# Secretary Problems. October 21, José Soto SPAMS

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1 Secretary Problems October 21, 2010 José Soto SPAMS

2 A little history 50 s: Problem appeared. 60 s: Simple solutions: Lindley, Dynkin : Generalizations. It became a field. ( Toy problem for Theory of Optimal Stopping) 80-now: Generalizations to Game Theory, Decision Theory, Combinatorial Optimization, etc... Other: Cayley s problem (1875). Gardner s Googol Problem (1960)

3 Rules. 1. Want to choose one object. 2. Number n of objects is known. 3. Objects appear sequentially in uniform random order. 4. Objects are rankable. 5. Accepted or rejected before the next object appears. 6. Decision depend only on the relative ranks. 7. Rejected objects can not be recalled. 8. Payoff: Only win if the best is selected.

4 Toy Examples. Secretary Problem: Hire a secretary among n candidates. Marriage Problem: Getting married with one of n possible people. Sultan s Dowry Problem: A Sultan granted a commoner a chance to marry one of his n daughters, each one has a different dowry. Random auction: Sell an item to one of n agents.

5 Variations / Break the rules. Normal Variation Select one Combinatorial restrictions. Known n Unknown n. Unif. Random order Dependent order. Object rankable Limited Confidence. Immediate decision Deferred decision. Rank-dependent decision Value or time dependent decision. Can not recall rejected They accept with certain prob. Win if best selected Other payoff functions. Classic variations: Cayley s problem: Urn with n balls labeled 1,..., n. Select at random, k chances. When to stop? Googol problem: Two player version.

6 Variations / Break the rules. Normal Variation Select one Combinatorial restrictions. Known n Unknown n. Unif. Random order Dependent order. Object rankable Limited Confidence. Immediate decision Deferred decision. Rank-dependent decision Value or time dependent decision. Can not recall rejected They accept with certain prob. Win if best selected Other payoff functions. Classic variations: Cayley s problem: Urn with n balls labeled 1,..., n. Select at random, k chances. When to stop? Googol problem: Two player version.

7 First solution for original problem. Observe half and find X=best. Select first element Y better than X. Win with probability at least 1/4. Can show (e.g. by Backwards Induction, or using properties of Markov Chains) that the optimum rule is of the form: For some r(n), reject r 1 elements. Accept the first element of relative rank 1 (RECORD) after that.

8 First solution for original problem. Observe half and find X=best. Select first element Y better than X. Win with probability at least 1/4. Can show (e.g. by Backwards Induction, or using properties of Markov Chains) that the optimum rule is of the form: For some r(n), reject r 1 elements. Accept the first element of relative rank 1 (RECORD) after that.

9 First solution for original problem. Observe half and find X=best. Select first element Y better than X. Win with probability at least 1/4. Can show (e.g. by Backwards Induction, or using properties of Markov Chains) that the optimum rule is of the form: For some r(n), reject r 1 elements. Accept the first element of relative rank 1 (RECORD) after that.

10 First solution for original problem. Observe half and find X=best. Select first element Y better than X. Win with probability at least 1/4. Can show (e.g. by Backwards Induction, or using properties of Markov Chains) that the optimum rule is of the form: For some r(n), reject r 1 elements. Accept the first element of relative rank 1 (RECORD) after that.

11 First solution for original problem. Observe half and find X=best. Select first element Y better than X. Win with probability at least 1/4. Can show (e.g. by Backwards Induction, or using properties of Markov Chains) that the optimum rule is of the form: For some r(n), reject r 1 elements. Accept the first element of relative rank 1 (RECORD) after that.

12 Finding the best r } {{ } r 1 }{{} i-th P i = Probability that BEST is in i-th position AND we select it. (We need that no element with relative rank 1 is after the wall.) { 0 if i r 1 P i = 1 n r 1 i 1 if i r. Prob. to WIN = n i=r P i = r 1 n n i=r 1 i 1. Maximized on the the maximum value of r such that (roughly r = n/e and Pr = 1/e 36.7%) n 1 r 1

13 Finding the best r } {{ } r 1 }{{} i-th P i = Probability that BEST is in i-th position AND we select it. (We need that no element with relative rank 1 is after the wall.) { 0 if i r 1 P i = 1 n r 1 i 1 if i r. Prob. to WIN = n i=r P i = r 1 n n i=r 1 i 1. Maximized on the the maximum value of r such that (roughly r = n/e and Pr = 1/e 36.7%) n 1 r 1

