Ordering uniforms

Suppose you are going to be shown three iid uniform rv, $U_i \sim U[0,1]$. Your task is to after seeing $U_1$, assign it lowest, middle, or highest. Then see $U_2$ and assign it the remaining two of lowest, middle, or highest. Then by default $U_3$ is assigned the remaining label. What should your strategy be to maximize the probability of labeling all 3 correctly?

Solution

Let $L_1$ be the event $U_1$ is the lowest, and $W_i$ be the event we win the game of guessing the relative order of $i$ uniforms.s If we pick lowest for $U_1=u$, our probability of winning is the probability of guessing the relative order of the last two correctly times the probability of the last two being larger than $u$. That is

\(\mathrm{Pr}(L_1 \cap W_3 \mid U_1=u) = (1-u)^2 \mathrm{Pr}(W_2)\)

The probability of winning the two uniform game is $3/4$. This can be intuitively seen by noting you can only be incorrect when they land on the same half, and in that case you are only wrong half the time by exchangeability.

If we instead pick middle for $U_1=u$ our probability of winning is $2u(1-u)$. Thus we need to find where these two are equal to decide our cutoff.

$2u(1-u) = 3(1-u)^2/4$

Solving this yields $u=3/11$. Therefore we should pick lowest for $u_1 < 3/11$, highest for $u_1 > 8/11$, and middle otherwise.