# How to Represent as a Payoff Matrix

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.

## 1 Answer

You have the right idea, make a 4 by 3 payoff matrix with fruit at the top and at the side weather/water yes/no, giving 12 different cells, then put the respective dollar payoffs.

You could also make another table and find the expected payoff by multiplying each payoff by its probability, although since all the probabilities are the same, the table wouldn’t show anything different/useful to the normal payoff matrix with nominal dollar values.

Then add up the values in each fruit column and the highest value will be the optimal strategy.

• Your use of the term "dominant strategy" seems to deviate from its commonly understood meaning in game theory. Also, fruits are in rows, not columns. – Herr K. Mar 25 at 18:21
• You’re right it should be optimal strategy not dominant strategy I’ve edited my answer . Also whether or not the fruits are in rows or columns depends on how you make the table. Technically the fruits are part of columns and rows regardless... adding up the values of the apple column is what I’m getting at. – Henry M Mar 25 at 18:27
• Well, you said the payoff matrix is 3 by 4, which would suggest that the rows represent fruits and columns states. (Again, with the conventional understanding that m by n refers to m rows and n columns.) – Herr K. Mar 25 at 19:13
• Ahhhh okay I see what you mean, I meant a 4 by 3 matrix not a 3 by 4 matrix. I’ve edited my answer thanks. I wasn’t aware that game theory style matrices followed the same convention. – Henry M Mar 25 at 19:17
• @HenryM Just out of curiosity, if I could split up my 100 dollar investment, would it still be rational to invest it all into the one fruit with the highest summed utility (as mentioned in the last sentence of your answer)? – Wivaviw Mar 26 at 1:54