I have $$ A_{100\times 2} = \left[ {\begin{array}{cccc} a_{1\ 1} & a_{12} & \\ a_{21} & a_{22} & \\ \vdots & \vdots & \\ a_{100\ 1} & a_{100\ 2} &\\ \end{array} } \right] $$ where (each column) $a_{i} \sim U(0, 500)$.
Let $$ V = |a_{1} - a_{2}|$$ be a utility function (which is just the absolute difference between two columns) that an algorithm $Q$ tries to "maximize". Algorithm is very simple in nature. It does a ranked selection (overall two rows) of the row(s) corresponding to the top values of either $a_1$, $a_2$ (case 1), both (case 2) or none of them (case 3, no ranked selection, purely random selection). In case 2 it simply takes one row per each column's top value
What I have been trying to find is the expected value of $Q$. I know there are three possible cases and I could calculate the two of them.
Case 1
When selecting either one of the columns (lets say $a_1$), by the rule of expectation using formulas from these questions here and here
$$E(Q) = | \frac{E(a_{1_{(100)}}) + E(a_{1_{(99)}})}{2} - E(a_2)| \approx 492.5 - 250 = 242.5 $$
Case 3. Purely random selection, so $$E(Q) = | E(a_1) - E(a_2) | = 250 - 250 = 0$$
Case 2.
This one is the tricky part. Because here the selections are somehow dependent on each other. I do not know how to calculate it in this case. Would appreciate any help.