Pay The Smalls

Alice and Bob play the following game: Bob generates a uniformly random number in $[n] = \{1,2,\dots,n\}$. Alice can select any number in $[n]$. Whoever has the larger number pays out the amount of the smaller number to the person with the smaller number. For example, with $n = 50$, if Alice says $30$ and Bob says $40$, then Bob pays Alice $30$. If there is a tie, no transaction occurs. Assuming Alice plays optimally, what is her expected payout when $n = 20$?