Each of two players shows one or two fingers (simultaneously) and $C$ pays to $R$ a sum equal to the total number of fingers shown. Write the game matrix. Show that the game is strictly determined and find the value and optimal strategies.
I'm brand new at game theory and just learned how to tell if a game is fair or strictly determined. I think I'm thinking about this wrong but if there are only two options and I need the game to be strictly determined wouldn't the answer be a matrix with all one's or all two's?
The specific arrangement of how you insert the information varies based on textbook, but a common way to write it out would be such as the following:
$\begin{array}{r|c|c|}&C~\text{shows}&C~\text{shows}\\&\text{1 finger}&\text{2 fingers}\\\hline R~\text{shows 1 finger}\\ \hline R~\text{shows 2 fingers}\\\hline \end{array}$
writing values into the table according to how much money $R$ would win (which is the same amount that $C$ would lose) if those selections were simultaneously made.
For example, in the top left entry with $R$ showing one finger and $C$ showing one finger, there would be a total of two fingers shown implying that $C$ would need to pay $R$ two dollars (or whatever currency they happen to be using).
As for deciding strategies... $C$ knows that if he chooses to show more fingers he'll lose more money than if he chose to show only one finger... so $C$'s strategy will be to _______.
Similarly, $R$ wants to make as much money as possible so the more fingers $R$ shows the more money he'll stand to make so $R$'s strategy will be to ______.
Since both players are dead-set on what strategies they'll be following, we can expect that every time they play this game the outcome will be identical every time... i.e. it is indeed a strictly determined game, and the outcome every time they play will be that $R$ wins ______ dollars from $C$.
Since the expected outcome of the game is not zero, the game is not fair.