REPRESENTING GAMES

In game theory, normal form or it is also called strategic form , is a description of a game. The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs. When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. 

A sample payoff matrix is shown below. There are two players, player A and player B with three strategies each i.e. 1, 2 and 3. The inner values in the matrix is the outcome of different combinations. If player A selects the 3rd strategy and the player B selects the 1st strategy, then the outcome will be 35, and if the player A selects the 2nd strategy and the player B also selects the 2nd strategy then the outcome will be 15.

If the outcome is positive, then it is a gain to player A and loss to player B. If the outcome is negative, then it is loss to player A and gain to player B. Consider the below payoff matrix, if the outcome is -25 then player A loses 25 points while player B gains 25 points. 

1. Pure and Mixed Strategies:

In a pure strategy, players adopt a strategy that provides the best payoffs. In other words, a pure strategy is the one that provides maximum profit or the best outcome to players. Therefore, it is regarded as the best strategy for every player of the game. 

On the other hand, in a mixed strategy, players adopt different strategies to get the possible outcome. For example, in cricket a bowler cannot throw the same type of ball every time because it makes the batsman aware about the type of ball. In such a case, the batsman may make more runs.

However, if the bowler throws the ball differently every time, then it may make the batsman puzzled about the type of ball, he would be getting the next time.

Therefore, strategies adopted by the bowler and the batsman would be mixed strategies, which are shown ion Table-2

In Table-2, when the batsman’s expectation and the bowler’s ball type are same, then the percentage of making runs by batsman would be 30%. However, when the expectation of the batsman is different from the type of ball he gets, the percentage of making runs would reduce to 10%. In case, the bowler or the batsman uses a pure strategy, then any one of them may suffer a loss.

Therefore, it is preferred that bowler or batsman should adopt a mixed strategy in this case. For example, the bowler throws a spin ball and fastball with a 50-50 combination and the batsman predicts the 50-50 combination of the spin and fast ball. In such a case, the average hit of runs by batsman would be equal to 20%.

This is because all the four payoffs become 25% and the average of four combinations can be derived as follows:

0.25(30%) + 0.25(10%) + 0.25(30%) + 0.25(10%) = 20%

However, it may be possible that when the bowler is throwing a 50-50 combination of spin ball and fastball, the batsman may not be able to predict the right type of ball every time. This would decrease his average run rate below 20%. Similarly, if the bowler throws the ball with a 60-40 combination of fast and spin ball respectively, and the batsman would expect either a fastball or a spin ball randomly. In such a case, the average of the batsman hits remains 20%.

The probabilities of four outcomes now become:

Anticipated fastball and fastball thrown: 0.50*0.60 = 0.30

Anticipated fastball and spin ball thrown: 0.50*0.40 = 0.20

Anticipated spin ball and spin ball thrown: 0.50*0.60 = 0.30

Anticipated spin ball and fastball thrown: 0.50*0.40 = 0.20

When we multiply the probabilities with the payoffs given in Table-2, we get

0.30(30%) + 0.20(10%) + 0.20(30%) + 0.30(10%) = 20%

This shows that the outcome does not depends on the combination of fastball and spin ball, but it depends on the prediction of the batsman that he can get any type of ball from the bowler.