Preference Elicitation in Combinatorial Auctions:

An Overview

Tuomas Sandholm

[For an overview, see review article by Sandholm & Boutilier in the textbook

Combinatorial Auctions, MIT Press 2006, posted on course home page]

Setting

Combinatorial auction: m items for sale• Private values auction, no allocative externalities

– So, each bidder i has value function, vi: 2m R

• Free disposal• Unique valuations (to ease presentation)

Another complex problem in combinatorial auctions: “Revelation problem”

• In direct-revelation mechanisms (e.g. VCG), bidders bid on all 2#items combinations– Need to compute the valuation for exponentially many

combinations• Each valuation computation can be NP-complete local planning problem

• For example if a carrier company bids on trucking tasks: TRACONET [Sandholm AAAI-93, …]

– Need to communicate the bids

– Need to reveal the bids• Loss of privacy & strategic info

Revelation problem …

• Agents need to decide what to bid on– Waste effort on counter-speculation

– Waste effort making losing bids

– Fail to make bids that would have won • Reduces economic efficiency & revenue

Clearing algorithm

What info is needed from an agent depends on what others have revealed

Elicitor

Conen & Sandholm IJCAI-01 workshop on Econ. Agents, Models & Mechanisms, ACMEC-01

Elicitor decides what to ask next based on answers it has received so far

$ 1,000 for

$ 1,500 for

? for

Conen & Sandholm IJCAI workshop-01, ACMEC-01

Elicitor skeleton

• Repeat:– Decide what to ask (and from which bidder)– Ask that and propagate the answer in data structures– Check whether you know the optimal allocation of

items to agents. If so, stop

Incentive to answer elicitor’s queries truthfully

• Elicitor’s queries leak information across agents• Thrm. Nevertheless, answering truthfully can be made

an ex post equilibrium [Conen&Sandholm ACMEC-01]

– Elicit enough to determine optimal allocation overall, and for each agent removed in turn

– Use VCG pricing

• Push-pull mechanism• If a bidder can endogenously decide which bundles for

which bidders to evaluate, then no nontrivial mechanism – even a direct revelation mechanisms - can 1) be truth-promoting, and 2) avoid motivating an agent to compute on someone else’s valuation(s) [Larson&Sandholm AAMAS-05]

First generation of elicitors

• Rank lattice based elicitors

[Conen & Sandholm IJCAI-01 workshop, ACMEC-01, AAAI-02, AMEC-02]

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

A search algorithm for the rank latticeAlgorithm PAR “PAReto optimal“

OPEN [(1,...,1)]while OPEN [] do

Remove(c,OPEN); SUC suc(c);if Feasible(c) then

PAR PAR c; Remove(SUC,OPEN)else foreach node SUC do

if node OPEN and Undominated(node,PAR)then Append(node,OPEN)

• Thrm. Finds all feasible Pareto-undominated allocations (if bidders’ utility functions are injective, i.e., no ties)

• Welfare maximizing solution(s) can be selected as a post-processor by evaluating those allocations – Call this hybrid algorithm MPAR (for “maximizing” PAR)

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]

Search algorithm family for the value-augmented rank lattice

Algorithm EBF “Efficient Best First“OPEN (1,...,1)loop if |OPEN| = 1 then c combination in OPEN else

M k OPEN | v(k) = maxnode OPEN v(node) if |M| 1 node M with Feasible(node) then return nodeelse choose c M such that c is not dominated by any node M

OPEN OPEN \ c if Feasible(c) then return c else foreach node suc(c) do

if node OPEN then OPEN OPEN node

• Thrm. Any EBF algorithm finds a welfare maximizing allocation• Thrm. VCG payments can be determined from the information already elicited

Best & worst case elicitation effort

• Best case: rank vector (1,...,1) is feasible – One bundle query to each agent, no value queries– VCG payments are all 0

• Thrm. Any EBF algorithm requires at worst (2#items #bidders – #bidders#items)/2 + 1 value queries– Proof idea. Upper part of the lattice is infeasible and not less in

value than the solution

• Not surprising because in the worst case, finding a provably (even approximately) optimal allocation requires exponentially many bits to be communicated no matter what query types are used and what query policy is used [Nisan&Segal J. Economic Theory 2006]

