So i was wondering how to optimize the following problem: basically its Rock Paper Scissors with the Probabilities of
$$P(R) = 0.4;\quad P(S) = 0.3;\quad P(Pp) = 0.3,$$ basically Player 1 is more likely to use Rock. What Probability should Player 2 use if he knows this? I came to the Conclusion:
$$P(R2) = r;\quad P(S2) = s;\quad P(Pp2) = pp;$$
$$P(\text{winning}) = 0.4 \cdot pp + 0.3 \cdot r + 0.3\cdot s $$
and $$ 1 = pp + r + s $$ $$ 0 \le pp \le 1; 0 \le r \le 1; 0 \le s \le 1 $$
Now how do I proceed this problem? Just trough testing I get that it has to be: $$ pp = 1;\quad r = 0;\quad s = 0; $$
but how do solve this. I cant use the derivative since it would give me the wrong answer.