is assumed that player A is always a gainer whereas player B a loser. Let 𝑎𝑖𝑗 be the

payoff which player A gains from player B if player A chooses strategy 𝑖 and player B

chooses strategy 𝑗, then the pay off matrix is:

Player B`s strategies

𝑩𝟏 𝑩𝟐 𝑩𝟑 𝑩𝒏

Player A`s strategies

𝑨𝟏 𝒂𝟏𝟏 𝒂𝟏𝟐 𝒂𝟏𝟑 𝒂𝟏𝒏

𝑨𝟐 𝒂𝟐𝟏 𝒂𝟐𝟐 𝒂𝟐𝟑 𝒂𝟐𝒏

𝑨𝟑 𝒂𝟑𝟏 𝒂𝟑𝟐 𝒂𝟑𝟑 𝒂𝟑𝒏

𝑨𝒎 𝒂𝒎𝟏 𝒂𝒎𝟐 𝒂𝒎𝟑 𝒂𝒎𝒏

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 𝑎𝑖𝑗 as large as possible

and B tries to minimize the same.

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.

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.

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.

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.

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

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.