14 Finding the best r } {{ } r 1 }{{} i-th P i = Probability that BEST is in i-th position AND we select it. (We need that no element with relative rank 1 is after the wall.) { 0 if i r 1 P i = 1 n r 1 i 1 if i r. Prob. to WIN = n i=r P i = r 1 n n i=r 1 i 1. Maximized on the the maximum value of r such that (roughly r = n/e and Pr = 1/e 36.7%) n 1 r 1

15 Finding the best r } {{ } r 1 }{{} i-th P i = Probability that BEST is in i-th position AND we select it. (We need that no element with relative rank 1 is after the wall.) { 0 if i r 1 P i = 1 n r 1 i 1 if i r. Prob. to WIN = n i=r P i = r 1 n n i=r 1 i 1. Maximized on the the maximum value of r such that (roughly r = n/e and Pr = 1/e 36.7%) n 1 r 1

16 Example, 100 daughter Sultan. Reject first 37, accept the first RECORD you see.

17 What to do if n is unknown.??? If n is random w/known distribution. Can we WIN? Example: n uniform from {1, 2,..., N}. Surprisingly, BEST STRATEGY is still to wait until a fixed position r(n) and select first RECORD you see. For large N, r N e 1/2 and Pr(WIN) 2e %. For general distributions the best strategy is of the form: Select the first RECORD in positions [a 1, b 1 ] [a 2, b 2 ] [a k, N].

18 What to do if n is unknown.??? If n is random w/known distribution. Can we WIN? Example: n uniform from {1, 2,..., N}. Surprisingly, BEST STRATEGY is still to wait until a fixed position r(n) and select first RECORD you see. For large N, r N e 1/2 and Pr(WIN) 2e %. For general distributions the best strategy is of the form: Select the first RECORD in positions [a 1, b 1 ] [a 2, b 2 ] [a k, N].

19 What to do if n is unknown.??? If n is random w/known distribution. Can we WIN? Example: n uniform from {1, 2,..., N}. Surprisingly, BEST STRATEGY is still to wait until a fixed position r(n) and select first RECORD you see. For large N, r N e 1/2 and Pr(WIN) 2e %. For general distributions the best strategy is of the form: Select the first RECORD in positions [a 1, b 1 ] [a 2, b 2 ] [a k, N].

20 What to do if n is unknown.??? If n is random w/known distribution. Can we WIN? Example: n uniform from {1, 2,..., N}. Surprisingly, BEST STRATEGY is still to wait until a fixed position r(n) and select first RECORD you see. For large N, r N e 1/2 and Pr(WIN) 2e %. For general distributions the best strategy is of the form: Select the first RECORD in positions [a 1, b 1 ] [a 2, b 2 ] [a k, N].

21 What if we only know n N.??? Can we WIN? (Optimal strategy is not simple.) Next one is almost optimal: Every time you see a RECORD, accept it with prob. p. What p should we choose? E[#Records(n)] = n i=1 1/i = H n H N. By choosing p = 1/H N we get Pr(WIN) 1/(eH N ).

22 What if we only know n N.??? Can we WIN? (Optimal strategy is not simple.) Next one is almost optimal: Every time you see a RECORD, accept it with prob. p. What p should we choose? E[#Records(n)] = n i=1 1/i = H n H N. By choosing p = 1/H N we get Pr(WIN) 1/(eH N ).

23 What if we only know n N.??? Can we WIN? (Optimal strategy is not simple.) Next one is almost optimal: Every time you see a RECORD, accept it with prob. p. What p should we choose? E[#Records(n)] = n i=1 1/i = H n H N. By choosing p = 1/H N we get Pr(WIN) 1/(eH N ).

24 What if we only know n N.??? Can we WIN? (Optimal strategy is not simple.) Next one is almost optimal: Every time you see a RECORD, accept it with prob. p. What p should we choose? E[#Records(n)] = n i=1 1/i = H n H N. By choosing p = 1/H N we get Pr(WIN) 1/(eH N ).

25 Example, selecting from at most 100 secretaries. The previous strategy has prob. 1/(eH 100 ) 7% to win. The best strategy has prob. 1/H % to win.

