Saw this interview question on a forum, I'm stuck in the way that I thought the order of rolling the four dice should not influence the expected value as they are independent anyway?
Question: There are four six-sided dice, rolled independently by four players, one after the other. Say the i-th person rolls $X_i$, i.e. $X_1$ is the result rolled by the first person, $X_2$ rolled second... etc.
Let $X_{(1)} \leq X_{(2)} \leq X_{(3)} \leq X_{(4)}$ represent the four rolls in increasing value, they are not necessarily rolled in this order, i.e. $X_{(1)}$ need not be $X_{1}$.
Your payoff of this game is calculated by $$\frac{X_{(1)}+X_{(4)} - X_{(2)} -X_{(3)}}{2}$$ that is, we group the highest and lowest roll, find their average, then subtract that by the average of the other group formed by the remaining rolls.
Given you are the second person to roll the dice, what value of $X_1$ would maximise your payoff?
Edit: clarity and context.