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What Is Game Theory? -making in situations where two or

more intelligent and rational opponents are involved under conditions of

conflict and competition.

The approach of game theory is to seek to

counter-

Basic Terminology The models in the theory of games can be classified depending upon the

following factors:

Number of Players: If a game involves only two players (competitors),

then it is called a two-person game. However, if the number of players is

more, the game is referred to as n-person game.

Payoff: Outcome of a game when different alternatives are adopted by

the competing players.

Strategy: A set of rules or alternative courses of action available to a

player in advance, by which he decides the course of action that he

should adopt.

Strategy may be of two types:

Pure strategy: If the players select the same strategy each time. In

this case each player knows exactly what the other is going to do.

Mixed strategy: When the players use a combination of strategies

and each player always kept guessing as to which course of action

is to be selected by the other player at a particular occasion.

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Optimum strategy: A course of action or play which puts the

player in the most preferred position, irrespective of the strategy

of his competitors.

Value of the Game: 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 unfair if

it is non-zero.

Payoff Matrix: The payoffs in terms of gains or losses, when players

select their particular strategies, presented in form of matrix.

Two-person Zero-sum Game: A game of two persons, in which the

gains of one player are the losses of the other player.

The algebraic sum of the gains to both the players after a play is

bound to be zero

Maximin Minimax Principle: The maximum of minimum gains is the

maximin value of the game and the corresponding strategy is the

maximin strategy. In a similar way, the minimum of maximum losses will

be called the minimax value of the game and the corresponding strategy

is the minimax strategy.

Saddle point - The position in the matrix where the maximin is

equal to the minimax.

Assumptions/ Characteristics of a Game The number of competitors called players is finite.

The players act rationally and intelligently

Each player has available to him a finite number of choices or possible

courses of action called strategies.

The number of choices need not to be the same for each player

All relevant information is known to each player in advance.

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Application of Game Theory in Real Life Situations Oligopolistic Strategy

Two Players A and B, play the coin

In Football Match

Politics

Methods to Find Value of a Game under Decision

Making Environment & Calculation Process

Methods:

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Calculation process:

Pure Strategy Games (With Saddle Point )

A pure strategy game is a game whereby the players select the same

strategy each time.

There is a deterministic situation and the objective of the player is to

maximize gains or to minimize losses.

The maximizing player arrives at his optimal strategy on the basis of the

maximin

minimax criterion.

The Saddle Point which is the solution of the game is derived at the

point where the maximin value equals the minimax value

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Example: For the following payoff matrix for Firm A, determine the optimal

strategies for both Firm A and Firm B and the value of the game (using

maximum-minimax principle)

the value of the game

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Analysis of Pure Strategy Games

Develop the payoff matrix.

maximin strategy (for the maximizing player).

Identify column maximums and select the smallest of these as the

for the minimizing player).

If the maximin value equals the minimax value, the game is a pure

strategy game and that value is the saddle point

Principle of Dominance

payoff matrix by eliminating a course

of action which is so inferior to another course of action that can be left out

of the set of choices.

The concept of dominance is especially useful for the evaluation of two-

person zero-sum games where a saddle point does not exist.

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Example (no saddle point):

After the matrix is reduced, it can now be solved further, using the mixed

strategy methods.

The rules of dominance to reduce the size of a payoff matrix:

When all elements 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, therefore, be deleted from the matrix.

When all elements in a column of a payoff matrix are greater than or

equal to the corresponding elements in another column, then the

former column is dominated by the latter and can, therefore, be deleted

from the matrix.

A pure strategy may be dominated if it is inferior to average of two or

more other pure strategies.

Nash Equilibrium

It is a set of strategies such that each player believes (correctly) that it is

doing the best it can, given the strategy of the opponents.

Since the player is satisfied that he has made the best decision possible,

he has no incentive to deviate from the chosen strategy. Thus, Nash

strategies are stable.

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It is possible only if we assume that all players understand the game and

are rational.

Illustration: PAYOFF MATRIX: ADVERTISING GAME

The Nash equilibrium for this game is Advertising by both Firm A and Firm B.

Distinction between Nash Equilibrium & Dominant-Strategy Equilibrium

In dominant strategy case, each player chooses his best strategy,

irrespective of the strategies of other players.

While in the case of Nash equilibrium, each player chooses a strategy

that is his best choice, subject to what strategies the opponent chooses

Mixed Strategy Games (Games without saddle point )

In a game without saddle point, the optimal policy is to use mixed strategies.

This is the combination of strategies which keep each player guessing on

what course of action is to be selected by the other player at a particular

occasion.

It is a selection among pure strategies with fixed probabilities.

To order to solve such a game, each player adopts the concept of chance move

and starts to play in a random manner and in such a way that his average

payoff over a large number of plays of the game should be optimal, even

though he may lose more in any individual play of the game.

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A mixed strategy can be solved by the following methods:

1. Algebraic/Arithmetic Method;

2. Graphical Method;

3. Matrix Method;

4. Linear Programming Method.

Algebraic Method (2 2 Strategies Game)

If this game is to have no saddle point, the two largest elements of the

matrix must constitute one of the diagonals.

We have assumed this and therefore both players use mixed strategies.

Our task is to determine the probability with which both players choose

their course of action.

Solve the following game:

, ) and for the player B, the

) respectively. The value of the game is 13

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Arithmetic Method (2 2 Strategies Game)

It is also known as odds method or shortcut method.

Example: Reduce the following game by dominance and find the game value.

, ) and for the player B, the

) respectively. The value of the game is

Steps in calculation:

Subtract the smaller payoff in each row from the larger one and the

smaller payoff in each column from the larger one.

Interchange each of these pairs of subtracted numbers found in Step 1.

Put each of the interchanged numbers over the sum of the pair of

number

Simplify the fraction to obtain the required strategies.

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Graphical Method (2 m or n 2 Strategies Game)

2, so that it can be possible to solve using Algebraic/Arithmetic Method.

Example: Obtain the optimal strategy to BOTH player and the value of the

game for two-person zero-sum game by using graphical method whose payoff

matrix is given as follows:

Optimal strategy for player A is: (0, 0, 0, ) and optimal strategy for player B

is: ( , ) and value of game, v =

Steps in Calculation:

State the expected payoff of the player with many strategies in line with

the other, if pure strategy is selected.

These expected payoffs will be plotted against the two strategy, they are

the parallel line in which the expected payoff line is drawn in.

For the maximum value, the innermost intersect on the graph and the

two expected payoff that results to it, will be extracted and use to form

a 22 matrix, similarly in terms of minimum value, the outermost point

is extracted and the payoffs are selected.

Finally, the two payoffs can be equated to get immediate probability (p

or q as the case maybe) and also the value of the game. Hence, the 22

matrix gotten will be used to solve for the remaining payoff.