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Game Theory Basic Definitions

We assume that players are economically rational i.e., a player can (i) assess outcomes,

(ii) choose actions that yield their most preferred outcomes, given the actions of the other players.

 

(i) Game : All situations in which at least one player can only act to maximize his utility

through anticipating the responses to his actions by one or more other players is called a game.

 

(ii) Strategy : A strategy is a possible course of action open to the player.

 

(iii) Pure strategy : A pure strategy is defined by a situation in which a course. of action is played with probability one.

 

(iv) Mixed strategy : A mixed strategy is defined by a situation in which no course of action is taken with probability one.

 

(v) Payoff matrix (or Reward matrix) : A payoff matrix is an array in which any (i, j)th entry shows the outcome. Positive entry is the gain and negative entry is the loss for the row- player.

Matrix games are referred to as 'normal form' or 'strategic form' games, and games as trees are referred to as 'extensive form' games. The two sorts of games are not equivalent.

 

(vi) Maximin criterion : This is a criterion in which a player will choose the strategies with the largest possible payoff given an opponent's set of minimizing countermoves.

 

(vii) Minimax criterion : This is a criterion in which a player will choose the strategies with the smallest possible payoff given an opponent's set of maximising countermoves.

 

(viii) Saddle point : If a payoff matrix has an entry that is simultaneously a maximum of row minima and a minimum of column maxima, then this entry is called a saddle point of the game and the game is said to be strictly determined.

 

(ix) Value of the game : If the game has a saddle point then the value at that entry is called the value of the game. If this value is zero then the game is said to be fair.

 

(x) Zero-sum game : A zero-sum game is a game in which the interests of the players are diametrically opposed i.e., what one player wins the other loses. When two person play such game then it is called two person zero-sum game.

 

In this chapter we shall consider only matrix games.

 

Note. If in a game the total payoff to be divided among players is invariant i.e., it does not depend upon the mix of strategies selected, then the game is called constant-sum game.