Game Theory

Meaning Game Theory is a body of knowledge which is concerned with the study of decision making in situation where two or more rational opponents are involved under condition of competition and conflicting interests.

Essential Features of Game Theory A competitive situation is called a game if it has following features: Finite number of competitors. Finite number of action. Knowledge of alternatives. Choice. Outcome or Gain. Choice of opponent.

Terms used Two Person Zero-sum game – It is the situation which involves two persons or players and gains made by one person is equal to the loss incurred by the other. n-persons game – A game involving n persons is called a n-person game. Pay offs - Outcome of a game due to adopting the different courses of actions by competing players in the form of gains or losses for each of the players is known as pay offs.

Pay off matrix – In a game, the gains or losses, resulting from different moves and counter moves, when represented in the form of a matrix are known as pay off matrix. Maximin Criteria – The maximizing player lists his minimum gains from each strategy and selects the strategy which gives the maximum out of these minimum gains. Minmax Criteria – The minimizing player lists his maximum loss from strategy and selects the strategy which gives him the minimum loss out of these maximum losses. Value of Game – The maximum guaranteed gain to the maximizing player if both the players use their best strategy.

Saddle Point Saddle point is the equilibrium point in the theory of games. It is the smallest value in its row and largest value in its column. Steps to find out Saddle Point:- Select the minimum value of each row & put a circle O around it. Select the maximum value of each column and put square □ around it. The value with both circle and square is the saddle point.

Strategy Types of Strategy Pure Strategy Mixed Strategy It is the pre-determined rule by which each player decides his course of action from list available to him. It is course of action taken by one of the participants in a game. Types of Strategy Pure Strategy Mixed Strategy

Pure Strategy It is predetermined course of action to be employed by the player. The players knew it in advance. It is usually represented by a number with which the course of action is associated.

Games with Pure Strategy In pure strategy, the maximizing player arrives at his optimal strategy on the basis of maximin criterion. The game is solved when maximin value equals minimax value. Firm Y Y1 Y2 3 X1 Firm X X2 3 6

Mixed Strategy In mixed strategy the player decides his course of action in accordance with some fixed probability distribution. Probabilities are associated with each course of action and the selection is done as per these probabilities. In mixed strategy the opponent can not be sure of the course of action to be taken on any particular occasion.

Games with Mixed Strategies All game problems where saddle point does not exist are taken as mixed strategy problems. Following are the methods of Mixed strategies:- ODDS Method (2*2 game without saddle point) Dominance Method Sub Games Method Equal Gains Method Linear Programming method – Graphic solution Algebraic Method Linear Programming – Simplex Method Iterative Method

1. ODDS Method (For 2*2 game) Possible only in case of games with 2*2 matrix. It should be ensured that sum of column odds and row odds should be equal. Method of finding out odds:- Find out the difference in the value of in cell (1,1) and the value in the cell (1,2) of the first row and place it in front of second row. Find out the difference in the value of in cell (2,1) and the value in the cell (2,2) of the second row and place it in front of first row. Find out the differences in the value of cell (1,1) and (2,1) of the first column and place it below the second column. Similarly find the difference between the value of the cell (1,2) and the value in the cell (2,2) of the second column and place it below the first column. The above odds or differences are taken as positive (ignoring the negative sign).

Mathematically, Y Odds X (b1-b2) (a1-a2) (a2-b2) (a1-b1) The value of game is determined by the following equation: Value of game (v) = a1(b1-b2) + b1(a1-a2) (b1-b2) + (a1-a2) Probabilities for X1 = b1 – b2 X2 = a1 – a2 Probabilities for Y1 = a2 – b2 (a2-b2) + (a1-b1) Y2 = a1 – b1 a1 a2 b1 b2

2. Dominance Method The Principle of Dominance states that if the strategy of a player dominates over the other strategy in all condition then the later strategy is ignored . Rules to be followed : If all the elements of a row ( say ith row) of a pay off matrix are less than or equal to (≤) the corresponding each element of the other row (say jth row) then the player A will never choose the ith strategy OR ith row is dominated by jth row. Then delete ith row. If all the element s of a column (say jth column) are greater than or equal to the corresponding elements of any other column (say jth column) then ith column is dominated by jth column. Then delete ith column. A pure strategy of a player may also be dominated if it is inferior to some convex combination of two or more pure strategies. By eliminating some of the dominated rows and columns, if the game is reduced to 2*2 form, it can be easily solved by odds method.

