Preference elicitation
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Transcript of Preference elicitation
Preference elicitationPreference elicitation
Communicational Burden Communicational Burden
by Nisan, Segal, Lahaie and Parkesby Nisan, Segal, Lahaie and Parkes
October 27th, 2004October 27th, 2004Jella PfeifferJella Pfeiffer
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Outline
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
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Outline
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
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Exponential number of bundles in the number of goods Communication of values Determination of valuations
Reluctance to reveal valuation entirely
minimze communication and information revelation*
* Incentives are not considered
Motivation
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Agenda
MotivationCommunication
BurdenProtocols
Lindahl pricesCommunication complexityPreference ClassesApplying Learning Algorithms to Preference elicitationApplicationsConclusionFuture Work
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Communication burden:
Minimum Number of messages
Transmitted in a protocol (nondeterministic)
Realizing the communication
Here: „worst-case“ burden = max. number
Communication burden
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Communication protocols
Sequential message sending
1. Deterministic protocol:Message send, determined by type and preceding messages
2. Nondeterministic protocol: Omniscient oracle
Knows state of the world ≽ andDesirable alternative x ∈ F(≽)
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Definition Nondeterministic protocol
A nondeterministic protocol is a triple Г = (M, μ, h) where M is the message set, μ: R M is the message correspondance, and h: MX‘ is the outcome function, and the message correspondance μ has the following two properties:
1.Existence: μ(≽) ≠ ∅ for all ≽ ∈ ℜ,2.Privacy preservation: μ(≽) = ∩i μi(≽i) for all ≽ ∈ ℜ, where μi: Ri M for all i ∈ N.
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Agenda
MotivationCommunication Lindahl prices
EquilibriaImportance of Lindahl prices
Communication complexityPreference ClassesApplying Learning Algorithms to Preference elicitationApplicationsConclusionFuture Work
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Lindahl Equilbria
Lindahl prices: nonlinear and non-anonymous
Definition: is a Lindahl equilibrium in state ≽ ∈ ℜ if
1. ≽i) for all i ∈ N, (L1)2. (L2)
Lindahl equilibrium correspondance: ↠
XxB XN 2)',(
,'(xLBi
iiBX
UE : XXN 2
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Importance of Lindahl prices
Protocol <M, μ, h> realizes the weakly Pareto efficient correspondence F* if and only if there exists an assignment of budget sets to messages such that protocol <M, μ, (B,h)> realizes the Lindahl equilibrium correspondance E.
Communication burden of efficiency
=
burden of finding Lindahl prices
XNMB 2:
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AgendaMotivationCommunication Lindahl pricesCommunication complexity
Alice and BobProof for Lower BoundCommunication complexity
Preference ClassesApplying Learning Algorithms to Preference elicitationApplicationsConclusionFuture Work
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Alice and Bob
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Communication Complexity (1)
Finding a lower bound from „Alice and Bob“:
Including auctioneer Larger number of biddersQueries to the biddersCommunicating real numbers Deterministic protocols
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The proof
Lemma: Let v ≠u be arbitrary 0/1 valuations. Then, the sequence of bits transmitted on inputs (v,v*), is not identical to the sequence of bits transmitted on inputs (u,u*).
(v*(S) = 1-v(Sc))
Theorem: Every protocol that finds the optimal allocation for every pair of 0/1 valuations v1, v2 must use at least bits of total communication in the worst case.
2/L
L
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Comments on the proofIn the main paper: Better allocation than auctioning off all objects as a bundle in a two-bidder auction needs at least
Holds for valuations with:No externalitiesNormalization
With L = 50 items, the number of bits is (about 500 Gigabytes of data)
2/L
L
14103.1
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Communication Complexity (2)Theorem*: Exact efficiency requires
communicating at least one price for each of the possible bundles. ( is the dimension of the message space)
*Holds for general valuations.
12 L
12 L
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Agenda
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
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Preference ClassesSubmodular valuations:
Dimension of message space in any efficient protocol is at least -1Homogenous valuations:
Agents care only about number of items recieved
Dimension LAdditive Valuations
Dimension L
)()()()( TvSvTSvTSv
2/L
L
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AgendaMotivationCommunication Lindahl pricesCommunication complexityPreference ClassesApplying Learning Algorithms to Preference elicitation
Learning algorithmsPreference elicitationParallels (polynomial query learnable/elicitation)Converting learning algorithms
ApplicationsConclusionFuture Work
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Applying Learning Algorithms
Learning theory
Preference elicitation
Membership Query Equivalence Query
Value Query Demand Query
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What is a Learning Algorithm?
