1. Optimal Stopping

2. Have you searched for something lately?
Can you give examples for what you’ve searcher for?

3. Searching What? Everything! Searching for a partner
Searching for a job
Searching for a product
Searching for a parking space
Searching for a java class (reuse)
Search for a thesis advisor …
The goal here is to optimize the process rather than ending up with the optimal search object

4. Example - Marriage Market legacy domain (search “pioneers”)
f(x)
Lifetime Utility

5. Statistics Reminder given a continuous random variable X, we denote:
The probability density function, pdf as f(x). (also known as the probability distribution function and the probability mass function)
The cumulative distribution function, cdf, as F(x).
The pdf and cdf give a complete description of the probability distribution of a random variable

6. PDF
The pdf of X, is a function f(x) such that for two numbers, a and b with a≤b:
That is, the probability that X takes on a value in the interval [a, b] is the area under the density function from a to b.

7. CDF The cdf is a function F(x), defined for a number x by:
That is, for a given value x, F(x) is the probability that the observed value of X will be at most x.

8. דוגמה: התפלגות אחידה
f(x)=0.01
200
300

9. התפלגות בדידה
במקום f(x) אנו מדברים על P(x)
למשל בהטלת קוביה, P(2)=1/6

10. Example - Marriage Market legacy domain (search “pioneers”)
f(x)
Lifetime Utility
Should I try to do better?

11. Can we do better? Yes we can! However, it has a cost
Thus a search strategy is needed
Strategy:
(opportunities, time, cost)->(terminate, resume)

12. Search Characteristics
A distribution of plausible opportunities
The searcher is interested in exploiting one opportunity
Unknown value of specific opportunities
Search costs

13. Searching What?
Application
Cost
Opportunity
Marriage Market
Time / money / loneliness
Better partner
Job Market
Time / money / confidence
Better job
Product
Time / money
Better price / performance
Parking
time
Closer parking space
Looking for a thesis advisor
Working with him a little
More interesting thesis …
Anyone searched for an apartment in her life? What made you take the one you are living in?
Anyone sold an apartment in her life? What made you accept the “winning” bid?
The key concept – don’t attempt to find the best opportunity, instead find the best policy

14. The search strategy After each draw, the searcher has a choice:
Keep what he has
Draw another opportunity from the distribution F(), at a cost c
Notice: the net profit is a random variable whose value depends both on the actual draws and on his decisions to accept or reject particular opportunities

15. The Goal Maximize the expected value of the net profit Application
Cost
Opportunity
Marriage Market
Time / money / loneliness
Better partner
Job Market
Time / money / confidence
Better job
Product
Time / money
Better price / performance
Parking
time
Closer parking space

16. The optimal strategy
Let V* be the expected profit if following the optimal strategy
If infinite decision horizon:
If rejecting an opportunity, he is in the same situation as a searcher who is starting anew: expect profit V*
Therefore the optimal strategy is threshold based
The reservation property of the optimal search rule is a consequence of the stationarity of the search problem (a searcher discarding an opportunity is in exactly the same position as before starting the search)

17. Example - Marriage Market
f(x)
Reservation Value - x
Lifetime Utility
Should I try to do better?
Ring the door bell - just before a blind date
In a simple infinite horizon model - doesn’t depend on history

18. Example - Marriage Market
f(x)
Resume Search - sample one more
Terminate Search
Reservation Value - x
Lifetime Utility
Should I try to do better?
Ring the door bell - just before a blind date
In a simple infinite horizon model - doesn’t depend on history

19. The optimal Reservation Value
f(x)
Resume Search - sample one more
Terminate Search
Lifetime Utility x
Distribution of utilities in the environment (p.d.f / c.d.f)
Expected utility when using reservation value x
Search cost
F(x)

20. The Reservation Value Concept
Distribution of utilities in the environment (p.d.f / c.d.f)
Expected utility when using reservation value x
Search cost
F(x)
What is x that maximizes V(x)?

