Game Theory in Operation Research ~ Introduction

Game theory 

It was developed for the purpose of analyzing competitive situations involving conflicting interests. In other words, game theory is used for decision making under conflicting situations where there are one or more opponents.

The game theory models can be classified into several categories. Some important categories are listed below.

Two-person & N-person games

If the number of players is two, it is known as two-person game. On the other hand, if the number of players is N, it is known as N-person game.

Zero sum & Non-zero sum game

In a zero sum game, the sum of the points won equals the sum of the points lost, i.e., one player wins at the expense of the other. To the contrary, if the sum of gains or losses is not equal to zero, it is either positive or negative, then it is known as non-zero sum game. An example of non-zero sum game is the case of two competing firms each with a choice regarding its advertising campaign. In such a situation, both the firms may gain or loose, though their gain or loss may not be equal.

Games of Perfect and Imperfect information

If the strategy of a player can be discovered by his competitor, then it is known as a perfect information game. In case of imperfect information games no player has complete information and tries to guess the real situation.

Pure & Mixed strategy games

If the players select the same strategy each time, then it is referred to as pure strategy games. If a player decides to choose a course of action for each play in accordance with some particularly probability distribution, it is called mixed strategy game.

Assumptions of Game Theory

• There are finite number of competitors (players).
• The players act reasonably.
• Every player strives to maximize gains and minimize losses.
• Each player has finite number of possible courses of action.
• The choices are assumed to be made simultaneously, so that no player knows his opponent's choice until he has decided his own course of action.
• The pay-off is fixed and predetermined.
• The pay-offs must represent utilities.

Basic Terminologies in Game Theory

Player

Each participant (interested party) is called a player.

Strategy

The strategy of a player is the predetermined rule by which a player decides his course of action from the list of courses of action during the game. A strategy may be of two types:

• Pure strategy. It is a decision, in advance of all plays, always to choose a particular course of action. In other words, if the best strategy for each player is to play one particular strategy throughout the game, it is called pure strategy.
• Mixed strategy. It is a decision, in advance of all plays, to choose a course of action for each play in accordance with some particular probability distribution. In other words, if the optimal plan for each player is to employ different strategies at different times, we call it mixed strategy.

Optimal strategy

The course of action which maximizes the profit of a player or minimizes his loss is called an optimal strategy.

Saddle point

A saddle point is an element of the matrix that is both the smallest element in its row and the largest element in its column. Furthermore, saddle point is also regarded as an equilibrium point in the theory of games.

Pay-off

The outcome of playing the game is called pay-off.

Pay-off Matrix

It is a table showing the outcomes or payoffs of different strategies of the game.

 Value of the Game

It refers to the expected outcome per play, when players follow their optimal strategy. It is generally denoted by V.

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