Learning an unknown function f: X Y via questions to an oracleKnown function class C Typically: , Y either {0,1} or ⊆ ℜManifest hypotheses: Size(f) with respect to presentation
Example: f: ;f(x) = 2 if x consists of m 1‘s, and f(x) = 0 otherwise.
1) a list of values2)
mX }1,0{f~
m2mxx 12
m1,0
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Learning Algorithm - Queries?Learner? !Oracle!
x ∈ X f(x)
YES, if
NO; counterexample x
such that
f~
)()(~
xfxf
ff ~
Mem
bership
Query
Equivalence
Query
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Preference elicitationAssumptions:
NormalizedNo externalitiesQuasi-linear utility functionPolynomial time for representation values of bundles
Goal:Sufficient set of manifest valuations to compute an optimal allocation.
nvv ~,...,~1
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Preference eliciation - Queries ?auctioneer? !agent!
S ⊆ M v(S)
YES, if S most
preferred at p
NO; presents more
preferred S‘
Value
Query
Dem
and
Query
Spm
,)2(
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1) Membership query Value query
2) Equivalence query ? Demand query
Lindahl prices are only a constant away from manifest valuationsOut of a preferred bundle S‘, counterexamples can be computed
Parallels: learning & eliciation pref.
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Polynomial-query learnableDefintion: The representation class C is
polymonial-query exactly learnable from membership and equivalence queries if there is a fixed polynomial and an algorithm L with access to membership and equivalence queries of an oracle such that for any target function f ∈ C, L outputs after at most p(size(f),m) queries a function such that for all instances x.
),( p
)()(~
xfxf Cf ~
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Polynomial-query elicited
Similar to definition for polynomial-query learnable but:
Value and demand queriesAgents‘ valuations are target functionsOutputs in p(size(v1,...,vn),m) an optimal allocationValuation functions need not to be determined exactly!
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Converting learning algorithms
Idea proved in paper:
If each representation class V1,…,V2 can be polynomial-query exactly learned from membership and equivalence queries
V1,…,V2 can be polynomial-query elicited from value and demand queries.
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Converted Algorithm
1) Run learning algorithms on valuation classes until each requires response to equivalence query
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Converted Algorithm
2) Compute optimal allocation S* and Lindahl prices L* with respect to manifest valuations
3) Represent demand query with S* and L*
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Converted Algorithm
4) Quit if all agents answer YES, otherwise give counterexample from agent i to learning algorithm i. goto 1
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AgendaMotivationCommunication Lindahl pricesCommunication complexityPreference ClassesApplying Learning Algorithms to Preference elicitationApplications
Polynomial representationXOR/DNFLinear-Threshold
ConclusionFuture Work
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PolynomialsT-spares, multivariate polynomials:
T-termsTerm is product of variables (e.g. x1x3x5)
„Every valuation function can be uniquely written as polynomial“ [Schapire and Selli]
Example: additive valuations Polynomials of size m (m = number of items)x1+…+xm
Learning algorithm:At most Equivalence queriesAt most Membership queries
2imt2/)3)(1( 2
iii ttmt
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XOR/DNF Representations (1)XOR bids represent valuations wich have free-disposal Analog in learning theory: DNF formulae
Disjunction of conjunctions with unnegated bitsE.g.Atomic bids in XOR have value 1
)()( BvAvBA
543421 xxxxxx
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XOR/DNF Representations (2)An XOR bid containing t atomic bids can be exactly learned with t+1 equivalence queries and at most tm membership queries
Each Equivalence query leads to one new atomic bidBy m membership queries (exluding bids out of the counteraxample which do not belong to the atomic bid)
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Linear-Threshold Representationsr-of-S valuationLet , r-of-k threshold functions:If r known: equivalence queries or demand queries
Sk rxx iki ...1
1ln1458 2 mkrkr
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Important Results by Nisan, SegalImportant role of prices (efficient allocation must reveal suppporting Lindahl prices)Efficient communication must name at least one Lindahl price for each of the bundlesLower bound:
no generell good communication designfocus on specific classes of preferences
12 L
2/L
L
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Important Results by Lahaie, ParkesLearning algorithm with membership and equivalence queries as basis for preference elicitation algorithmIf polynomial-query learnable algorithm exists for valuations, preferences can be efficiently elicited whith queries polynomial in m and size(v1,…,vn)
solution exists for polynomials, XOR, linear- threshold
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Future Work
Finding more specific classes of preferences which can be elicited efficientlyAddress issue of incentivesWhich Lindahl prices may be used for the questions
Thank you for your Thank you for your attenttionattenttion
Any Questions?Any Questions?