There are $n$ men and $n$ women. They are all located in a square randomly and uniformly. Each man is married to a woman such that there is no unmarried pair that both are closer to each other than to its spouse. Note that there is just one marriage satisfying this condition.
I am computing the expected distance to the married partner. Say, if $n=2$
$E$(dist to spouse)= $\frac{3}{4} X_1 + \frac{1}{4} X_2$
where $X_i$ denotes the (distance to the) i-closest partner, i.e. $X_1$ is the closest, $X_2$ the second closest and so on. The 3/4 comes because there is half chance that my preferred partner likes me back. In that case I marry $X_1$. But I also marry her if she prefers the other guy, but that guy prefers the other girl over her, which happens with probability 1/2*1/2=1/4. So 1/2 + 1/4 = 3/4.
I thought the distance can be computed using the formula, say for $n=3$
$E$(dist to spouse)= $\underbrace{\frac{1}{3}}_{X_1 \text{likes me first}} X_1 +$
$\underbrace{\frac{1}{3}}_{X_1 \text{likes me second}} (\overbrace{\frac{2}{3}}^{\text{her first choice didn't like her}} X_1+ \overbrace{\frac{1}{3}}^{\text{her first choice did like her}} (\frac{3}{4}X_2+\frac{1}{4}X_3)]+$
$\underbrace{\frac{1}{3}}_{X_1 \text{likes me third}} [\frac{1}{3} X_1+ \frac{2}{3} (\frac{3}{4}X_2+\frac{1}{4}X_3)]$
If that reasoning would be correct, then the expected distance to the spouse with $n$ women present would be given by (call exp distance $A$)
$A(n)= \sum_{i=1}^n \frac{n+2-i}{n2^i} X_i$.
However, computational simulations show that this guess is a bit wrong - it underestimates the distance to the partner by a little bit.
Any ideas of what is the correct expression?