26 Another case: unknown n arriving in a fix time period. Assume people arrive in interval [0, 1] independently. May assume also uniformly. Can NOT beat probability 1/e to win. Can we achieve it? Fix a wall at time T and select first RECORD after T. Let t be the time at which BEST arrives. [ ] T 1 T Pr(Win) E t 1 {t > T} = dt = T ln T. t t Optimum at T = 1/e, with Pr(Win) = 1/e 36.7%. T

27 Another case: unknown n arriving in a fix time period. Assume people arrive in interval [0, 1] independently. May assume also uniformly. Can NOT beat probability 1/e to win. Can we achieve it? Fix a wall at time T and select first RECORD after T. Let t be the time at which BEST arrives. [ ] T 1 T Pr(Win) E t 1 {t > T} = dt = T ln T. t t Optimum at T = 1/e, with Pr(Win) = 1/e 36.7%. T

28 Another case: unknown n arriving in a fix time period. Assume people arrive in interval [0, 1] independently. May assume also uniformly. Can NOT beat probability 1/e to win. Can we achieve it? Fix a wall at time T and select first RECORD after T. Let t be the time at which BEST arrives. [ ] T 1 T Pr(Win) E t 1 {t > T} = dt = T ln T. t t Optimum at T = 1/e, with Pr(Win) = 1/e 36.7%. T

29 Example: Best age to consider getting married. A 1 A 2 Looking from age A 1 to age A 2, roughly uniform. Only getting married once, and only happy with the best choice. Must decide over a candidate before the next arrive. Best strategy: Sample until A 1 + (A 2 A 1 )/e and then choose first Record. Win with prob. 1/e 36.7%. Example: If A 1 = 18, A 2 = 40, then wall is at /e 26.

30 Example: Best age to consider getting married. A 1 A 2 Looking from age A 1 to age A 2, roughly uniform. Only getting married once, and only happy with the best choice. Must decide over a candidate before the next arrive. Best strategy: Sample until A 1 + (A 2 A 1 )/e and then choose first Record. Win with prob. 1/e 36.7%. Example: If A 1 = 18, A 2 = 40, then wall is at /e 26.

31 Same game. Different utility function. 0 1 a Utility = b c marry the BEST stay single marry other. with a b c. Where should we put the wall T? Pr(marry the BEST) = T ln T. Pr(stay single) = T. Pr(marry other) = 1 (T T ln T). Hence E[Utility] = at ln T + bt c(1 T + T ln T) Optimum at T = exp( (a b)/(a c)).

32 Same game. Different utility function. 0 1 a Utility = b c marry the BEST stay single marry other. with a b c. Where should we put the wall T? Pr(marry the BEST) = T ln T. Pr(stay single) = T. Pr(marry other) = 1 (T T ln T). Hence E[Utility] = at ln T + bt c(1 T + T ln T) Optimum at T = exp( (a b)/(a c)).

33 Same game. Different utility function marry the BEST Utility = 0 stay single 1 marry other. with a b c. Optimum: A = / e Pr(marry the BEST) = 1/(2 e) 30.33%. Pr(stay single) = 1/ e 60.65%. Pr(marry other) = 1 3/(2 e) 9.02%. And E[Utility] = 1 + 2/ e

34 Other utility functions. Second Best Suppose (for some reason) that we WIN only if we select the 2nd one. Optimal strategy: Sample first (n + 1)/2 candidates, and then choose first element with relative rank 2. Pr(Win) 1/4 as n. Note that 1/4 < 1/e, hence getting second one is HARDER.

35 Other utility functions. Minimum Rank. Want to Minimize the expected rank of the element selected. Optimal strategy is complex. For each position i, define a threshold H n (i) a priori and select i if and only if the relative rank of it is at least H n (i) For the best choices of H n (i), E[rank] as n. H n (i) = i 1 n + 1 R n(i + 1). R n (i) = i H n(i) R n (i + 1) + n + 1 i i(i + 1) Hn(i)(H n (i) + 1). 2 H n (n) = n, R n (n) = n

36 Other utility functions. Minimum Rank. Want to Minimize the expected rank of the element selected. Optimal strategy is complex. For each position i, define a threshold H n (i) a priori and select i if and only if the relative rank of it is at least H n (i) For the best choices of H n (i), E[rank] as n. H n (i) = i 1 n + 1 R n(i + 1). R n (i) = i H n(i) R n (i + 1) + n + 1 i i(i + 1) Hn(i)(H n (i) + 1). 2 H n (n) = n, R n (n) = n

37 Game of Googol. 1. Player I writes n diff. numbers (in [0, 1] or [0, ]). 2. Player II uncovers them at random. 3. Player II wins if he stops on biggest. O.w, Player I wins. Player II has 1/e prob of winning for any strategy of Player I. Player II can now SEE the numbers, can he do better?

