On the multi-item full-information Secretary Problem – max expectation criterion

1. On the multi-item full-information Secretary Problem – max expectation criterion

2. Simple Full-Info Secretary is Expected-Value DP 2.336 2.17 1.9

3. The classical "House-Selling" problem |>---×------×---×------| 0 T Renewal offers with inter-arrival dist G i.i.d positive offers X with (known) dist F One unit to be sold by time T No recall, max Expectation.

4. Karlin (1962) Assume finite expectation of offer value. Optimal policy: Stop iff {x > y(t)}. If G is exp(l) (Poisson arrival): t measured backwards.

5. I have a number of tickets for the play-off of Maccabee, it is two weeks ahead of the game. People who wish to by tickets from me call me about 5 times a day. The bids for a ticket are distributed the same and I know this distribution function and have to decide if I accept a bid or reject it. If I reject a bid it becomes unavailable later. How would I know whether to accept a bid or to reject it?

6. n - number of units to sell. y1 (t), y2 (t), … , yn (t) - optimal threshold functions. En (t) −value function at time t assuming that an offer has just been acted upon. (because max{ x + En-1(t) , En(t)})

7. A recursive system of equations for the threshold curves Theorem: The yn's are differentiable and for We let Initial conditions:

9. Exponential offer-value (discovered before by Sakaguchi, 1976)

10. 1 q2 qk C1 C2 Ck-1 Ck Finitely-many values per offer

12. Critical times i = 1, …, k – 1 j = 1, …, n 500 400 300 200 100 0 sell ticket j for Ci (or more) iff t < tj (i) 0 2 4 6 8 10 12 14

13. Manual solution with n = 2 , k = 2 Example: l = 1, n = 2, c1 = 1, c2 = 3, p1 = p2 = 0.5. Q(y) = 2 – y for yc1, Q(y) = (3-y)/2 for y > c1. Using t1 < t2,

14. j =1 y j=2 t2 t1 t

17. Type of solution: In case of n items and |C| =k the value function (and the last threshold function) is composed of n×(k-1) expressions relative to n×(k-1) sub-intervals. A typical such expression is (here n=10, k=2): 2-1/181440*exp(-x)*x^9-1/20160*exp(-x)*x^8-1/2520*exp(-x)*x^7-1/360*exp(-x)*x^6-1/60*exp(-x)*x^5-1/12*exp(-x)*x^4-1/3*exp(-x)*x^3-exp(-x)*x^2-2*exp(-x)*x-2*exp(-x)

18. Way of solving for the y(t)’s and the tj(i)’s • Matlab offers a recursive symbolic solution for differential equations. • Matlab can “revive” the symbolic solution so as to solve scalar equations based on this solution. Still, Matlab is presently restricted to cases with n(|C|–1)<13.

19. Overcoming the SYMBOLIC restriction Numerical solution (Matlab’s ODE functions). Advantage: once the threshold function (numerical array) for the (j-1)th unit is built, there is no need for any lower-index function. The number of segments in a threshold curve is |C|, non-increasing with n.

20. Example Suppose we hold 7 tickets for the Maccabee game. Offers arrive in values of 100, 200, 300, 400, 500 with probabilities 0.1, 0.2, 0.2, 0.3, 0.2 respectively. 2 offers arrive, on the average, every day. There are two weeks remaining to the game.

21. 500 400 300 200 100 0 0 2 4 6 8 10 12 14

22. 500 400 300 200 100 0 28 0 2 4 6 8 10 12 14

23. 500 400 300 200 100 0 5 0 2 4 6 8 10 12 14