Question

An investor is considering purchasing one of three stocks. Stock A is regarded as conservative, stock B as speculative, and stock C as highly risky. If the economic growth during the coming year is strong, then stock A should increase in value by $3000, stock B by $6000, and stock C by $15,000. If the economic growth during the next year is average, then stock A should increase in value by $2000, stock B by $2000, and stock C by $1000. If the economic growth is weak, then stock A should increase in value by $1000 and stocks B and C decrease in value by $3000 and $10,000, respectively.

(a) Give the pay off matrix for this problem and decide if the game is strictly determined or not.

(b) What is the optimal strategy for the investor?

(c) What is the value or expected value of the game?

Answer 1

(a) The following table is the pay off matrix, in this problem we assume that the investor is playing against the economic growth which doesn't want to us to earn money, wheras we want to earn as much as possible.

	Stock A	Stock B	Stock C
Strong	+3000	+6000	+15000
Average	+2000	+2000	+1000
Weak	+1000	-3000	-10000

The investor can choose which stock to go for and the economic growth can choose how the economic growth will be.

Worst case for each stock(i.e when the investor chooses a stock) is higlighted in green.(Selected in a column)
Worst case for economic growth(i.e when economic growth chooses the trend) is given a pink background. (Selected in a row)

A game is said to be strictly determined if atleast one saddle point exists. A saddle point is a point which is max in it's row and minimum in it's column(Assuming we want the column player to win).

We see that there is a saddle point in our payoff matrix, therefore the game is strictly determined. Basically in a strictly determined, game both of the players tend to fix to a specific option to minimize their loses and because of this we can directly determine what may happen.

(b) Since we are assuming that the economic growth is playing against the investor, it will always choose the economic growth to be in such a way that he will have maximum loss. That means it will choose the economic growth to be weak, therefore the optimal strategy i.e the strategy using which we will have minimum loss(Maximum profit), for the investor is to choose stock A.  

(c)The value of the game is defined to be the minimum value that our player will gain if he sticks to the optimal strategy. Since the investor will always choose stock A, the minimum value he will gain in any situation of the economic growth is $1000. This value, $1000 is the value of the game.