38 Game of Googol. 1. Player I writes n diff. numbers (in [0, 1] or [0, ]). 2. Player II uncovers them at random. 3. Player II wins if he stops on biggest. O.w, Player I wins. Player II has 1/e prob of winning for any strategy of Player I. Player II can now SEE the numbers, can he do better?

39 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

40 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

41 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

42 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

43 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

44 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

45 Game of Googol (for Player II and n large) What if P.II. knows that... P.I. selects the numbers {1,..., n}? He WINS with prob. 1 P.I. selects numbers uniformly in [0, 1] i.i.d.? Simple strategy: Select any number better than 1 t/n. t 1.5 leads to Pr(WIN) 51.7%. Best strategy: For every i, selects a threshold t(i), Best selection gives Pr(WIN) 58%. P.I. selects number i.i.d. from known distribution? Same as for uniform. Can Player I decrease the probability of winning of P.II?

46 Game of Googol (for Player I and n large) What if P.II. knows that... P.I. selects i.i.d. from an UNKNOWN distribution? Normal N(µ, 1). Uniform [θ 1/2, θ + 1/2]..

47 Game of Googol (for Player I and n large) What if P.II. knows that... P.I. selects i.i.d. from a PARAMETRIC FAMILY of dist? Normal N(µ, 1). Uniform [θ 1/2, θ + 1/2]..

48 Game of Googol (for Player I and n large) What if P.II. knows that... P.I. selects i.i.d. from a PARAMETRIC FAMILY of dist? Normal N(µ, 1). P. II. estimate µ with tiny sample. WINS with prob. 58%. Uniform [θ 1/2, θ + 1/2]..

49 Game of Googol (for Player I and n large) What if P.II. knows that... P.I. selects i.i.d. from a PARAMETRIC FAMILY of dist? Normal N(µ, 1). P. II. estimate µ with tiny sample. WINS with prob. 58%. Uniform [θ 1/2, θ + 1/2]. P. II can still find decision thresholds, such that he WINS with prob. 43.5%.

50 Game of Googol (for Player I and n large) What if P.II. knows that... P.I. selects i.i.d. from a PARAMETRIC FAMILY of dist? Normal N(µ, 1). P. II. estimate µ with tiny sample. WINS with prob. 58%. Uniform [θ 1/2, θ + 1/2]. P. II can still find decision thresholds, such that he WINS with prob. 43.5%. Combined distribution: Select θ Pareto(α, 1), and n numbers from Uniform[0, θ].

51 Game of Googol (for Player I and n large) What if P.II. knows that... P.I. selects i.i.d. from a PARAMETRIC FAMILY of dist? Normal N(µ, 1). P. II. estimate µ with tiny sample. WINS with prob. 58%. Uniform [θ 1/2, θ + 1/2]. P. II can still find decision thresholds, such that he WINS with prob. 43.5%. Combined distribution: Select θ Pareto(α, 1), and n numbers from Uniform[0, θ]. ε > 0, n, there is α such that Pr(P.II. wins) 1/e + ε.

52 Selecting more than one object Suppose we want to select more than one object. Example: At most k object out of the n candidates. Reasonable objective?

53 Selecting more than one object Suppose we want to select more than one object. Example: At most k object out of the n candidates. Reasonable objective? Expected total value of the objects.

54 Selecting more than one object Suppose we want to select more than one object. Example: At most k object out of the n candidates. Reasonable objective? Expected total value of the objects. If k = 1 we can get 1/e fraction of the Optimum. What if k 2?

55 Selecting more than one object Suppose we want to select more than one object. Example: At most k object out of the n candidates. Reasonable objective? Expected total value of the objects. If k = 1 we can get 1/e fraction of the Optimum. What if k 2? Simple strategy: Each of top k elements is selected w.p. 1/e.

56 Selecting more than one object Suppose we want to select more than one object. Example: At most k object out of the n candidates. Reasonable objective? Expected total value of the objects. If k = 1 we can get 1/e fraction of the Optimum. What if k 2? Simple strategy: Each of top k elements is selected w.p. 1/e. Best strategy: Can recover 1 θ(1/ k) fraction of the optimum.

57 Extension: Combinatorial objects. Want to select a maximum weight set of balls of the same color spanning tree of linearly independent vectors.

58 Extension: Combinatorial objects. Want to select a maximum weight set of balls of the same color. ω(log n/ log log n)-comp spanning tree. O(1)-comp of linearly independent vectors. O(log dim.)-comp OPEN: Find an O(1)-comp. algorithm.

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