Question: A statistician has to decide on the basis of two observations whether the parameter ? of a binomial distribution is $\frac{1}{4}$ or $\frac{1}{2}$; his loss (a penalty that is deducted from his fee) is $160$ if he is wrong.
a) construct a table showing four possible value of the loss function
b) list the eight possible decision functions
c) show that three of the decision function are not admissible
d) find the decision that is best according to the minimax criterion
My work: I already have done part a) and I know how to do c) and d). My book has already given be the answer for b) as:
$d_1(0) = \frac{1}{4}$ $d_1(1) = \frac{1}{4}$ $d_1(2) = \frac{1}{4}$
$d_2(0) = \frac{1}{4}$ $d_2(1) = \frac{1}{4}$ $d_2(2) = \frac{1}{4}$
$...$
I am trying to understand where the input of $2$ comes in for $d_i(2)$ when there are only two possible choices someone can make and only two observations. I feel like, $d_i(0)$ and $d_i(1)$ should be the only decision functions where $0$ represents choosing $\frac{1}{4}$ and $1$ choosing $\frac{1}{2}$ under a given circumstance.
Secondly, I don't understand the literal meaning behind what each of these decisions are and as a result it is making it difficult to list them out and understand what each one represents.