Elicitation in combinatorial
auctions with restricted
preferences and bounded
interdependency between items
Vincent Conitzer, Tuomas Sandholm, Carnegie Mellon University
Paolo Santi, Pisa University
Corresponding to papers:
Santi, Conitzer, Sandholm, “Towards a Characterization of
Polynomial Preference Elicitation in CAs” (COLT-04)
Conitzer, Sandholm, Santi, “Combinatorial Auctions with
k-wise Dependent Valuations” (Draft)
Introduction
Combinatorial auction
• Can bid on combinations of items
– Bidder’s perspective:
• Allows bidder to express what she really wants
– Avoids exposure problems
– No need for lookahead / counterspeculation
– Auctioneer’s perspective:
• Automated optimal bundling
• Winner determination problem:
– Label bids as winning or losing so as to maximize sum of bid prices
» Each item can be allocated to at most one bid
– If approximating, watch incentives
– => Better allocations of items than in noncombinatorial auctions
Another complex problem in
combinatorial auctions:
• In direct-revelation mechanisms (e.g. VCG), bidders bid
on all 2m combinations
– Need to compute the valuation for exponentially many
combinations
• Each valuation computation can be NP-complete local planning problem
• E.g. carrier company bidding on trucking tasks: TRACONET [Sandholm
AAAI-93]
– Need to communicate the bids
– Need to reveal the bids => Loss of privacy & strategic info
• Bidding languages [Sandholm 98, 99; Nisan 00; Hoos &
Boutilier 01] do not solve the problem
? for
$ 1,000 for
$ 1,500 for
What info is needed from
Elicitor
an agent depends on what
Clearing algorithm
others have revealed
Elicitor decides what to ask
next based on answers it
has received so far
Conen & Sandholm IJCAI-01 workshop on Econ. Agents, Models & Mechanisms, ACMEC-01
Related research
• Nondeterministic (i.e., oracle) models
–
–
–
–
–
–
Bikhchandani & Ostroy JET-02
Gul & Stacchetti JET-00
Conitzer & Sandholm AAAI-02
Parkes AMEC-02
Nisan & Segal 03
Segal 04
• Deterministic models
– Ascending CAs, e.g. Parkes 99; Wurman & Wellman 00; Ausubel &
Milgrom 02; Kwasnicka, Ledyard, Porter, DeMartini 04
– General elicitation framework
• General preferences (no externalities, free disposal)
– Conen & Sandholm IJCAI-01 workshop, ACMEC-01, AAAI-02, AMEC-02
– Hudson & Sandholm AMEC-02, AAMAS-03, AAMAS-04
• Restricted valuation classes [Techniques from computational learning theory]
– Zinkevich, Blum, Sandholm ACMEC-03
– Blum, Jackson, Sandholm, Zinkevich COLT-03, JMLR-04
– Lahaie & Parkes ACMEC-04
Partial vs. full elicitation
•
In general, can achieve savings in elicitation by basing
queries to one agent on answers from others
•
Here, will assume that auctioneer will want to know each
agent’s entire preference function
–
•
So can focus on eliciting one agent’s function
Will assume that agent’s valuation function is drawn
from a restricted class of functions
Model
•
•
•
•
•
Set of items I for sale
Bidder has true valuation function v: 2I  
Elicitor knows class of functions C with v  C
Elicitor’s goal is to identify v
Elicitor can ask bidder for v(B) for any bundle B
–
•
Counts as one (value) query
Distinguish between eliciting using
– polynomial #queries
– polynomial time
•
May take significant time to compute which query to ask
Some examples of polynomialquery elicitable classes
Read-once valuations
[Zinkevich, Blum, Sandholm 03]
Valuations are represented by a tree
Leaf nodes correspond to items and their values
Nonleaf nodes (gates) perform operations including:
SUM: computes the sum of its children
MAXc: computes sum of the c highest inputs
ATLEASTc: returns sum of inputs if at least c
nonzero
PLUS
MAX
RO+M: only MAX and
SUM allowed
ALL
ALL
1000
500
400
200
100
150
Toolt (=ToolboxDNF)
[Zinkevich, Blum, Sandholm 03]
Valuation represented by polynomial with items as variables
Using only t monomials
3A + 5AB + 2AC + 4D
All coefficients must be nonnegative
A
B
C
3
+
2
D
=
Can be elicited in O(mt) queries
5
Tool-t (slight variation)
Here, weights on monomials with  2 items must be negative
3A + 6B + 2C + D - 2AB - AC
A
3
B
-
C
1 + 2
D
=
4
Thrm. Can be elicited in O(mt) queries
Proof: First ask all singletons. Then, discover monomials one by
one. Only need to find minimal subset of items that has value
less than sum of contained monomials discovered so far. So,
start by querying grand bundle and remove items one by one.
