CP nets

Toby Walsh

NICTA and UNSW

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

Representing preferences

Quantitative My preference for bourbon is 0.8, and for

whisky is 0.6 E.g. soft constraints

Qualitative Ordering relation:

Bourbon > Whisky E.g. CP nets

CP nets

Qualitative, conditional factored representation of preferences

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

Ceteris paribus statements

Simple syntax Features: X, Y, Z, … Assignment: X=x,Y=-y, Z=z… Conditional statement:

X=x : Y=y > Y=-yX=-x: Y=-y > Y=y

Compact qualitative specification of complex preference function Exploits independence like Bayesian network

CP net example

Unconditional

Main=fish > Main=meat

Conditional

Main=fish :

Wine=white > Wine=red

Main=meat :

Wine=red > Wine=white

CP nets

Parent feature Condition that preference depends on E.g. Main course is a parent feature of Wine in:

Main=meat : Wine=red > Wine=white

Defines directed feature graph Not necessarily acyclic

Reasoning with CP nets

Worsening flip Changing value of a feature so that it is less

preferred in some statement E.g. Main=fish, Wine=white to

Main=fish, Wine=red as

Main=fish : Wine=white > Wine=red

Reasoning with CP nets

Ordering on outcomes A is preferred to B (A>B) iff there is a sequence of

worsening flips from A to B

Partial order A and B can be incomparable

Example: Flying to Australia

Airline

Class

Business classEconomy class

Variables and Domains:

SABA

bus eco

Flying to Australia

If I fly Singapore, I prefer Economy to Business since I can save money and have enough room

SA : eco > bus

Flying to Australia

If I fly Singapore, I prefer Economy to Business since I can save money and have enough room

If I fly British, I prefer Business to Economy since there is not enough room

SA : eco > bus

BA: bus > eco

Flying to Australia

If I fly Singapore, I prefer Economy to Business since I can save money and have enough room

If I fly British, I prefer Business to Economy since there is not enough room

If I fly Business, I prefer Singapore to British since it hasbetter service

SA : eco > bus

BA: bus > eco

bus: SA > BA

Flying to Australia

If I fly Singapore, I prefer Economy to Business since I can save money and have enough room

If I fly British, I prefer Business to Economy since there is not enough room

If I fly Business, I prefer Singapore to British since it hasbetter service

If I fly Economy, I prefer British to Singapore since I collect British Airlines miles

SA : eco > bus

BA: bus > eco

bus: SA > BA

eco: BA > SA

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 > bus”

Flying to Australia

Parent Order

BA bus>eco

SA eco>bus

Airline

Class

Parent Order

bus SA>BA

eco

BA>SA

≥ ≥

≥≥

BA busBA eco

SA ecoSA bus

≥

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

Preferences of multiple agents

mCP-nets

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

QuickTime™ and aTIFF (Uncompressed) decompressor

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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

Preference aggregation

Represent preferences of each agent mCP-net

For each agent, (partial) CP net

Soft constraints …

Each agent votes Is A > B?

How do we add up the votes? Run an election!

Voting semantics

Pareto order A >p B iff A>B or A indifferent to B for all agents

Majority order A >maj B iff

#better > (#worse + #incomparable) Ignore agents who are indifferent

Max order A >max B iff

#better > max(#worse,#incomparable)

Voting semantics

Lex order A >lex B iff

For agent 1, A>B Or agent 1 is indifferent between them and for agent 2, A > B

or …

Rank order A >r B iff sum of ranks(A) < sum of ranks(B) Rank = minimal #worsening flips to optimal

Basic properties

Ordering >p and >lex are strict

partial ordersTransitive, irreflexive

and antisymmetric >maj and >max are not

Only irreflexive and antisymmetric

>r is total order

Basic properties

OptimalityA is >-optimal iff no B with B > A

Existence of optimal outcome? Pareto-optimal, majority-optimal, max-optimal,

lex-optimal, rank-optimal outcomes always exist

Fairness of aggregation?

Arrow’s 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 B>A or B indifferent to A to A>B

makes A more preferred Non-dictatorial

Final ordering depends on more than one agent

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

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) Arrow’s theorem, this

cannot be fair

Bibliography

1. Reasoning with conditional ceteris-paribus preference statements. C. Boutilier, R. Brafman, H. Hoos and D. Pooel, Proceedings of UAI-99

2. 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 Arrow’s theorem to partial orders)