The optimal marriage

2,298 views

Published on

My tea-time talk about

Published in: Education, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
2,298
On SlideShare
0
From Embeds
0
Number of Embeds
1,373
Actions
Shares
0
Downloads
9
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

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

×