Alice and Bob are playing a game where Bob will give Alice $4$ natural numbers, one by one, all less than $100$. Alice has to put them in an order. Alice's score will be the number of correct ordering between any two numbers. The rule of order is: the greater number will be on the left of the lesser. For example, a score of $33,56,57,89$ is 0, a score of $89,54,90,99$ is $1$. In the second example, Bob gave her the numbers in the following order: $54,89,90,99$. It is easy to show Alice can score at least $1$. What is the expectation of score, $E(s)$? Does Alice have any strategy to score more than $1$, and does Bob have his optimal strategy to restrict the score of Alice at $1$?
PS: I thought about this problem from a recent trend of blind ranking of footballers after following a course on expected value by Prof. Tom Leighton.
