preferences toby walsh nicta and unsw tw/teaching.html

Preferences

Toby Walsh

NICTA and UNSW

www.cse.unsw.edu.au/~tw/teaching.html

Outline

May 5,15:00-17:00 Introduction, soft constraints

May 6, 10:00-12:00 CP nets

May 7, 15:00-18:00 Strategic games, CP-nets, and soft constraints Voting theory

May 8, 15:00-18:00 Manipulation, preference elicitation

May 9, 10:00-12:00 Matching problems, stable marriage

Motivation

Preferences are everywhere! Alice prefers not to

meet on Monday morning

Bob prefers bourbon to whisky

Carol likes beach vacations more than activity holidays

…

Major questions

Representing preferences Soft CSPs, CP nets, …

Reasoning with preferences What is the optimal outcome? Do I prefer A to B? How

do we combine preferences from multiple agents? …

Eliciting preferences Users don’t want to answer lots of questions! Are users going to be truthful when revealing their

preferences? …

Preference formalisms

Psychological relevance Can it express your preferences?

Quantitative: I like wine twice as much as beer Qualitative: I prefer wine to beer Conditional: if we’re having meat, I prefer red wine to

white …


Expressive power What types of ordering over outcomes can it

represent? Total Partial Indifference Incomplete …


Succinctness How succinct is it compared to other formalisms?

Can it (compactly) represent all that another formalism can?

…

Complexity How difficult is it to reason with?

What is the computationally complexity of ordering two choices?

What is the computationally complexity of finding the most preferred choice?

…

Utilities

Map preferences onto a linear scale Typically reals, naturals, …

Issues Cardinal or ordinal utility?

Numbers meaningful or just ordering? Different agents have different utility scales Incomparability Combinatorial domains

First course x Main dish x Sweet x Wine x …

Ordering relation

I prefer A to B (written A > B) Transitive or not: if A > B and B > C then is A > C? Total or partial: is every pair ordered? Strict or not: A > B or A ≥ B …

Issues Elicitation requires ranking O(m2) pairs Combinatorial domains …

Case study: combinatorial auction

Auctioneer Puts up number of items

for sale Agents

Submit bids for combinations of items

Winner determination Decide which bids to

accept Two agents cannot get

the same item Maximize revenue!


Why are bids not additive? Complements

v(A & B) > v(A) + v(B) Left shoe of no value without right shoe

Substitutes v(A & B) < v(A) + v(B) As you can only drive one car at a time, a second Ferrari

is not worth as much as the first

Auction mechanism that simply assigns items in turn may be sub-optimal

How you value item depends on what you get later


Winner determination problem Deciding if there is a solution achieving a given

revenue k (or more)

NP-complete in general Even if each agent submits jut a single bid And this bid has value 1


Winner determination problem Membership in NP

Polynomial certificate Given allocation of goods, can compute revenue it

generates


Winner determination problem NP-hard

Reduction from set packing Given S, a collection of sets and a cardinality k, is

there a subset of S of disjoint sets of size k? Items in sets are goods for auction One agent for each set in S, value 1 for goods in

their set, 0 otherwise One other agent who bids 0 for all goods


Winner determination problem NP-hard

One agent for each set in S, value 1 for goods in their set, 0 otherwise

One special agent who bids 0 for all goods Allocation may not correspond to set packing

• Agents may be allocated goods with 0 value (ie outside their desired set)

• But can always move these goods over to special agent Revenue equal to cardinality of the subset of S


Winner determination problem Tractable cases

Conflict graph: vertices = bids, edges = bids that cannot be accepted together

If conflict graph is tree, then winner determination takes polynomial time

• Starting at leaves, accept bid if it is greater than best price achievable by best combination of its children


Winner determination problem Intractable cases

Integer programming Heuristic search

• States = accepted bids• Moves = accept/reject bid• Initial state = no bids accepted• Heuristics

Bid with high price & few goods Bid that decomposes conflict graph


Bidding languages Used for agents to express their preferences over goods If there are m goods, there are 2m possible bids

Many possibilities Atomic bids OR bids XOR bids OR* bids with dummy items …


Bidding languages: assumptions

Normalized v({})=0

Monotonic v(A) ≤ v(B) iff A B

Implies valuations are non-negative!