– We will prove this later

EBF minimizes feasibility checks

• Def: An algorithm is admissible if it always finds a welfare maximizing allocation

• Def: An algorithm is admissibly equipped if it only has– value queries, and– a feasibility function on rank vectors, and– a successor function on rank vectors

• Thrm: There is no admissible, admissibly equipped algorithm that requires fewer feasibility checks (for every problem instance) than an (arbitrary) EBF algorithm

MPAR minimizes value queries

• Thrm. No admissible, admissibly equipped algorithm (that calls the valuation function for bundles in feasible rank vectors only) will require fewer value queries than MPAR

• MPAR requires at most #bidders#items value queries

Differential-revelation• Extension of EBF• Information elicited: differences between valuations

– Hides sensitive value information

• Motivation: max ∑ vi(Xi) min ∑ [vi(r-1(1)) – vi(Xi)]

– Maximizing sum of value Minimizing difference between value of best ranked bundle and bundle in the allocation

• Thrm. Differences suffice for determining welfare maximizing allocations & VCG payments

• 2 low-revelation incremental ex post incentive compatible mechanisms ...

Differential elicitation ...

• Questions (start at rank 1)– “tell me the bundle at the current rank”– “tell me the difference in value of that bundle and

the best bundle“• increment rank

• Natural sequence: from “good” to “bad” bundles

Policy-independent elicitor algorithms

What query should the elicitor ask next ?

• Simplest answer: value query– Ask for the value of a bundle vi(b)

• How to pick b, i?

Hudson & Sandholm AMEC-02, AAMAS-04

• Asks randomly chosen value queries whose answer cannot yet be inferred

• Thrm. If the full-revelation mechanism makes Q value queries and the best value-elicitation policy makes q queries, random elicitation makes on average

value queries

– Proof idea: We have q red balls, and the remaining balls are blue; how many balls do we draw before removing all q red balls?

Random elicitation

Random elicitation

• Not much better than theoretical bound

agentsitems3 4 5 6 7 8 92 10 2 3 4 5 6

queries

1

10

100

1000

20

40

60

80

queries

Full revelation

Queries

2 agents 4 items

Querying random allocatable bundle-agent pairs only…

• Bundle-agent pair (b,i) is allocatable if some yet potentially optimal allocation allocates bundle b to agent i

• How to pick (b,i)?– Pick a random allocatable one

• Asking only allocatable bundles means throwing out some queries

• Thrm. This restriction causes the policy to make at worst twice as many expected queries as the unrestricted random elicitor. (Tight)– Proof idea: These ignored queries are either

• Not useful to ask, or• Useful, but we would have had low probability of asking it, so no big difference in

expectation

Querying random allocatable bundle-agent pairs only…

• Much better– Almost (#items / 2) fewer queries than unrestricted random– Vanishingly small fraction of all queries asked !– Subexponential number of queries

agentsitems3 4 5 6 7 8 92 10 2 3 4 5 6

queries

1

10

100

1000

20

40

60

80

queries

Full revelation

Queries

Hudson & Sandholm AMEC-02, AAMAS-04

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

2 agents

3 agents

4 agents

Number of items for sale

Fraction of values queried before provably optimal allocation found

Omniscient elicitor

Optimal elicitor implementable, but utterly intractable.

Best value query elicitation policy so farFocus on allocations that have highest upper bound.Ask a (b,i) that is part of such an allocation and among them, pick the one that affects (via free disposal) the largest number of bundles in such allocations.

Worst-case number of bits transmitted (nondeterministic model)

• Exponential (even to approximately optimally allocate the items within ratio better than 1/2) [Nisan & Segal JET-06; see also CS-friendly version from Nisan’s home page]

Proof.

