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### Preference elicitation

### Thank you for your attenttion

Communicational Burden

by Nisan, Segal, Lahaie and Parkes

October 27th, 2004

Jella Pfeiffer

Outline

- Motivation
- Communication
- Lindahl prices
- Communication complexity
- Preference Classes
- Applying Learning Algorithms to Preference elicitation
- Applications
- Conclusion
- Future Work

Outline

- Motivation
- Communication
- Lindahl prices
- Communication complexity
- Preference Classes
- Applying Learning Algorithms to Preference elicitation
- Applications
- Conclusion
- Future Work

Motivation

- Exponential number of bundles in the number of goods
- Communication of values
- Determination of valuations

- Reluctance to reveal valuation entirely
minimzecommunication and informationrevelation*

* Incentives are not considered

Agenda

- Motivation
- Communication
- Burden
- Protocols

- Lindahl prices
- Communication complexity
- Preference Classes
- Applying Learning Algorithms to Preference elicitation
- Applications
- Conclusion
- Future Work

Communication burden

Communication burden:

- Minimum Number of messages
- Transmitted in a protocol (nondeterministic)
- Realizing the communication
Here: „worst-case“ burden = max. number

Communication protocols

Sequential message sending

- Deterministic protocol:
Message send, determined by type and preceding messages

- Nondeterministic protocol: Omniscient oracle
- Knows state of the world ≽ and
- Desirable alternative x ∈ F(≽)

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:

- Existence: μ(≽) ≠ ∅ for all ≽ ∈ ℜ,
- Privacy preservation: μ(≽) = ∩i μi(≽i) for all ≽ ∈ ℜ, where μi: Ri M for all i ∈ N.

Agenda

- Motivation
- Communication
- Lindahl prices
- Equilibria
- Importance of Lindahl prices

- Communication complexity
- Preference Classes
- Applying Learning Algorithms to Preference elicitation
- Applications
- Conclusion
- Future Work

Lindahl Equilbria

Lindahl prices: nonlinear and non-anonymous

Definition: is a Lindahl equilibrium in state ≽ ∈ ℜ if

- ≽i) for all i ∈ N, (L1)
- (L2)
Lindahl equilibrium correspondance: ↠

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

!

!

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

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

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.

Comments on the proof

- In 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 externalities
- Normalization

- With L = 50 items, the number of bits is
(about 500 Gigabytes of data)

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.

Agenda

- Motivation
- Communication
- Lindahl prices
- Communication complexity
- Preference Classes
- Applying Learning Algorithms to Preference elicitation
- Applications
- Conclusion
- Future Work

Preference Classes

- Submodular valuations:
Dimension of message space in any efficient protocol is at least -1

- Homogenous valuations:
Agents care only about number of items recieved

Dimension L

- Additive Valuations
Dimension L

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

Equivalence Query

Value Query

Demand Query

Applying Learning AlgorithmsLearning theory

Preference elicitation

What is a Learning Algorithm?

- Learning an unknown function f: X Y via questions to an oracle
- Known function class C
- Typically: , Y either {0,1} or ⊆ ℜ
- Manifest 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 values

2)

- Example: f: ;f(x) = 2 if x consists of m 1‘s, and f(x) = 0 otherwise.

Preference elicitation

Assumptions:

- Normalized
- No externalities
- Quasi-linear utility function
- Polynomial time for representation values of bundles
Goal:

Sufficient set of manifest valuations to

compute an optimal allocation.

Parallels: learning & eliciation pref.

- Membership query Value query
- Equivalence query ? Demand query
- Lindahl prices are only a constant away from manifest valuations
- Out of a preferred bundle S‘, counterexamples can be computed

Polynomial-query learnable

Defintion: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.

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!

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.

Converted Algorithm

1) Run learning algorithms on valuation classes until each requires response to equivalence query

Converted Algorithm

- Compute optimal allocation S* and Lindahl prices L* with respect to manifest valuations
- Represent demand query with S* and L*

Converted Algorithm

4) Quit if all agents answer YES, otherwise give counterexample from agent i to learning algorithm i. goto 1

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

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

- Example: additive valuations
- Learning algorithm:
- At most Equivalence queries
- At most Membership queries

XOR/DNF Representations (1)

- XOR bids represent valuations wich have free-disposal
- Analog in learning theory: DNF formulae
- Disjunction of conjunctions with unnegated bits
- E.g.
- Atomic bids in XOR have value 1

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)

Linear-Threshold Representations

- r-of-S valuation
- Let , r-of-k threshold functions:
- If r known: equivalence queries or demand queries

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 bundles
- Lower bound:
no generell good communication design

focus on specific classes of preferences

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, linear- threshold

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

Any Questions?

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