Suppose there are $N$ red dots and $M$ blue dots uniformly distributed in a square region with side $S$. Each red dot finds its closest blue dot and each blue dot finds its closest red dot. What is the probability that these closest peers will be mutual? For example, if $N_i = \mathrm{closest}(M_j)$ then the probability that $M_j = \mathrm{closest}(N_i)$?
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Closest in term of Euclidean distance I guess? – Frostic Apr 20 '18 at 20:19
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yes you are right. – Eyup Apr 20 '18 at 20:24
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Do you mean the probability that all pairs are mutual? Or that a certain amount of them are mutual? – user Apr 20 '18 at 20:55
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$p=1$? ... why wouldn't nearness be mutual? – phdmba7of12 Apr 20 '18 at 20:58
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I actually need the ratio of this mutual case happening? Assume we find and match the closest red dot for each blue dot. Out of these total M red dots (some maybe the same) matched to blue dots, how many consider its matching red dot as the closest among all blue dots? – Eyup Apr 20 '18 at 21:04
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Not sure this is all that helpful but here's a related question (which asked for a different probability): https://math.stackexchange.com/questions/2363841/matching-red-to-blue-dots – antkam Apr 20 '18 at 21:06
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Is there any reason you expect this problem to have a nice answer? Any context you could give might help us solve it. – Mike Earnest Apr 20 '18 at 21:27
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Context will confuse things. But I use it in stable matching where the preference lists are defined based on the distance of other set users. I noticed that pretty good amount of users (50-56% with N=M) are matched to their first preferences mutually. I want to analytically prove it. – Eyup Apr 20 '18 at 21:59