1. Advanced Topics in Search Theory 1 - Introduction

2. In Today’s Class In Today’s Class  Course procedures  What is economic search?  Characteristics of economic search  Classical models in Search Theory: –One Sided –Two-Sided –Mediated Search  Reservation-Value based search 2

3. Goal  Get familiar with the concept of “economic search”  Learn and master the main principles of economic search: –One-sided –Two-sided 3

4. Course Procedures  Course web-site can be found here: http://www.cs.biu.ac.il/~sarned/Courses/search/  Teacher: David Sarne (sarned@cs.biu.ac.il)sarned@cs.biu.ac.il  Office hours: Tue11:00 (building 216, room 2)  Course exercises – 20%  Course final exam – 80% 4

5. Course Plan 5 WeekTopicReadings 1Introduction to Search Theory 2Pandora’s Problem 3One-Sided Search – principles and optimal strategy 4One sided search with unknown distribution 5Concurrent search 6Cooperative Search 7The secretary Problem 8Market throughput in one-sided search 9Two-Sided Search with no search costs 10Two-Sided Search with search costs multi-type 11Two-Sided Search with search costs with one and two types 12Throughput in two-sided search 13Two-sided search with mediators

6. Disclaimer…  Search in AI: deals with finding nodes having certain properties in a graph (find an optimal path from the initial node to a goal node if one exists) –Branch and bound –A* –Hill climbing –…  This is not what we are interested in (at least in this course)  We deal with economic search 6

7. Have you searched for something lately?  Can you give examples for what you’ve searcher for? 7

8. 8 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

9. How about the “secretary problem”? (also known as the marriage problem, the sultan's dowry problem, the fussy suitor problem)  There is a single secretarial position to fill.  There are n applicants for the position, and the value of n is known.  The applicants can be ranked from best to worst with no ties.  The applicants are interviewed sequentially in a random order, with each order being equally likely.  After each interview, the applicant is accepted or rejected.  The decision to accept or reject an applicant can be based only on the relative ranks of the applicants interviewed so far.  Rejected applicants cannot be recalled.  The object is to select the best applicant. The payoff is 1 for the best applicant and zero otherwise. 9

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

11. 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 11

12. 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. 12

13. CDF  Thecdf 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. 13

14. דוגמה : התפלגות אחידה 14 200300 f(x)=0.01

15. התפלגות בדידה  במקום f(x) אנו מדברים על P(x)  למשל בהטלת קוביה, P(2)=1/6 15

16. Sampling from the distribution  Draw a random value from a uniform distribution  Take the value for which the CDF equals the value drawn 16 P1 P2 P3 P4P4 t f(t) f4 f3 f1 f2 x1x2x3x4x5x

17. Fitting a Distribution  Visualize the Observed Data (decide on how to divide date to bins)  Come up with possible theoretical distributions  Test goodness-of-fit and p-values based on the empirical distribution function (EDF): –Kolmogorov-Smirnov –Chi-Square –Anderson-Darling 17 measures of discrepancy between the empirical distribution function and the cumulative distribution function based on a specified distribution

18. 18

19. 19 Example - Marriage Market legacy domain (search “pioneers”) Lifetime Utility Should I try to do better? f(x)

20. 20 Can we do better?  Yes we can!  However, it has a cost  Thus a search strategy is needed Strategy: (opportunities, time, cost)->(terminate, resume)

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

22. Searching What? ApplicationCostOpportunity Marriage MarketTime / money / loneliness Better partner Job MarketTime / money / confidence Better job ProductTime / moneyBetter price / performance ParkingtimeCloser 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

23. 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 23

24. The Goal  Maximize the expected value of the net profit 24 ApplicationCostOpportunity Marriage MarketTime / money / loneliness Better partner Job MarketTime / money / confidence Better job ProductTime / moneyBetter price / performance ParkingtimeCloser parking space

