preferences n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Preferences PowerPoint Presentation
Download Presentation
Preferences

Loading in 2 Seconds...

play fullscreen
1 / 35

Preferences - PowerPoint PPT Presentation


  • 123 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Preferences' - cortez


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
preferences

Preferences

Toby Walsh

NICTA and UNSW

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

outline
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
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
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
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
preference formalisms1
Preference formalisms
  • Expressive power
    • What types of ordering over outcomes can it represent?
      • Total
      • Partial
      • Indifference
      • Incomplete
preference formalisms2
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?
utilities
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
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
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!
case study combinatorial auction1
Case study: combinatorial auction
  • 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
case study combinatorial auction2
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
case study combinatorial auction3
Case study: combinatorial auction
  • Winner determination problem
    • Membership in NP
      • Polynomial certificate
      • Given allocation of goods, can compute revenue it generates
case study combinatorial auction4
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
case study combinatorial auction5
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
case study combinatorial auction6
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
case study combinatorial auction7
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
case study combinatorial auction8
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
case study combinatorial auction9
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
case study combinatorial auction10
Case study: combinatorial auction
  • Bidding languages: assumptions
    • Normalized
      • v({})=0
    • Monotonic
      • v(A) ≤ v(B) iff A  B
      • Implies valuations are non-negative!
case study combinatorial auction11
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
case study combinatorial auction12
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 }
case study combinatorial auction13
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
case study combinatorial auction14
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
case study combinatorial auction15
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) }
case study combinatorial auction16
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
case study combinatorial auction17
Case study: combinatorial auction
  • 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)

case study combinatorial auction18
Case study: combinatorial auction
  • 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
case study combinatorial auction19
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})

case study combinatorial auction20
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
case study combinatorial auction21
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

case study combinatorial auction22
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
case study combinatorial auction23
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
case study combinatorial auction24
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
conclusions
Conclusions
  • Wide variety of formalisms for representing preferences
    • Several dimensions along which to analyse them
      • Completeness
      • Succinctness
      • Complexity of reasoning