L is the number of items

Restricted preferences

Even worst-case number of queries is polynomial when agents’ valuation functions fall within

certain natural classes…

Zinkevich, Blum & Sandholm ACMEC-03

Read-once valuations

• Thrm. If an agent has a read-once valuation function, the number of value queries needed to elicit the function is polynomial in items

• Thrm. If an agent’s valuation function is approximable by a read-once function (with only MAX and PLUS nodes), elicitor finds an approximation in a polynomial number of value queries

PLUS

ALL

MAX

ALL

500 400 200 100

1000

150

GATEk,c

Returns sum of c highest-valued inputs if at least k inputs are positive, 0 otherwise

Zinkevich, Blum & Sandholm ACMEC-03

Toolbox valuations

• Items are viewed as tools• Agent can accomplish multiple goals

– Each goal has a value & requires some subset of tools – Agent’s valuation for a package of items is the sum of

the values of the goals that those tools allow the agent to accomplish

• E.g. items = medical patents, goals = medicines• Thrm. If an agent has a toolbox valuation function,

it can be elicited in O(#items #goals) value queries

Zinkevich, Blum & Sandholm ACMEC-03

Computational complexity of finding an optimal allocation after elicitation

• Thrm. Given one agent with an additive valuation fn and one agent with a read-once valuation fn, allocation requires only polynomial computation

• Thrm. With 2 agents with read-once valuations (even with just MAX, SUM, and ALL gates), it is NP-hard to find an allocation that is better than ½ optimal

• Thrm. Given 2 agents with toolbox valuations having s1 and s2 terms respectively, optimal allocation can be done in computation time poly(m, s1+s2)

Conitzer, Sandholm & Santi Draft-03, AAAI-05

0+1+2 = 3

2-wise dependent valuations

• Prop. If an agent has a 2-wise dependent valuation function, elicitor finds it in m(m+1)/2 queries

• Thrm. If an agent’s valuation function is approximately 2-wise dependent, elicitor finds an approximation in m(m+1)/2 queries

– Thrm. Every super-additive valuation function is approximately 2-wise dependent

• Thrm. These results generalize to k-wise dependent valuationsusing O(mk) queries

1

3

3-2

0

2

1Node = item

m items

Conitzer, Sandholm & Santi Draft-03

Gk = k-wise dependent valuations

• G1 G2 … Gm

• G1 = linear valuations: Easy to elicit & allocate

• Gk where k ≥ 2 is a constant: Easy to elicit, NP-hard to allocate – if graph cycle free (i.e. forest), allocation polytime

• Gg(m) where g(m) is an arbitrary (sublinear) fn s.t. g(m) approaches infinity as m approaches infinity: Hard to elicit & NP-hard to allocate

• Gm contains all valuation fns

Santi, Conitzer, Sandholm COLT-04

Combining polynomially elicitable classes

• Thrm. If class C1 (resp. C2) is elicitable using p1(m) (resp. p2(m)) queries, then C1 union C2 is elicitable in p1(m) + p2(m) + 1 queries. Tight

Blum, Jackson, Sandholm & Zinkevich JMLR-04

In some settings, learning only a tiny part of valuation fns suffices to allocate optimally

• Consider 2 agents with valuations f and g– Each has some subsets of items that he likes– Each such subset is of size log m– Agent’s valuation is 1 if he gets a set of items that he likes, 0 otherwise

• Since there are bundles of size log m, some members of this class cannot be represented in poly(m) bits => can require super-polynomial number of queries to learn an agent’s valuation fn

• But… Thrm. Optimal allocation can be determined in poly(m) queries– Proof: Try random partitions of items into two equal-sized sets– Derandomization: A set of assignments to m boolean variables is (m,k)-universal if

for every subset of k variables, the induced assignments to those variables cover all 2k settings. Naor and Naor (1990) give efficient constructions of such sets using only 2O(k) log m assignments. We can use k = O(log m), so the construction is polynomial time and space. Each of these assignments corresponds to a partition of items, and we ask f and g for their valuations on each one and take the best.

Blum, Jackson, Sandholm & Zinkevich JMLR-04

In some settings, learning only a tiny part of valuation fns suffices to allocate optimally…

• There can be super-polynomial power even when valuation fns have short descriptions

• Let each agent have some distinguished bundle S’• Agent’s valuation is

1 for all bundles of size ≥ |S’|, except for S’ itself0 otherwise

• Prop. It can take value queries to learn such a valuation fn

• Thrm. With two agents with such valuation fns, the optimal allocation can be determined in 4 + log2 m value queries– Proof. First find |S’| in log2 m + 1 queries using binary search. Then

make 3 arbitrary queries of size |S’|. At most 1 of them can return 0. Call the other two sets T and T’. We then query the other agent for M-T; if it returns 1, then T, M-T is an optimal allocation. Otherwise, T’, M-T’ is optimal.

