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Presentation Transcript

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
- …

Preference formalisms

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

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

- 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!

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 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 auction

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

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 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 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 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 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 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 auction

- Bidding languages: assumptions
- Normalized
- v({})=0
- Monotonic
- v(A) ≤ v(B) iff A B
- Implies valuations are non-negative!

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 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) | 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 }

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 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 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) | X1X}
- B1 XOR B2
- v(X) = max { v1(X), v2(X) }

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 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 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 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 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 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 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 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 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

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

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