Game theory is a type of decision theory in which one’s choice of action is determined after taking into account all possible alternatives available to an opponent player the same game.
This type of problems is fundamentally based upon the minimax / maximin criterion. This implies the assumption of rationality from which it is argued that each player will act so as to maximize his minimum gain or minimize his maximum loss.
Game refers to the general situation of conflict and competition in which two or more competitors are involved in decision- making activities in anticipation of certain outcomes over a period of time.
The set of rules defines the game.
There can be a various types of game. They can be classified on the basis of the following characteristics.
Chance of strategy- if in a game, activities is determined by skill, it is said to be game of strategy. If they are determined by chance, it is a game of chance.
Number of players – a game is called an n number game if the number of persons playing is n. the person’s means an individual or a group aiming at a particular objective.
Number of activities – there may be finite or infinite.
Number of alternative available to each person in a particular activity may also be finite or infinite.
Information to the players about the past activities of other player is completely available, partly available or not available at all.
Payoff- a quantitative measure of satisfaction a person gets at the end of each play is called a payoff. It is a real valued function of variables in the game
Competitive game – a competitive situation is called a competitive game. It has following properties-
The number of competitors called players is finite.
The players act intelligently and rationally
Each player has a list of finite number of choice or possible courses of action called strategy.
All relevant information i.e, the different strategies of each player and the amount of gain or loss of an individuals move are known to each player in advance.
The players select their respective courses of action/ strategy simultaneously.
The player make individual decisions without direct communication
The maximizing player attempts to maximize his gains and the minimizing player tries to minimize his losses.
The payoff is fixed and determined in advance
Two persons zero-sum game – a game of two player, in which the gains of one is equal to the losses of the other player is called a two player zero sum game.
Strategy – a rule for decision making in advance of all the plays by which he decides the activities he should adopt. It is a course of action taken by one of the participants in a game. It is a set of rules that specifies which of the available course of action he should make at each play.
Pure strategy – if a player know what exactly the other player is gong to do, a deterministic situation is obtained and objective function is to maximize the gains, therefore, the pure strategy is a decision rule always to select a particular course of action.
Mixed strategy – if a player is guessing as to which activity is to be selected by other on any particular occasion, a probabilistic situation is obtained and objective function is to maximize the expected gain.
Pay off matrix- the payoff is the result or the outcome of the strategy.
Suppose the player A has m activities and the player B has n activities. Then a pay-off matrix can be formed by adopting the following rules :
Row designations for each matrix are activities available to player A
Column designations for each matrix are activities available to player B
Cell entry ‘Vij’ is the payment to player A in A’s payoff matrix when A chooses the activity i & B chooses activity j.
With a zero-sum, 2 person game, the cell entry in the player B’s payoff matrix will be –ve of the corresponding cell entry Vij in the player A’s payoff matrix so that sum of payoff matrixes for player A and player B is ultimately Zero.
Maximin principle- the player decides to play the strategy which corresponds to the maximum of the minimum gains from his different course of action.
Minimax principles- the player selects that strategy which corresponds to the minimum of the maximum losses from his different course of action and this is known as the minimax principles.
Optimal strategy –a course of action / play which puts the player in the most preferred position, irrespective of the strategy of his competitors is called an optimal strategy. If the payoff matrix has the saddle point the player are said to have optimal strategies.
Saddle point – a saddle point of a payoff matrix is the position of such an element in the payoff matrix which is minimum in its row and maximum in its column.
Value of game – the payoff at the saddle point is called the value of game and it is obviously equal to the maximin and minimax value of the game. The game is called fair if the value of the game is zero and unfair if it is non-zero.