cp nets toby walsh nicta and unsw. representing preferences unfactored not decomposable into parts...
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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
are needed to see this picture.
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
QuickTime™ and aTIFF (Uncompressed) decompressor
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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)