Minimal Preference Elicitation in Combinatorial Auctions
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Transcript of Minimal Preference Elicitation in Combinatorial Auctions
Minimal Preference Elicitation in Combinatorial Auctions
Wolfram Conen Tuomas Sandholm Xonar GmbH Carnegie Mellon University
Computer Science Department
Outline
• Combinatorial auctions for multi-item auctions• “The revelation problem”
– Previous approaches
– Our approach: Elicitor “agent”• Topological observations that motivate elicitation
• Different elicitation queries
• Policy dependent elicitor algorithms
• General policy independent elicitor framework (with data structures & assimilation algorithms) & specific elicitor algorithms
• Making the elicitor incentive compatible
Combinatorial auction (CA)
• Can bid on combinations of items [Rassenti,Smith & Bulfin 82]...
– Bidder’s perspective
• Allows bidder to express what she really wants– No need for lookahead / counterspeculationing of items
– Auctioneer’s perspective:
• Automated optimal bundling
• Binary 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– NP-complete [Rothkopf et al 98 using Karp 72]
– Inapproximable [Sandholm IJCAI-99 using Hastad 99]
Another complex problem in (single-shot) combinatorial auctions:
“The revelation problem”• Bidders may need to bid on all 2#items combinations
– Need to compute the valuation for each combination• Each valuation computation can be NP-complete
• For example if a carrier company bids on trucking tasks: TRACONET [Sandholm AAAI-93]
– Need to communicate the bids– Need to reveal the bids
Setting
Combinatorial auction: m items for sale• Private values auction, no allocative externalities
• Each bidder i has value function, vi: 2m R
• Unique valuations (to ease presentation)
Can avoid unnecessary computation/revelation/communication of valuations!
Approaches for tackling the revelation problem• Classic single-shot full revelation mechanims (Vickrey-Clarke-Groves)
– Exponentially many valuations revealed
• (Ascending) mechanisms with price feedback (iBundle, [Parkes et al 1999] , akBa [Wurman et al. 2000] , etc.)– Can save revelation– Need exponential revelation in worst case [Nisan 2001]
• Our new approach: an elicitor “agent”– Knows things that individual bidders don’t (others’ bids so far)– Asks non-redundant questions from bidders to focus their revelation – Can save revelation– Thrm. Exponential revelation in worst case if only value and order queries are
allowed (even with 1 bidder)– Can be combined with price feedback mechanisms
Elicitor algorithms
• Query policy dependent elicitor algorithms– Algorithm & query policy are intertwined– Based on search algorithms where each search step
involves asking a bidder a question
• Policy independent elicitor algorithms– General framework & specific algorithms– Can support any query policy– Use exponential memory (in worst case)
• Note: Query policies are online control policies, i.e. contingency plans
Terminology
(X1,...,X#bidders) is a collection
• Bundle Xi is earmarked for agent i
• An allocation is a feasible collection (i.e., collection where Xi’s don’t overlap in items)
Objectives: (1) Find Pareto efficient allocation(s)(2) Find social welfare maximizing
allocation(s)
Rank Lattice
[1,1]
[1,2] [2,1]
[2,3]
[3,1]
[3,2]
[2,4]
[3,4] [4,3]
[3,3] [4,2]
[4,4]
[1,4] [4,1]
[2,2][1,3]
Infeasible
Feasible
Dominated
Rank of Bundle Ø A B ABfor Agent 1 4 2 3 1for Agent 2 4 3 2 1
Query Policy Independent Elicitation Algorithms: Computing Pareto Optima
s=(1,...,1); PAR = []; OPEN = [s];
while OPEN ≠ [] do
Remove(c,OPEN); SUC = suc(c);
if Feasible(c) then
PAR = PAR {c}; Remove(SUC,OPEN);
else foreach node є SUC do
if node OPEN and Undominated(node,PAR)
then Append(node,OPEN)
Value-Augmented Rank Lattice
Value of Bundle Ø A B ABfor Agent 1 0 4 3 8for Agent 2 0 1 6 9
17
14 13
9 10 12
98
[1,1]
[1,2] [2,1]
[2,3]
[3,1]
[3,2]
[2,4]
[3,4] [4,3]
[3,3] [4,2]
[4,4]
[1,4] [4,1]
[2,2][1,3]
Query Policy Independent Elicitation Algorithms: Computing Welfare Maxima
s=(1,...,1); OPEN = {s}; CLOSED = Ø;
while OPEN ≠ Ø do
c = arg maxc є OPEN Σi є N vi(ci)
OPEN = OPEN \ {c};
if Feasible(c) then return(c);
CLOSED = CLOSED {c}; SUC = suc(c);
foreach n є SUC do
if node OPEN and node CLOSED
then OPEN = OPEN {node}
Policy independent elicitor algorithms
Our Query Types for Agent Interrogation
• Order information: Which bundle do you prefer, A or B?