25. The optimal strategy  Let V* be the expected profit if following the optimal strategy  Clearly the searcher should never accept an opportunity with a value less than V*  If he rejects the opportunity, he is in the same situation as a searcher who is starting anew: expect profit V*  Therefore: 25

26. 26 Example - Marriage Market Lifetime Utility Should I try to do better? f(x) Reservation Value - x In a simple infinite horizon model - doesn’t depend on history

27. What is a reservation value?  It’s a threshold for decision making!  Example: “Krovim Krovim”  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) 27

28. 28 Example - Marriage Market Lifetime Utility Should I try to do better? f(x) Reservation Value - x Terminate Search Resume Search - sample one more In a simple infinite horizon model - doesn’t depend on history

29. 29 Terminate Search Resume Search - sample one more The optimal Reservation Value Lifetime Utility f(x) x Distribution of utilities in the environment (p.d.f / c.d.f) Search cost Expected utility when using reservation value x F(x)

30. 30 The Reservation Value Concept Distribution of utilities in the environment (p.d.f / c.d.f) Search cost Expected utility when using reservation value x What is x that maximizes V(x)? F(x)

31. 31 The Reservation Value Concept

32. 32 Example - Marriage Market Lifetime Utility Should I try to do better? f(x) Reservation Value - x Terminate Search Resume Search - sample one more 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

33. Some more interesting interpretations 33

34. Some more interesting interpretations (2) 34 Stop searching and keeping x* Searching exactly one more time

35. Myopic rule  Important property of the optimal search rule – myopic: –The searcher will never decide to accept an opportunity he has rejected beforehand  Searcher cares only about whether or not he wants the opportunity now  Therefore, we don’t care for the recall option 35

36. Also notice that…  and: 36 Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure".

37. Calculating the optimal RV 37 Notice that:

38. Calculating the optimal RV 38 Therefore:

39. When trying to minimize expense 39

40. When trying to minimize expense 40

41. CS economic search domains  CSAs  Job scheduling  Searching for free space in disks  Searching for media in P2P  Classical tradeoff – time it takes to process vs. time it takes to find a strong processor 41

42. The Scheduling Problem Proxy Price quote (q) Processor 1 Processor 2 Processor N Price quote (q) Scheduling Process c1 c2 cN

43. WorkFlow  Receive a job  Contact proxy to learn about available processors  Query processors by using the proxy –Each query delays you in c_i seconds –Each query will return the temporary load on the server (this value will not change as long as current job is not scheduled)  Keep on querying until you are ready to schedule your job

44. The Goal is…  To schedule the job in a way that minimizes the EXPECTED overall delay –Overall delay = all delays due to queries + the time job waits in queue of the selected processor

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 200-300 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. Solution 200300 f(x) 0.01 Sequential search: x cost of search marginal benefit

47. Find the minimum cost

48. Verification  V(x)=x?  Mean number of merchants visited:  Mean payment to merchant: 214.14-7.14=207 (notice it’s less than minimum of sampling 7 merchants) V

49. Alternative Solution 200300 f(x) 0.01 Sequential search: x cost of search marginal benefit

50. 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 50 Stoppingterminating

51. 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 51

52. 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 52

53. 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. 53

54. 54 Lifetime Utility f(x) Reservation Value - x

55. Comparison Shopping Agents (CSAs)  Shopbots and Comparison Shopping –automatically query multiple vendors for price information –Growing market, growing interest comparison- shopping agents

56. Comparison Shopping Agents (CSAs) Offline - central DB of prices (daily updated): DB Requests UI Query Timely Updates Real-time querying upon receiving a request: Requests UI Query

57. Real-Time Querying (CSAs) Ever-increasing frequency of price updates Dynamic pricing theories (based on competitors’ prices) [Greenwald and Kephart, 1999] “Hit and run” sales strategies (short term price promotions at unpredictable intervals) [Baye et al, 2004] Assumption: Future CSAs will use real-time (costly) querying