21. The Reservation Value Concept

22. Example - Marriage Market
f(x)
Resume Search - sample one more
Terminate Search
Reservation Value - x
Lifetime Utility
The expected utility from accepting only “better” partner than the optimal reservation value woman will yield an expected overall utility equal to the “lowest’ partner I’m willing to accept
Should I try to do better?
Ring the door bell - just before a blind date
At the end of the evening the question is: “should I try and do better” (I.e. try to find a better wife)

23. Some more interesting interpretations

24. Some more interesting interpretations (2)
Stop searching and keeping x*
Searching exactly one more time

25. Myopic rule
Searcher cares only about whether or not he wants the opportunity now
Therefore, we don’t care for the recall option

26. Also notice that…
and:
Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure".

27. Problem 1 You are about to purchase an iPod touch over the internet
You estimate the price distribution of the product over the different sellers to be uniform between dollars
You can search by yourself, by visiting different web-sites – the cost of time for obtaining a price quote is $1
How will you search? What will be your expected cost? What’s the mean of the number of merchants you’ll visit?

28. Solution
f(x)
0.01
200
300

29. Find the minimum cost

30. Verification V(x)=x? Mean number of merchants visited:
Mean payment to merchant: =207
(notice it’s less than minimum of sampling 7 merchants) V

31. The finite case Assume we need to choose within N periods of time
We’ll use backward induction – start at period N:
Now consider N-1
Stopping
resuming

32. The finite case (2) If we use x*>x then:
For any best value so far x<z<x* we are supposed to search now (and necessarily terminate after)
Therefore:
However this cannot hold because z>x

33. The finite case (2) If we use x*<x then:
For any best value so far x*<z<x we are supposed to stop now (and necessarily search after)
In that case, a dominating strategy would be to search earlier than later

34. The finite case (4) Expected benefit:
Either we managed to find something above the reservation value x (with probability (1-F(x)^N)
Or, we have searched all N opportunities and everything was below x.

35. f(x)
Reservation Value - x
Lifetime Utility

36. מתי נעדיף לדגום N בו-זמנית?
מועמד השולח בקשות למספר מוסדות
חברה הפונה למספר ספקים לקבל הצעות
תהליך הבודק מספר שרתים בו זמנית

37. דוגמה
המוכרים מתחלקים (באופן שווה) בין כאלה המבקשים 2$ עבור המוצר וכאלה המבקשים 3$
מהי התפלגות המחיר המינימלי אם דוגמים מספר מוכרים?
0.75*2+0.25*3
0.875* *3

38. Distribution of the Minimum Price
fN(x)
fN(x)

39. תוחלת מינימום המחיר
דוגמה – uniform distribution
f(x) 1
P=1 1
F(x) 1 1

40. תוחלת מינימום המחיר (המשך)
You can easily check this with Excel…
Can you guess what is EN(x) for the maximum of a sample?

41. התועלת שבדגימה נוספת
יורד בקצב קטן (הוכחה על-ידי Robert Solow)

42. עלות החיפוש בד"כ פרופורציונאלית למספר המוכרים שנדגום
כמות החיפוש האופטימלית תתקבל על-ידי השוואת הרווח השולי מחיפוש נוסף ועלות החיפוש
cost of search
marginal benefit N

43. דוגמה מחירים מתפלגים בצורה אחידה בין 50 ל- 100 (רציף)
עלות כל מחיר נוסף – 1 דולר
f(x)
0.02
N=5: E_5+5=63.33
N=6: E_6+6=63.14
N=7: E_7+7=63.25 50
100

44. EN+N N

45. Problem 1 You are about to purchase an iPod touch over the internet
You estimate the price distribution of the product over the different sellers to be uniform between dollars
You can search by yourself, by visiting different web-sites – the cost of time for obtaining a price quote is $1
How will you search? What will be your expected cost? What’s the mean of the number of merchants you’ll visit?

46. Problem 1 (cont.)
Alternatively, you can access one of the comparison-shopping web-sites over the internet
Web-site “A” offers you searching 100 web-sites for a total of $10 (average of 10 cent per searched site).
Web-site “B” offers you searching 20 web-sites for a total of $2 (average of 10 cent per searched site)

47. Solution
The minimum of a sample of size 100 can be calculated using:

48. Solution (cont.) Calculating the expected minimum:
Integration by parts:

49. Solution (cont.) 214.14>200.99+10 So we better take this offer…
and here is a simpler way to come up with that:
The minimum of 100 quotes is very close to the 200 lower bound
In fact, we know for the uniform distribution that:
f(x)
0.01
200
300
In the 201 vicinity

50. It can become even simpler…
f(x)
In our problem:
0.01
200
300
f(x) 1 1

51. Solution for web-site B
Conclusion: it’s better to use web-site B