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The optimal marriage

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The optimal marriage

1. 1. Theory Applications Experiments The optimal marriage Ferenc Huszár Computational and Biological Learning Lab Department of Engineering, University of Cambridge May 14, 2010optimal marriage - tea talk CBL
2. 2. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
3. 3. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1. you have to choose one partner to marry 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
4. 4. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1. you have to choose one partner to marry 2. The number of potential partners, N, is ﬁnite and known 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
5. 5. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1. you have to choose one partner to marry 2. The number of potential partners, N, is ﬁnite and known 3. the N partners are “tried” sequentially in a random order1 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
6. 6. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1. you have to choose one partner to marry 2. The number of potential partners, N, is ﬁnite and known 3. the N partners are “tried” sequentially in a random order1 4. There is a clear ranking of partners, the decision is either accept or reject based only on the relative ranking of partners “tried’ ’ so far 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
7. 7. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1. you have to choose one partner to marry 2. The number of potential partners, N, is ﬁnite and known 3. the N partners are “tried” sequentially in a random order1 4. There is a clear ranking of partners, the decision is either accept or reject based only on the relative ranking of partners “tried’ ’ so far 5. once rejected a partner cannot be called back 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
8. 8. Theory Applications ExperimentsThe standard marriage problema.k.a. the standard secretary problem Marriage as an optimal stopping problem: 1. you have to choose one partner to marry 2. The number of potential partners, N, is ﬁnite and known 3. the N partners are “tried” sequentially in a random order1 4. There is a clear ranking of partners, the decision is either accept or reject based only on the relative ranking of partners “tried’ ’ so far 5. once rejected a partner cannot be called back 6. you are satisﬁed by nothing but the best (0-1 loss) 1 uniform distribution over permutationsoptimal marriage - tea talk CBL
9. 9. Theory Applications ExperimentsThe optimal strategyin the standard marriage problemoptimal marriage - tea talk CBL
10. 10. Theory Applications ExperimentsThe optimal strategyin the standard marriage problem there is no point of accepting anyone who is not the best so faroptimal marriage - tea talk CBL
11. 11. Theory Applications ExperimentsThe optimal strategyin the standard marriage problem there is no point of accepting anyone who is not the best so far P[#r is the best |#r is the best in ﬁrst r ] = 1/N = N 1/r roptimal marriage - tea talk CBL
12. 12. Theory Applications ExperimentsThe optimal strategyin the standard marriage problem there is no point of accepting anyone who is not the best so far P[#r is the best |#r is the best in ﬁrst r ] = 1/N = N 1/r r ∗ the optimal strategy is a cutoﬀ rule with threshold r : reject ﬁrst r ∗ − 1, then accept the ﬁrst, that is best-so-faroptimal marriage - tea talk CBL
13. 13. Theory Applications ExperimentsThe optimal strategyin the standard marriage problem there is no point of accepting anyone who is not the best so far P[#r is the best |#r is the best in ﬁrst r ] = 1/N = N 1/r r ∗ the optimal strategy is a cutoﬀ rule with threshold r : reject ﬁrst r ∗ − 1, then accept the ﬁrst, that is best-so-far determining r ∗ : φN (r ∗ ) = P[you win with threshold r ∗ ] N = P[#j is the best and you select it] j=r ∗ N N 1 r∗ − 1 r∗ − 1 1 = = j=r ∗ N j −1 N j=r ∗ j − 1optimal marriage - tea talk CBL
14. 14. Theory Applications ExperimentsThe optimal strategyin the standard marriage problem there is no point of accepting anyone who is not the best so far P[#r is the best |#r is the best in ﬁrst r ] = 1/N = N 1/r r ∗ the optimal strategy is a cutoﬀ rule with threshold r : reject ﬁrst r ∗ − 1, then accept the ﬁrst, that is best-so-far determining r ∗ : φN (r ∗ ) = P[you win with threshold r ∗ ] N = P[#j is the best and you select it] j=r ∗ N N 1 r∗ − 1 r∗ − 1 1 = = j=r ∗ N j −1 N j=r ∗ j − 1 r ∗ (N) = argmaxr φN (r )optimal marriage - tea talk CBL
