CP nets

1
CP nets

• Toby Walsh
• NICTA and UNSW

2
Representing preferences

• Unfactored
• Not decomposable into parts
• E.g. assign utility to each outcome
• Factored
• Large number of outcomes
• Decompose preference function
• Exploit (conditional) independence

3
Representing preferences

• Quantitative
• My preference for bourbon is 0.8, and for whisky
  is 0.6
• E.g. soft constraints
• Qualitative
• Ordering relation
• Bourbon gt Whisky
• E.g. CP nets

4
CP nets

• Qualitative, conditional factored representation
  of preferences

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CP nets

• Conditional preferences
• If main course is meat then I prefer red wine to
  white
• Ceteris paribus
• All other things being equal
• E.g. the dessert, if it is the same in both
  meals, is irrelevant to our preference on the
  main course
• Binary valued in what follows
• Everything usually generalizes easily to multiple
  valued features

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Ceteris paribus statements

• Simple syntax
• Features X, Y, Z,
• Assignment Xx,Y-y, Zz
• Conditional statement
• Xx Yy gt Y-y
• X-x Y-y gt Yy
• Compact qualitative specification of complex
  preference function
• Exploits independence like Bayesian network

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CP net example

• Unconditional
• Mainfish gt Mainmeat
• Conditional
• Mainfish
• Winewhite gt Winered
• Mainmeat
• Winered gt Winewhite

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CP nets

• Parent feature
• Condition that preference depends on
• E.g. Main course is a parent feature of Wine in
• Mainmeat Winered gt Winewhite
• Defines directed feature graph
• Not necessarily acyclic

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Reasoning with CP nets

• Worsening flip
• Changing value of a feature so that it is less
  preferred in some statement
• E.g. Mainfish, Winewhite to
• Mainfish, Winered as
• Mainfish Winewhite gt Winered

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Reasoning with CP nets

• Ordering on outcomes
• A is preferred to B (AgtB) iff there is a sequence
  of worsening flips from A to B
• Partial order
• A and B can be incomparable

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Example Flying to Australia
Variables and Domains
SA
BA
Airline
bus
eco
Class
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Flying to Australia
If I fly Singapore, I prefer Economy to Business
since I can save money and have enough room
SA eco gt bus
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Flying to Australia
If I fly Singapore, I prefer Economy to Business
since I can save money and have enough room
SA eco gt bus
If I fly British, I prefer Business to Economy
since there is not enough room
BA bus gt eco
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Flying to Australia
If I fly Singapore, I prefer Economy to Business
since I can save money and have enough room
SA eco gt bus
If I fly British, I prefer Business to Economy
since there is not enough room
BA bus gt eco
If I fly Business, I prefer Singapore to British
since it has better service
bus SA gt BA
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Flying to Australia
If I fly Singapore, I prefer Economy to Business
since I can save money and have enough room
SA eco gt bus
If I fly British, I prefer Business to Economy
since there is not enough room
BA bus gt eco
If I fly Business, I prefer Singapore to British
since it has better service
bus SA gt BA
If I fly Economy, I prefer British to Singapore
since I collect British Airlines miles
eco BA gt SA
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Reasoning with CP nets

• Worsening flip
• Changing value of a feature so that it is less
  preferred in some statement
• E.g. Singapore in economy is preferred to
  Singapore in business since
• SA eco gt bus

17
Flying to Australia
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Reasoning with CP nets

• Is A better than B?
• Hard, may be exponential chain of worsening flips
• PSPACE-complete
• Is A optimal?
• Easy for acyclic CP nets, linear time sweep
  algorithm
• NP-hard for cyclic CP nets

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Preferences of multiple agentsmCP-nets
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A dinner party

• Agents have individual preferences
• Alice Bob prefer fish to meat
• Carol prefers meat to fish
• Preferences can be conditional
• If it is fish, Alice prefers white wine to red
• If is is meat, Alice prefers red wine to white

21
A dinner party

• Several notions of optimality
• Meat is Pareto optimal
• Changing to fish would be worse for Carol
• Fish is majority optimal
• Majority of guests prefer fish to meat

22
Preference aggregation

• Represent preferences of each agent
• mCP-net
• For each agent, (partial) CP net
• Soft constraints
• Each agent votes
• Is A gt B?
• How do we add up the votes?
• Run an election!

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Voting semantics

• Pareto order
• A gtp B iff AgtB or A indifferent to B for all
  agents
• Majority order
• A gtmaj B iff
• better gt (worse incomparable)
• Ignore agents who are indifferent
• Max order
• A gtmax B iff
• better gt max(worse,incomparable)

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Voting semantics

• Lex order
• A gtlex B iff
• For agent 1, AgtB
• Or agent 1 is indifferent between them and for
  agent 2, A gt B or
• Rank order
• A gtr B iff sum of ranks(A) lt sum of ranks(B)
• Rank minimal worsening flips to optimal

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Basic properties

• Ordering
• gtp and gtlex are strict partial orders
• Transitive, irreflexive and antisymmetric
• gtmaj and gtmax are not
• Only irreflexive and antisymmetric
• gtr is total order

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Basic properties

• Optimality
• A is gt-optimal iff no B with B gt A
• Existence of optimal outcome?
• Pareto-optimal, majority-optimal, max-optimal,
  lex-optimal, rank-optimal outcomes always exist
• Fairness of aggregation?

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Arrows theorem

• Free
• Transitive
• Independent to irrelevant alternatives
• Monotonic
• Non-dictatorial
• No electoral system on total orders with 2 or
  more voters 3 or more outcomes can satisfy all
  5 fairness properties

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Five fairness properties

• Free
• Any final ordering is possible
• Transitive
• Independent to irrelevant alternatives
• Final ordering of two outcomes only depends on
  how agents vote on these two outcomes
• Monotonic
• One agent changing from BgtA or B indifferent to A
  to AgtB makes A more preferred
• Non-dictatorial
• Final ordering depends on more than one agent

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Some examples

• Pareto order
• All agents are dictators
• Majority and Max orders
• Not transitive
• Lex order
• First agent is a dictator
• Rank order
• Not independent to irrelevant alternatives

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Conclusions

• Representing preferences
• Factored methods like CP nets
• Flipping semantics
• Can extend CP nets to combine the preferences of
  multiple agents
• But based on a (generalization of) Arrows
  theorem, this cannot be fair

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Bibliography

• Reasoning with conditional ceteris-paribus
  preference statements. C. Boutilier, R. Brafman,
  H. Hoos and D. Pooel, Proceedings of UAI-99
• mCP-nets representing and reasoning with
  preferences of multiple agents. Francesca Rossi,
  Brent Venable and Toby Walsh. Proceedings of
  AAAI-2004
• See my web pages for others (e.g. generalization
  of Arrows theorem to partial orders)