Preferences Toby Walsh NICTA and UNSW tw/teaching.html.
-
Upload
diana-little -
Category
Documents
-
view
226 -
download
4
Transcript of Preferences Toby Walsh NICTA and UNSW 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 …
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 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
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 redBut then monochromatic valuation is not possible
One XOR is blue and redBut 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