Interval bids
Items are ordered on a line
Value of bundle = sum of values of disjoint components
A
v({A}) = 1
B
C
v({B}) = 2
v({C}) = 2
v({A, B}) = 4
IMPLIED: v({A, C}) =
v({A})+v({C}) = 3
v({B, C}) = 3
v({A, B, C}) = 5
Thrm. Can be elicited using m(m+1)/2 queries if ordering is known
Thrm. Can be elicited using m2 – m + 1 queries if ordering is not
known, but v({x, y}) > v({x}) + v({y}) iff x and y are adjacent
Proof: Ask all singletons and pairs to find adjacencies (m(m+1)/2),
then ask remaining components (m(m-1)/2 – (m-1)), for total of
m2 – m + 1 queries
Tree bids require exponential queries
Natural generalization: tree such that value of bundle = sum of
values of disjoint components
Requires exponentially many queries:
…
There are 2m/2 such connected bids
Bounded interdependency
G2 = 2-wise dependent valuations
Value of bundle = sum of values of nodes/edges in bundle
3
Node = item
0
3
1
-2
1
0+1+2 = 3
2
Gk = k-wise dependent valuations
Value of bundle = sum of values of nodes/edges/multiedges in bundle
For example, k=3:
-2
3
Node = item
3
1
1
0
1
2
3-edge
Gk basic elicitation results
•
Thrm. Every valuation function has a unique Gm representation
– Proof: Suppose we have found the unique weights for
multiedges up to size j. Then weight of multiedge over S (with
|S| = j+1) must be v(S) – S’Sw(S’)
•
Thrm. A function in Gk can be elicited in O(mk) queries
– Proof: Query all bundles of size k or less. Again, weight of
multiedge over S (with |S| = j+1  k) must be v(S) –
S’Sw(S’), so can use dynamic programming
Optimal clearing is still hard in G2
• Pf: reduces from EXACT-COVER-BY-3-SETS
1
1
1
1
1
1
• Can get total value of 2m/3 if and only if an exact
3-cover exists
Special case: union of graphs is forest
2
1
9
7
1
3
6
• Thrm. Can solve clearing problem to optimality
by dynamic programming in time O(mn)
Approximating with G2 or Gk
•
Thrm. Suppose there exists some v’ in Gk such that for
any bundle S, |v(S) – v’(S)| ≤ δ. Then, using O(mk)
queries, we can construct a function g in Gk such that for
any bundle S, |v(S) – g(S)| ≤ δ(1+(|S| choose k)).
– Bound is tight for G2
•
Thrm. Suppose that all the weights in v’s Gm graph are
positive. Then, using m(m+1)/2 queries, we can construct
a function g in G2 such that for any S, |v(S)-g(S)| ≤
(M(v)/2) ((|S|(|S|+1)/2) (1+ |(|S|-1)/2)
–
here M(v) is a measure of the function’s disagreement with the
same function without any multiedges
Unions of classes
Polynomial-query elicitable valuation
classes closed under pairwise union
•
Let C1, C2 be valuation classes that can be elicited with
polynomial #queries
–
•
Consider the following simple algorithm for C1  C2
1.
2.
3.
4.
5.
