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

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Deterministic strategy

• Let U be a set of possible choices
  u1,u2,...,un.
• Let L U?R 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
  u1,u2,...,un.
• Let L U?R be a utility function.
• Let p U?R 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.

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Where is top-k?

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

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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

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An Example of Nature
L U x T? R
U
T
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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

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Taxonomy of Decision Making

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

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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

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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?

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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.

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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

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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.

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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

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An example of modeling

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

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How to estimate?

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

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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