Atomic bids (B,p)

“I want set of items B for price p” v(X) = p if X B otherwise 0

Note this valuation is monotonic

Very limited range of preferences expressible as atomic bids

Cannot express even simple additive valuations


OR bids Disjunction of atomic bids

(B1,p1) OR (B2,p2) Value is max. sum of disjoint bundles

v(X) = max { v1(X1) + v2(X \ X1) | X1X} Not complete

Can only express valuations without substitutes v(X u Y) ≥ v(X) + v(Y) Suppose you want just one item?

• v(S) = max{ vj | j S }


XOR bids Disjunction of atomic bids but only one is

wanted (B1,p1) XOR (B2,p2)

Value is max. of two possible valuations v(X) = max {v1(X), v2(X)}

Complete Can express any monotonic valuation Just list out all the differently valued sets of goods Hence XORs are more expressive than ORs


XOR bids Disjunction of atomic bids but only one is

wanted (B1,p1) XOR (B2,p2)

Additive valuation requires O(2k) XORs But only O(k) Ors

Thus, XORs are more expressive but less succinct than ORs


OR/XOR bids Arbitrary combinations of ORs and XORs Bid := (B,p) | Bid OR Bid | Bid XOR Bid

Recursively define semantics as before B1 OR B2

v(X) = max { v1(X1) + v2(X \ X1) | X1X} B1 XOR B2

v(X) = max { v1(X), v2(X) }


Two special casesOR of XOR

Bid := XorBid | XorBid OR XorBid XorBid := (B,p) | (B,p) XOR XorBid

XOR of OR Bid := OrBid | OrBid XOR OrBid OrBid := (B,p) | (B,p) OR OrBid


Downward sloping symmetric valuation Items symmetric

Only their number, k matters Diminishing returns

v(k)-v(k-1) ≥ v(k+1)-v(k)

Using OR of XOR, such a valuation over n items is O(n2) in size Let pk = v(k)-v(k-1) Then v(k) is

({x1},p1) XOR .. XOR ({xn},p1) OR

({x1},p2) XOR .. XOR ({xn},p2) OR .. OR

({x1},pn) XOR .. XOR ({xn},pn)


Downward sloping symmetric valuation Items symmetric

Only their number, k matters Diminishing returns

v(k)-v(k-1) ≥ v(k+1)-v(k)

Using XOR of ORs (or OR) such a valuation is exponential in size Need to represent all subsets of size k

OR of XORs is exponentially more succinct than XOR of ORs


Monochromatic valuations n/2 red and n/2 blue items Want as many of one colour as possible

v(X) = max {|X Red|, |X Blue|}

With such a valuation XOR of ORs is O(n) in size

({red1,p}) OR .. OR ({redn/2 },p) XOR

({blue1,p}) OR .. OR ({bluen/2,p})




With such a valuation OR of XORs is O(2n/2) in size

Atomic bids in OR of XORs only need be monochromatic• Removing non-monochromatic atomic bids will not change

valuation of a monochromatic allocation Atomic bids need to have price equal to their cardinality

• Anything higher or lower will only value a monochromatic allocation incorrectly





There can be only a single XOR• Suppose there are two (or more) XORs• There are two cases:

One XOR is just blue, other is just redBut then monochromatic valuation is not possible

One XOR is blue and redBut then again monochromatic valuation is not possible





There can be only a single XOR This must contain all O(2n/2) blue and O(2n/2) red subsets

XOR of ORs and OR of XORs are incomparable in succinctness


OR* bids Can modify OR bids so they can simulate XOR bids

Recall that OR bids are not complete But XOR bids can be exponentially more succinct Get best of both worlds?

Introduce dummy items (which cannot be shared) to OR bids to make them simulate XOR

(B u {dummy},p1) OR (C u {dummy},p2) is equivalent to (B,p1) XOR (C,p2)

Since XOR bids are complete, so are OR* bids


OR* bids Any OR/XOR bid of size O(s) can be represented as an

OR* bid of size O(s) Homework exercise: prove this!

This bidding language still has limitations Majority valuation requires exponential sized OR* bid

• Any allocation of m/2 or more of the items has value 1• Any smaller allocation has value 0

No non-zero atomic bid in the OR* bid can have less than m/2 items

• Otherwise we could accept this set and violate majority valuation

So we must have every nCn/2 possible subset of size n/2

Conclusions

Wide variety of formalisms for representing preferences Several dimensions along which to analyse

them Completeness Succinctness Complexity of reasoning …

preferences toby walsh nicta and unsw tw/teaching.html

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