•
•
Using algorithms A1, A2 with query bounds p1(n), p2(n)
f1A1 , f2A2
If f1 = f2, return it
Otherwise, find bundle S such that f1(S)  f2(S)
Query v(S)
If f1(S) = v(S), return f1, otherwise f2
At most p1(n) + p2(n) + 1 queries
Gives no bound on computation: checking identity of
functions in steps 2, 3 may take lot of computation
p1(n) + p2(n) + 1 bound is tight
•
•
•
•
•
•
Consider the following classes:
C1 = {fs} where
– fs(B) = 0 if B is empty or B = {s}
– fs(B) = 2 if B = I
– fs(B) = 1 otherwise
To elicit C1, simply ask v({s}) for every s
– Need at most m-1 queries
C2 = {f-s} where
– f-s(B) = 0 if B is empty
– f-s(B) = 2 if B = I or B = I – {s}
– f-s(B) = 1 otherwise
1
0
1
{a}
{b}
{c}
{}
I
0
2
I-{a}
I-{b}
I-{c}
1
2
1
To elicit C2, simply ask v(I-{s}) for every s
– Need at most m-1 queries
To elicit C1  C2 , need to find {s} or I-{s} with value different from 1
– Need 2m-1 = 2(m-1) + 1 queries
Does taking the union ever make
computation harder?
• Answer: yes. Consider following class:
• G2U: valuation is given by graph from G2 (with
positive edge weights) + upper bound u on value
A 2
3
1 B
3
u=6
v({A, C}) = 6
C 2
• Easy to elicit:
– ask all singletons, all pairs to get graph
– ask grand bundle to get u
Taking the union may make
computation harder…
• Now consider the following class:
• G2UH: same as G2U except no more than half of
bundle’s value can come from edges
– require: no edge worth more than sum of endpoints
A 2
3
1 B
3
u = 20
v({A, B, C}) = 10
C 2
• Again, easy to elicit:
– ask all singletons, all pairs to get graph
– ask grand bundle to get u
How computationally hard is it to
elicit G2U  G2UH?
• Thrm. It is coNP-complete to determine whether
a function from G2U and another from G2UH
(represented by their graphs and u) are identical
– That is, it is NP-complete to find a bundle whose
query would distinguish them
Proof of hardness
• Reduction from CLIQUE problem
every vertex:
weight 1
every edge: weight
(k+) / (k choose 2)
Required clique size:
k (say, 3)
u = 2k + 
• Clique of size k would have k vertex weight and
k+  edge weight
– So, G2UH at-most-half-from-edges constraint would be
binding
• Cannot happen when there are fewer edges
• For larger sets, the u-constraint is binding
Optimized polynomial-time elicitation algorithm for
RO+M  Tool-t  Toolt  G2  INT
Thrm. Runs in polynomial time and uses at most O(m(m+t)) queries
Towards characterizing easily
elicitable valuation functions
Polynomial inferability
•
•
•
Inferring a bundle = ascertaining its value from queries
on other bundles
Bundle is polynomially noninferable (strongly
polynomially noninferable) wrt C if for some (any)
function in C, the bundle’s valuation cannot be inferred
using polynomially many queries
Thrm. There exists a class of functions where
–
–
–
•
exponentially many bundles are polynomially noninferable
no bundles are strongly polynomially noninferable
the class cannot be elicited using polynomially many queries.
Proof uses [Angluin 88] idea, functions of the form:
…
v(B) = 1 iff B contains all
items corresponding to a
color
Conclusions
•
•
Focused on learning full valuation function in restricted
classes
New easy-to-elicit classes of valuations
–
•
Clearing for G2 is NP-complete
–
•
•
Tool-t, Interval, Gk
But easy if union of graphs is forest
Approximation with functions from G2 or Gk
Polyquery elicitable classes closed under pairwise union
– But computation required may go from polynomial to NP-hard
– Efficient algorithm for union of most of the classes studied
•
Even classes without strongly polynomially noninferable
bundles may require exponentially many queries for
elicitation
Future research
•
Can Interval class be elicited with polynomially many
queries without knowing the order?
•
Can we come up with a more general characterization of
what makes valuation functions easy to elicit?
•
What if we have a restricted class of valuations and we
only need to elicit enough to allocate (or compute VCG
payments)?
Thank you for your attention!