Eliminating Dominated Strategies Solve the game Column a b 9 4 a Row b 5

Eliminating Dominated Strategies For any column action, Maximizing player will prefer a. a b 9 4 a b 5

Eliminating Dominated Strategies Given that row will pick a, column will pick b. (a,b) is the unique Nash equilibrium. a b 9 4 a b 5

3. Sub – game Method (In case od 2*n or m*2 Matrices) A game where one of the players has two alternatives while the other player has more than two alternatives. This method is used when there is no saddle point or it can not be reduced by dominance method. This method is suitable when the number of alternatives is limited say 4 or 5. In case of large number of alternatives , the solution will be lengthy.

Procedure of Sub-Games Method Divide the m*2 or 2*n game matrix into as many 2*2 sub games as possible. Taking each game one by one and finding out the saddle point of each game (if exists) and then that sub game has pure strategies. In case there is no saddle point, then that sub game should be solved by odds method. Select the best sub game from the point of view of the players who has more than two alternatives. The strategies for this selected sub game will hold good for both the players for the whole game and the value of the so selected sub game will be the value of complete game.

4. Equal Gains Method Solution of 2*2 matrix without saddle point. As players are rational in their approach, the selection of their combination of strategies will be done in such a way that the net gain is not influenced by the selection of any combination of strategy by the opponent. In this, players select each of the available strategies for certain proportion of the time i.e., each player selects a strategy with some probability. This method is applicable only in case of square matrices. This method is not suitable when the rule of game is negative.

5. LPP – Graphic Method Graphic method is applicable only to those games in which one of the players has two strategies only (2*m or n*2 games). The following are the steps involved in this method:- The game matrices of 2*m or n*2 is divided into 2*2 sub matrices. Next taking the probabilities of the two alternatives of the first player say A as p1 and (1- p1) the net gain of A from the different alternative strategies of B is expressed with equation. The boundaries of the two alternative strategies of the first player are shown by two parallel line shown on the graph.

The gain equation of different sub games are then plotted on the graph. In case of maximizing player A, the point is identified where minimum expected gain is maximized. This will be the highest point – out the inter section of the gain lines in the ‘Lower Envelop’. In case of minimizing player B, the point where maximum loss is minimized is justified. This will be the lowest point at the intersection of the equations in the ‘Upper Envelop’.

6. Algebraic Method Algebraic Method is used in the game of 3*3 matrix and game does not have any saddle point. Another condition is that game cannot be reduced to 2*2 matrix by the principle of dominance. Use the formula of 2 3 1 2 for the cross multiplication.

7. Linear Programming – Simplex Method It is sometimes difficult to solve a game problem in the m*n pay off matrix having neither a saddle point nor any dominant row or column and m and n are more than 2. Simplex method of LPP is a general method for all types of game problems particularly when all the players have 3 or more strategies.

8. Iterative/Approximation Method This method can be applied to solve 3*3 or higher games which cannot be easily solved by any other method. The following steps are used for iterative method: Player A chooses the superior strategy over the other and places that row below the matrix. Player B examines this row and chooses a column corresponding to the smallest number in that row. This column is placed to the right of the matrix.

Player A examines this column and selects a row corresponding to the largest number in this column. This row is then added to the row last chosen and then sum of the two rows is placed below the previous row selected. Player B chooses a column corresponding to the smallest number in the row and adds this column to the last chosen and place it below the previous column selected. In case of tie, the player should choose the row or column different from his last choice. The procedure is repeated for a number of iterations. The upper limit of the game is calculated by dividing the highest number in the last column by the total number of iterations. Similarly the lower limit of the game is determined by dividing the lowest number in the last row by the total number of iterations.

Limitations of Game Theory Infinite number of strategy Knowledge about strategy Zero outcomes Risk & uncertainty Finite number of competitors Certainty of Pay offs Rules of game.