15. 15. Theory Applications ExperimentsAssymptotic behaviourin the standard marriage problemoptimal marriage - tea talk CBL
16. 16. Theory Applications ExperimentsAssymptotic behaviourin the standard marriage problem r introduce x = limN→∞ N N r −1 N 1 φN (r ) = N j=r j −1 N 1 1 →x dt = −x log x =: φ∞ (x ) x toptimal marriage - tea talk CBL
17. 17. Theory Applications ExperimentsAssymptotic behaviourin the standard marriage problem r introduce x = limN→∞ N N r −1 N 1 φN (r ) = N j=r j −1 N 1 1 →x dt = −x log x =: φ∞ (x ) x t this is maximised by x ∗ = 1 e ≈ 0.37optimal marriage - tea talk CBL
18. 18. Theory Applications ExperimentsAssymptotic behaviourin the standard marriage problem r introduce x = limN→∞ N N r −1 N 1 φN (r ) = N j=r j −1 N 1 1 →x dt = −x log x =: φ∞ (x ) x t this is maximised by x ∗ = 1 e ≈ 0.37 probability of winning is also φ∞ (x ∗ ) = 1 eoptimal marriage - tea talk CBL
19. 19. Theory Applications ExperimentsReal-world applicationﬁnding a long-term relationship in Hungaryoptimal marriage - tea talk CBL
20. 20. Theory Applications ExperimentsReal-world applicationﬁnding a long-term relationship in Hungary total population of Hungary: 10,090,330optimal marriage - tea talk CBL
21. 21. Theory Applications ExperimentsReal-world applicationﬁnding a long-term relationship in Hungary total population of Hungary: 10,090,330 single/widowed/divorced women,aged 20-29: 533,142 = Noptimal marriage - tea talk CBL
22. 22. Theory Applications ExperimentsReal-world applicationﬁnding a long-term relationship in Hungary total population of Hungary: 10,090,330 single/widowed/divorced women,aged 20-29: 533,142 = N r ∗ (533, 142) ≈ 196, 132optimal marriage - tea talk CBL
23. 23. Theory Applications ExperimentsReal-world applicationﬁnding a long-term relationship in Hungary total population of Hungary: 10,090,330 single/widowed/divorced women,aged 20-29: 533,142 = N r ∗ (533, 142) ≈ 196, 132 probability of ﬁnding the best is around 0.37optimal marriage - tea talk CBL
24. 24. Theory Applications ExperimentsReal-world applicationﬁnding a long-term relationship in Hungary total population of Hungary: 10,090,330 single/widowed/divorced women,aged 20-29: 533,142 = N r ∗ (533, 142) ≈ 196, 132 probability of ﬁnding the best is around 0.37 “try” and reject 200,000 partners before even thinking of marriageoptimal marriage - tea talk CBL
25. 25. Theory Applications ExperimentsHuman experimentsoptimal marriage - tea talk CBL
26. 26. Theory Applications ExperimentsHuman experiments Kahan et al (1967): absolute value instead of rankingoptimal marriage - tea talk CBL
27. 27. Theory Applications ExperimentsHuman experiments Kahan et al (1967): absolute value instead of ranking Rapoport and Tversky (1970): absolute values drawn Gaussian valuesoptimal marriage - tea talk CBL
28. 28. Theory Applications ExperimentsHuman experiments Kahan et al (1967): absolute value instead of ranking Rapoport and Tversky (1970): absolute values drawn Gaussian values Kogut (1999): lowest price of an item with known price distributionoptimal marriage - tea talk CBL
29. 29. Theory Applications ExperimentsHuman experiments Kahan et al (1967): absolute value instead of ranking Rapoport and Tversky (1970): absolute values drawn Gaussian values Kogut (1999): lowest price of an item with known price distribution Seale and Rapoport (1997): the standard marriage problemoptimal marriage - tea talk CBL
30. 30. Theory Applications ExperimentsHuman experiments Kahan et al (1967): absolute value instead of ranking Rapoport and Tversky (1970): absolute values drawn Gaussian values Kogut (1999): lowest price of an item with known price distribution Seale and Rapoport (1997): the standard marriage problem all studies found that subjects stopped earlier than optimaloptimal marriage - tea talk CBL
31. 31. Theory Applications ExperimentsHuman experiments Kahan et al (1967): absolute value instead of ranking Rapoport and Tversky (1970): absolute values drawn Gaussian values Kogut (1999): lowest price of an item with known price distribution Seale and Rapoport (1997): the standard marriage problem all studies found that subjects stopped earlier than optimal explained with a constant cost of evaluaing an optionoptimal marriage - tea talk CBL