• Value information: What is your valuation for bundle A? (Answer: Exact or Bounds)
• Rank information: – What is the rank of bundle b?
– What bundle is at rank x?
– Given bundle b, what is the next lower (higher) ranked bundle?
General Algorithmic Framework for Elicitation
Algorithm Solve(Y,G) while not Done(Y,G) do
o = SelectOp(Y,G) Choose QuestionI = PerformOp(o,N) Ask bidderG = Propagate(I,G) Update GraphY = Candidates(Y,G) Curtail set of candidate
collections / allocations
Input: Y – set of collections or allocationsG – partially augmented order graph
Output: Y – set of optimal solutions
(Partially) Augmented Order Graph
Ø B A AB∞ ∞ ∞ ∞
0 0 0 0
Ø A B AB0 0 1 6
4 0 3 6 2 6 1 9
Agent1
Agent2
A
B
Ø
B>
∞
1
∞
1
Allocations
AB6
1 9
Rank Upper Bound
Lower Bound
[1,1]
[1,2] [2,1]
[2,3]
[3,1]
[3,2]
[2,4]
[3,4] [4,3]
[3,3] [4,2]
[4,4]
[1,4] [1,4]
[2,2][1,3]
Some interesting procedures for combining different types of info
We present algorithms that use any combination of value, order
& rank queries
• If value queries are used, all social welfare maximizing allocations are guaranteed to be found
• Otherwise, all Pareto efficient allocation are guaranteed to be found
• We propose several query policies that are geared toward reducing the number of queries needed
Incentive compatibility of the different approaches
• Classic single-shot full revelation mechanims (Vickrey-Clarke-Groves)– Can be made dominant strategy incentive compatible
• (Ascending) mechanisms with price feedback (iBundle, akBa, etc.)– Can be made incentive compatible in weaker equilibrium notions
• Our new approach: an elicitor “agent”– Elicitor’s questions leak information about others’ preferences– Can be made incentive compatible in weaker equilibrium notions
• Ask enough questions to determine VCG prices• Could interleave these “extra” questions with real questions
– To avoid lazyness; Not necessary from an incentive perspective
Conclusions• Combinatorial auctions are desirable & winner determination algorithms now scale to
the large• Another problem: “The Revelation Problem”
– Valuation computation / revelation / communication
• Presented the design for an elicitor for combinatorial auctions that focuses revelation – Can save revelation– Provably find the welfare maximizing or Pareto efficient allocations
• Policy dependent search algorithms for elicitation– Based on topological observation
• Policy independent general elicitation framework– Any combination of value, order & rank queries– Several algorithm instantiations in the paper– Several query policies
• Presented a way to make the elicitor incentive compatible• Elicitor can be combined with price feedback mechanisms
Future research
• Evaluating the elicitor– Savings in revelation (how many queries needed ?)
• In general case / in cases with special preference structure
• Worst / average case
• Generalizing the elicitor– To (combinatorial) exchanges– To (combinatorial) markets with side constraints– To (combinatorial) markets with multiattribute
features
For combinatorial auctions