A Survey of Preference for Ranking


1
A Survey of Preference for Ranking

• Bin He

2
Preferences

• Preference in top-k query
• score-based preference
• partial-ordering base preference
• Preference in other scenarios
• Decision making
• Utility theory (Skip in this talk)
• Conjoint Analysis
• Voting (Skip in this talk)

3
Strategy of Decision Making

• Deterministic strategy
• Select the only one among many
• Given u1 u2 u3 select u2 w.r.t. the utility
  function
• Randomized strategy
• Select a distribution
• Given u1 u2 u3 select u2 is equal to select
  0 1 0

4
Deterministic strategy

• Let U be a set of possible choices
  u1u2...un.
• Let L UR be a utility function.
• We want to select a ui in U that maximizes L(ui).

5
Randomized strategy

• Let U be a set of possible choices
  u1u2...un.
• Let L UR be a utility function.
• Let p UR be the probability of selecting a
  particular choice ui denoted by p(ui) pi.
• We want to select a u in U that maximizes EL
  SL(ui)pi.

6
Where is top-k

• Currently top-k is deterministic strategy
• no idea whether top-k fits the randomized
  scenario

7
A Game Against Nature

• Incomplete knowledge about the utility function
  L.
• There may be uncertainty involved
• How to describe this uncertainty
• Introduce a special decision-maker called Nature
• T the set of choices for nature
• in T a particular choice by nature

8
An Example of Nature
L U x T R
U
T
9
Strategy with Nature

• The best strategy to adopt depends on what model
  we have of what nature will do
• Nondeterministic
• choose the column with the least maximum value
• or choose the column with the least average loss.
• Probabilistic
• use Bayesian analysis to calculate a probability
  distribution P() of the actions of nature and
  use that to make decisions

10
Where is top-k

• What is the Nature
• It seems that T can be the parameters (cr cs) in
  the cost model
• C crnr csns
• Sampling can be used as get P()
• So we can use the probabilistic strategy

11
Taxonomy of Decision Making

• Normative Interaction
• Things ought to be
• Descriptive Interaction
• Things are
• Prescriptive Interaction
• Things might to be

12
Normative Interaction

• What is a good preference
• Satisfying axioms built by experts
• If A B and B C A C (Transitivity)
• Axioms are rational and intelligent

13
Descriptive Interaction

• The axioms may not be correct
• The transitivity may not hold
• In reality it is possible A B B C and C A
• Then what is a good preference
• Satisfying empirical behavior
• Hard to modelize

14
Prescriptive Interaction

• What is a good preference
• Hard to say depends on how to sell
• Given A and C
• if the expert finds a B so that A B B C
  then we prefer A
• Maybe there is also a B so that C B B A
  then
• The preference can be affected by
• The way to present A and C
• The way to find such a B (or B)

15
What the top-k should be

• Now normative
• In reality can be prescriptive
• Consider the scenario to find a house
• The distance presented in miles and kilometers
  may affect users choice
• The way to present the payment ratio may also
  affect
• But how to model it in mathematical way

16
Conjoint Analysis

• Analyze the value of each factor (attribute
  component predicate).
• Play an important role in marketing.
• In the design of new products it is valuable to
  know which components carry what kind of utility
  for the customer.

17
A simple example

• A car producer plans to introduce a new car with
  two features
• of doors 2 4 5
• of air bags 1 2
• By asking a user we get the ranking as

18
A simple example (cont.)

• What do we want to know
• Elementary utilities
• Utility of 2-door
• Utility of 4-door
• Utility of 5-door
• Utility of 1-airbag
• Utility of 2-airbag
• Conjoint analysis aims at explaining the rank
  order given by the test person as a function of
  these elementary utilities.

19
Estimation of Preference Orderings

• Conjoint analysis uses an additive model
• Xj denote the features xjl are the levels of
  each Xj
• ßjl are the elementary utilities
• the constant µ denotes an overall level
• Yk is the observed preference for each situation

20
An example of modeling

• Given the table
• We have
• 1 ß11ß21µ
• 3 ß11ß22µ
• 2 ß12ß21µ
• 6 ß12ß22µ
• 4 ß13ß21µ
• 5 ß13ß22µ

21
How to estimate

• There are two types of solutions to estimate the
  elementary utilities
• metric solution
• non-metric solution

22
Relation to top-k

• Conjoint analysis looks like query by example
• Given a set of simple examples and let the user
  choose
• We will know what she wants and then query for her