Power of interleaving queries among agents

• Observation: In general (not just in combinatorial auctions), we can elicit without interleaving within a number of queries that is exponential in q– where q is the number of queries used when

eliciting with interleaving.

• Proof: Contingency plan tree is (merely) exponential in the number of queries

Other results on elicitation

• Interleaving value & order queries [Hudson & Sandholm AMEC-02, AAMAS-04]

• Bound-approximation queries [Hudson & Sandholm AMEC-02, AAMAS-04]

• Elicitation in exchanges (for multi-robot task allocation) [Smith, Sandholm & Simmons AAAI-02 workshop]

• Eliciting bid-taker’s non-price preferences in (combinatorial) reverse auctions [Boutilier, Sandholm, Shields AAAI-04]

Demand queries

“If the prices (on items or some bundles) were p, which bundle would you buy?”

Value queries vs. demand queries

• A value query can be simulated by a polynomial number of demand queries [Blumrosen&Nisan EC-05, see also their SIAM J. Computing 2010 paper]– Proof. Elicit value of adding one item at a time into the bundle. The marginal

value of each such addition is done via binary search on that item’s price.

• A demand query cannot be simulated in a polynomial number of value queries [Blumrosen&Nisan EC-05]

• There exists restricted CAs where optimal allocation can be found in poly bits, but exponential number of demand (and thus value) queries are needed [Nisan & Segal TARK-05]

Ascending combinatorial auctions

• Demand queries– Per-item prices vs. bundle prices– Discriminatory vs. nondiscriminatory prices

• Exponential communication complexity, but polynomial in special classes (e.g., when items are substitutes) [Nisan-Segal JET-06]– To allocate optimally, enough info has to be elicited to

determine the minimal competitive equilibrium prices [Parkes; Nisan-Segal JET-06]

• Could also use descending prices

Ascending combinatorial auctions…• Thm [Blumrosen & Nisan JET-10]. To achieve efficiency, the number of trajectories in

an ascending item-price CA may have to be exponential in the number of items– even if the trajectories can be interleaved and what is done on a trajectory can depend on what happened on

other trajectories

• Thm [Blumrosen & Nisan JET-10]. Any anonymous ascending (even bundle-price) CA may fail to find an efficient allocation

XOR-bidding language [Sandholm ICE-98, IJCAI-99]

• (umbrella, $4) XOR (raincoat, $5) XOR (umbrella,raincoat, $7) XOR …

• Bidder’s valuation is the highest-priced term, of the terms whose bundle the bidder receives

Power of bundle prices

• Thrm. [Lahaie & Parkes ACMEC-04] Using bundle-price demand queries (even when only poly(m) bundles are priced) and value queries, an XOR-valuation can be learned in O(m2 #terms) queries

• Thrm. [Blum, Jackson, Sandholm, Zinkevich COLT-03, JMLR-04] If the elicitor can use value queries and item-price demand queries only, then 2Ω(√m) queries are needed in the worst case– even if each agent’s XOR-valuation has only O(√m)

terms

Conclusions on preference elicitation in combinatorial auctions

• Reduces the number of local plans needed

• Capitalizes on multi-agent elicitation

• Truth-promoting push-pull mechanism

Future research on preference elicitation

• Scalable general elicitors (in queries, CPU, RAM)

• New polynomially elicitable valuation classes

• More powerful queries, e.g. side constraints• Using models of how costly it is to answer

different queries [Hudson & Sandholm AMEC-02, AAMAS-04]

– Strategic deliberation [Larson & Sandholm]

• Other applications (e.g. voting [Conitzer & Sandholm AAAI-02,

EC-04])

Tradeoffs between

1. Agent’s evaluation complexity

2. Amount revealed to the auctioneer (crypto)

3. Amount revealed to other agents (vs. to elicitor)

4. Bits communicated

5. Elicitor’s computational complexity (knowing when to terminate, what to ask next)

6. Elicitor’s memory usage (e.g., implicit candidate list)

7. Designer’s objective• Designing for specific prior & eliciting using the prior

• Terminating before optimal allocation, …