Preferences

1
Preferences

• Toby Walsh
• NICTA and UNSW
• www.cse.unsw.edu.au/tw/teaching.html

2
Outline

• May 5,1500-1700
• Introduction, soft constraints
• May 6, 1000-1200
• CP nets
• May 7, 1500-1800
• Strategic games, CP-nets, and soft constraints
• Voting theory
• May 8, 1500-1800
• Manipulation, preference elicitation
• May 9, 1000-1200
• Matching problems, stable marriage

3
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

4
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 dont want to answer lots of questions!
• Are users going to be truthful when revealing
  their preferences?

5
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 were having meat, I prefer red
  wine to white

6
Preference formalisms

• Expressive power
• What types of ordering over outcomes can it
  represent?
• Total
• Partial
• Indifference
• Incomplete

7
Preference formalisms

• 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?

8
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

9
Ordering relation

• I prefer A to B (written A gt B)
• Transitive or not if A gt B and B gt C then is A gt
  C?
• Total or partial is every pair ordered?
• Strict or not A gt B or A B
• Issues
• Elicitation requires ranking O(m2) pairs
• Combinatorial domains

10
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!

11
Case study combinatorial auction

• Why are bids not additive?
• Complements
• v(A B) gt v(A) v(B)
• Left shoe of no value without right shoe
• Substitutes
• v(A B) lt 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

12
Case study combinatorial auction

• 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

13
Case study combinatorial auction

• Winner determination problem
• Membership in NP
• Polynomial certificate
• Given allocation of goods, can compute revenue it
  generates

14
Case study combinatorial auction

• 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

15
Case study combinatorial auction

• 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

16
Case study combinatorial auction

• 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

17
Case study combinatorial auction

• 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

18
Case study combinatorial auction

• 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

19
Case study combinatorial auction

• 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

20
Case study combinatorial auction

• Bidding languages assumptions
• Normalized
• v()0
• Monotonic
• v(A) v(B) iff A ? B
• Implies valuations are non-negative!

21
Case study combinatorial auction

• 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

22
Case study combinatorial auction

• 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) X1?X
• 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

23
Case study combinatorial auction

• 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

24
Case study combinatorial auction

• 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

25
Case study combinatorial auction

• 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) X1?X
• B1 XOR B2
• v(X) max v1(X), v2(X)

26
Case study combinatorial auction

• Two special cases
• OR 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

27
Case study combinatorial auction

• Downward sloping symmetric valuation
• Items symmetric
• Only their number, k matters
• Diminishing returns
• v(k)-v(k-1) v(k1)-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)

28
Case study combinatorial auction

• Downward sloping symmetric valuation
• Items symmetric
• Only their number, k matters
• Diminishing returns
• v(k)-v(k-1) v(k1)-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

29
Case study combinatorial auction

• 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)

30
Case study combinatorial auction

• 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
• 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

31
Case study combinatorial auction

• 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
• OR of XORs is O(2n/2) in size
• 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 red
• But then monochromatic valuation is not possible
• One XOR is blue and red
• But then again monochromatic valuation is not
  possible

32
Case study combinatorial auction

• 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
• OR of XORs is O(2n/2) in size
• 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

33
Case study combinatorial auction

• 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

34
Case study combinatorial auction

• 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

35
Conclusions

• Wide variety of formalisms for representing
  preferences
• Several dimensions along which to analyse them
• Completeness
• Succinctness
• Complexity of reasoning