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?