I'm trying to represent the following as a pay-off matrix.
I have 100 dollars to invest in one agricultural stocks with a choice of apples, pears or grapes. Return on investment relies on whether enough water is available (with scenarios of 'yes there is enough' and 'no there isn't enough' occurring with equal probability) and whether the weather is ideal (again with scenarios of 'yes its ideal' and 'no it isnt ideal' occurring with equal probability). Both the availability of water and weather are independent to each other.
- With ideal weather and enough water, I will receive 130 dollars from apples, 110 dollars from pears and 130 dollars from grapes.
- With ideal weather and not enough water, I will receive 120 dollars from apples, 110 dollars from pears and 120 dollars from grapes.
- With unideal weather and enough water, I will receive 110 dollars from apples, 120 dollars from pears and 105 dollars from grapes.
- With unideal weather and not enough water, I will receive 100 dollars from apples, 105 dollars from pears and 100 dollars from grapes.
I know this is a decision problem under uncertainty, so the states of nature would be one of the 'players' in a normal form game and then the decisions would be another. So would it be appropriate for the table to have three rows corresponding to investing in each fruit and then four columns corresponding to each possible combination of the weather and water outcomes? And if so, how exactly would I bring the probabilities of each occurring into the mix? The four possible outcomes of the weather and water are all equally likely to occur ($p=0.25$), so could I just multiply all the dollar return amounts and put expected returns in the table instead?
Any help would be greatly appreciated.