Today at an interview I considered the following game:
Suppose there are 3 d6’s, each with the numbers: i) $1,2,3,4,5,6$; ii) $1,1,1,1,1,6$; and iii) $2,2,2,2,2,5$ respectively. Two pick a d6 each and roll it, and whoever rolls the larger number wins. What should the strategy (going first or second, and which dice to pick) be?
I’m not sure how to move forward with this game. My intuition tells me that the d6 ii) is just not as useful as d6 i), but this is merely a guess. I am also unsure how to approach this — do we do the usual assignment of probabilities of choosing each d6, and calculating the probability of attaining each number and setting them to be equal?
You compute the chance each die wins against the two others. For example if I have $222225$ against $123456$:
A $2$ (probability $\frac 56$) wins $\frac 16$ and loses $\frac 46$
A $5$ (probability $\frac 16$) wins $\frac 46$ and loses $\frac 16$
Clearly I am losing. I win $\frac 9{36}$ and lose $\frac {21}{36}$
You can check the other two pairs the same way. I believe it will be clear you want $123456$, so want to pick first.
There are sets of nontransitive dice where the second to choose has the advantage.