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