Advanced Topics in Search Theory

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Advanced Topics in Search Theory

• 1 - Introduction

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In Todays 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

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Goal

• Get familiar with the concept of economic
  search
• Learn and master the main principles of economic
  search
• One-sided
• Two-sided

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Course Procedures

• Course web-site can be found here
• http//www.cs.biu.ac.il/sarned/Courses/search/
• Teacher David Sarne (sarned_at_cs.biu.ac.il)
• Office hours Thu 1500-1600 (building 216, room
  2)
• Course exercises 20
• Course final exam 80

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

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Have you searched for something lately?

• Can you give examples for what youve searcher
  for?

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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
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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.

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Example - Marriage Marketlegacy domain (search
pioneers)
f(x)

• Lifetime Utility

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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

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PDF

• The pdf of X, is a function f(x) such that for
  two numbers, a and b with ab
• 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.

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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.

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????? ??????? ?????
f(x)0.01
200
300
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??????? ?????

• ????? f(x) ??? ?????? ?? P(x)
• ???? ????? ?????, P(2)1/6

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Sampling from the distribution
f(t)
f4
P2
f3
P4
f2
P1
f1
P3
t
x1
x2
x3
x4
x5
x

• Draw a random value from a uniform distribution
• Take the value for which the CDF equals the value
  drawn

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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

measures of discrepancy between the empirical
distribution function and the cumulative
distribution function based on a specified
distribution
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Comparison Shopping Agents (CSAs)

• Shopbots and Comparison Shopping
• automatically query multiple vendors for price
  information
• Growing market, growing interest

comparison-shopping agents
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Comparison Shopping Agents (CSAs)
Offline - central DB of prices (daily updated)
Real-time querying upon receiving a request
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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
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Exercise

• Select 5 different products (preferable
  electronics, computers etc.)
• Collect Prices for these products over the
  internet build their empirical distribution (at
  least 50 prices for each)
• Fit to a know distribution or describe the
  empirical distribution obtained
• Calculate the optimal search rule
• Send all the data with your file

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Example - Marriage Marketlegacy domain (search
pioneers)
f(x)

• Lifetime Utility

Should I try to do better?
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Can we do better?

• Yes we can!
• However, it has a cost
• Thus a search strategy is needed

Strategy (opportunities, time,
cost)-gt(terminate, resume)
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Search Characteristics

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

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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 dont attempt to find the best
opportunity, instead find the best policy
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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

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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
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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

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Example - Marriage Market
f(x)
Reservation Value - x

• Lifetime Utility

Should I try to do better?
In a simple infinite horizon model - doesnt
depend on history
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What is a reservation value?

• Its 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)

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Example - Marriage Market
f(x)
Terminate Search
Resume Search - sample one more
Reservation Value - x

• Lifetime Utility

Should I try to do better?
In a simple infinite horizon model - doesnt
depend on history
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The optimal Reservation Value
f(x)
Terminate Search
Resume Search - sample one more

• Lifetime Utility

x
Distribution of utilities in the environment
(p.d.f / c.d.f)
Expected utility when using reservation value x
Search cost
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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
What is x that maximizes V(x)?
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The Reservation Value Concept
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Example - Marriage Market
f(x)
Terminate Search
Resume Search - sample one more
Reservation Value - x

• Lifetime Utility

Should I try to do better?
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 Im willing to accept
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Some more interesting interpretations
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Some more interesting interpretations (2)
Stop searching and keeping x
Searching exactly one more time
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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 dont care for the recall option

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Also notice that

• and

Bernoulli trial is an experiment whose outcome is
random and can be either of two possible
outcomes, "success" and "failure".
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Calculating the optimal RV
Notice that
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Calculating the optimal RV
Therefore
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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

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

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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

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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? Whats the mean of the number of merchants
  youll visit?

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Solution
f(x)
0.01
200
300

• Sequential search

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Find the minimum cost
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Verification

• V(x)x?
• Mean number of merchants visited
• Mean payment to merchant 214.14-7.14207
• (notice its less than minimum of sampling 7
  merchants)

V
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Alternative Solution
f(x)
0.01
200
300

• Sequential search

marginal benefit
cost of search
x