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=-y X=-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 class Economy class Variables and Domains: SABA buseco

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 has better 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 has better 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 ParentOrder BAbus>eco SAeco>bus Airline Class ParentOrder busSA>BA ecoBA>SA BA bus BA eco SA eco SA 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

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 orders Transitive, irreflexive and antisymmetric >maj and >max are not Only irreflexive and antisymmetric >r is total order

Basic properties Optimality A 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?

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

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) Arrows 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 Arrows theorem to partial orders)