Preference elicitation
Communicational Burden
by Nisan, Segal, Lahaie and Parkes
Jella Pfeiffer
October 27th, 2004
Outline
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
2
Outline
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
3
Motivation
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
4
Agenda
Motivation
Communication
Burden
Protocols
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
5
Communication burden
Communication burden:
Minimum Number of messages
Transmitted in a protocol (nondeterministic)
Realizing the communication
Here: „worst-case“ burden = max. number
6
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 ≽ and
Desirable alternative x ∈ F(≽)
7
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: MX‘ 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
Motivation
Communication
Lindahl prices
Equilibria
Importance of Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
9
Lindahl Equilbria
Lindahl prices: nonlinear and non-anonymous
Definition: ( B, x' )  2 XN  X is a Lindahl equilibrium in
state ≽ ∈ ℜ if
1. Bi  L(x' ,≽i) for all i ∈ N,
(L1)
2. X   B
(L2)
i
i
Lindahl equilibrium correspondance: E : U ↠ 2 XN  X
10
Importance of Lindahl prices
Protocol <M, μ, h> realizes the weakly Pareto efficient
correspondence F* if and only if there exists an
assignment B : M  2 XN 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
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Agenda
Motivation
Communication
Lindahl prices
Communication complexity
Alice and Bob
Proof for Lower Bound
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
12
Alice and Bob
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Communication Complexity (1)
Finding a lower bound from „Alice and Bob“:
 Including auctioneer
 Larger number of bidders
 Queries to the bidders
 Communicating real numbers
 Deterministic protocols
14
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
 L 

 bits of total communication in the worst case.
L / 2


15
Comments on the proof
In the main paper: Better allocation than auctioning off
all objects as a bundle in a two-bidder auction needs
 L 
at least  L / 2 
Holds for valuations with:
No externalities
Normalization
With L = 50 items, the number of bits is 1.3  1014
(about 500 Gigabytes of data)
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Communication Complexity (2)
Theorem*: Exact efficiency requires communicating at
L
2
 1 possible
least one price for each of the
bundles. ( 2 L  1 is the dimension of the message
space)
*Holds for general valuations.
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Agenda
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Conclusion
Future Work
18
Preference Classes
Submodular valuations: v( S  T )  v( S  T )  v( S )  v(T )
Dimension of message space in any efficient
protocol is at least  LL/ 2  -1
Homogenous valuations:
Agents care only about number of items recieved
Dimension L
Additive Valuations
Dimension L
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Agenda
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Learning algorithms
Preference elicitation
Parallels (polynomial query learnable/elicitation)
Converting learning algorithms
Applications
Conclusion
Future Work
20
Applying Learning Algorithms
Learning theory
Membership Query
Equivalence Query
Value Query
Demand Query
Preference elicitation
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What is a Learning Algorithm?
Learning an unknown function f: X  Y via questions
to an oracle
Known function class C
m
Typically: X  {0,1} , Y either {0,1} or ⊆ ℜ
~
Manifest hypotheses: f
Size(f) with respect to presentation
Example: f: 0,1m   ;f(x) = 2 if x consists of m 1‘s,
and f(x) = 0 otherwise.
1) a list of 2 m values
2) 2 x1    xm
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Learning Algorithm - Queries
x∈X
f(x)
Equivalence
NO; counterexample x
~
such that f ( x )  f ( x )
Query
~
f
~
YES, if f  f
Membership
!Oracle!
Query
?Learner?
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Preference elicitation
Assumptions:
Normalized
No externalities
Quasi-linear utility function
Polynomial time for representation  values of
bundles
Goal:
~ ~
Sufficient set of manifest valuations v1 ,..., vn to
compute an optimal allocation.
24
Preference eliciation - Queries
S⊆M
v(S)
NO; presents more
preferred S‘
Demand
,S
Query
p 
( 2m )
YES, if S most
preferred at p
Value
!agent!
Query
?auctioneer?
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Parallels: learning & eliciation pref.
1) Membership query
2) Equivalence query
Value query
?
Demand query
Lindahl prices are only a constant away from
manifest valuations
Out of a preferred bundle S‘,
counterexamples can be computed
26
Polynomial-query learnable
Defintion: The representation class C is polymonial-query
exactly learnable from membership and equivalence
queries if there is a fixed polynomial p(,) 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 ~f  C such that f ( x )  f ( x )
for all instances x.
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Polynomial-query elicited
Similar to definition for polynomial-query
learnable but:
Value and demand queries
Agents‘ valuations are target functions
Outputs in p(size(v1,...,vn),m) an optimal
allocation
Valuation 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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Agenda
Motivation
Communication
Lindahl prices
Communication complexity
Preference Classes
Applying Learning Algorithms to Preference elicitation
Applications
Polynomial representation
XOR/DNF
Linear-Threshold
Conclusion
Future Work
33
Polynomials
T-spares, multivariate polynomials:
T-terms
Term 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 mti  2 Equivalence queries
At most (mti  1)( ti2  3ti ) / 2 Membership queries
34
XOR/DNF Representations (1)
XOR bids represent valuations wich have freedisposal A  B  v( A)  v( B)
Analog in learning theory: DNF formulae
Disjunction of conjunctions with unnegated bits
E.g. x1 x2  x4  x3 x4 x5
Atomic bids in XOR have value 1
35
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 bid
By m membership queries (exluding bids out of the
counteraxample which do not belong to the atomic bid)
36
Linear-Threshold Representations
r-of-S valuation
Let k  S , r-of-k threshold functions: xi1  ...  xik  r
If r known: 8r 2  5k  14kr ln m  1 equivalence
queries or demand queries
37
Important Results by Nisan, Segal
Important role of prices (efficient allocation must
reveal suppporting Lindahl prices)
Efficient communication must name at least one
Lindahl price for each of the 2 L  1 bundles
Lower bound:  LL/ 2 


no generell good communication design
focus on specific classes of preferences
38
Important Results by Lahaie, Parkes
Learning algorithm with membership and equivalence
queries as basis for preference elicitation algorithm
If 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, linearthreshold
39
Future Work
Finding more specific classes of preferences
which can be elicited efficiently
Address issue of incentives
Which Lindahl prices may be used for the
questions
40
Thank you for your
attenttion
Any Questions?