$A$ and $B$ play the following game: $A$ writes down either number $1$ or number $2$, and $B$ must guess which one. If the number that $A$ has written down is $i$ and $B$ has guessed correctly, $B$ receives $i$ units from $A$. If $B$ makes a wrong guess, $B$ pays $3/4$ unit to $A$. If $B$ randomizes his decision by guessing $1$ with probability $p$ and $2$ with probability $1 − p$, determine his expected gain if
$(a)$ $A$ has written down number $1$.
$(b)$ $A$ has written down number $2$.
What value of $p$ maximizes the minimum possible value of $B$’s expected gain, and what is this maximin value? (Note that $B$’s expected gain depends not only on $p$, but also on what $A$ does.)
Consider now player $A$. Suppose that she also randomizes her decision, writing down number $1$ with probability $q$. What is $A$’s expected loss if
$(c)$ $B$ chooses number $1$?
$(d)$ $B$ chooses number $2$?
What value of $q$ minimizes $A$’s maximum expected loss? Show that the minimum of $A$’s maximum expected loss is equal to the maximum of $B$’s minimum expected gain.
My attempt (just the results)
I don't understand the what the "maximize minimum" and "minimize maximum" part of the question means.