Preference elicitation. 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
by Nisan, Segal, Lahaie and Parkes
October 27th, 2004
minimzecommunication and informationrevelation*
* Incentives are not considered
Here: „worst-case“ burden = max. number
Sequential message sending
Message send, determined by type and preceding messages
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:
Lindahl prices: nonlinear and non-anonymous
Definition: is a Lindahl equilibrium in state ≽ ∈ ℜ if
Lindahl equilibrium correspondance: ↠
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
Finding a lower bound from „Alice and Bob“:
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.
Holds for valuations with:
(about 500 Gigabytes of data)
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.
Dimension of message space in any efficient protocol is at least -1
Agents care only about number of items recieved
Demand QueryApplying Learning Algorithms
1) a list of values
Sufficient set of manifest valuations to
compute an optimal allocation.
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.
Similar to definition for polynomial-query
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.
1) Run learning algorithms on valuation classes until each requires response to equivalence query
4) Quit if all agents answer YES, otherwise give counterexample from agent i to learning algorithm i. goto 1
no generell good communication design
focus on specific classes of preferences
solution exists for polynomials, XOR, linear-threshold