Title: Preference Elicitation in Combinatorial Auctions: An Overview Tuomas Sandholm [For an overview, see review article by Sandholm
1Preference Elicitation in Combinatorial
AuctionsAn OverviewTuomas SandholmFor an
overview, see review article by Sandholm
Boutilier in the textbook Combinatorial Auctions,
MIT Press 2006, posted on course home page
2Setting
- 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)
3Another complex problem in combinatorial
auctions Revelation problem
- In direct-revelation mechanisms (e.g. VCG),
bidders bid on all 2items 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
4Revelation 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
5What info is needed from an agent depends on what
others have revealed
Elicitor
Clearing algorithm
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
6Elicitor 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
7Incentive to answer elicitors queries truthfully
- Elicitors queries leak information across agents
- Thrm. Nevertheless, answering truthfully can be
made an ex post equilibrium ConenSandholm
ACMEC-01 - Elicit enough to determine optimal allocation
overall, and for each agent removed in turn - Use externality pricing Vickrey-Clarke-Groves
(VCG) - 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 elses
valuation(s) LarsonSandholm AAMAS-05
8First generation of elicitors
- Rank lattice based elicitors
Conen Sandholm IJCAI-01 workshop, ACMEC-01,
AAAI-02, AMEC-02
9Rank Lattice
Rank of Bundle Ø A B AB for Agent 1
4 2 3 1 for Agent 2 4
3 2 1
1,1
1,2
2,1
3,1
2,2
1,3
2,3
3,2
1,4
4,1
2,4
3,3
4,2
3,4
4,3
4,4
Infeasible
Feasible
Dominated
10A search algorithm for the rank lattice
- Algorithm 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)
11Value-Augmented Rank Lattice
Value of Bundle Ø A B AB for Agent 1
0 4 3 8 for Agent 2 0 1 6 9
17
1,1
14
13
1,2
2,1
10
12
9
3,1
2,2
1,3
8
9
2,3
3,2
1,4
4,1
2,4
3,3
4,2
3,4
4,3
4,4
12Search 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 node - else 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
13Best worst case elicitation effort
- Best case rank vector (1,...,1) is feasible
- One bundle query to each agent, no value queries
- (VCG payments 0)
- Thrm. Any EBF algorithm requires at worst
(2items bidders biddersitems)/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 NisanSegal 03
14EBF 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
15MPAR 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 biddersitems value queries
16Rank lattice based elicitation
- Go down the rank lattice in best-first order (
EBF) - Performance not as good as value-based why?
- nodes in rank lattice is 2agents items
- feasible nodes is only agentsitems
queries
queries
80
1000
Full revelation
60
100
40
Queries
10
20
1
2
3
4
5
6
4
6
8
2
10
12
agents
items
17Differential-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 ...
18Differential 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
19Differential elicitation ...
- Variation Bitwise decrement mechanism
- Is the difference in value between the best
bundle and the bundle at the current rank greater
than d? - if yes increment d, requires min. Increment
- allows establishing a bit stream (yes/no
answers)
20Differential-revelation Algorithm
- Like EBF algorithms, except in step 3,
determination of the set of combinations that are
considered for expansion - M k?OPEN Tight(k) ? ?k ?d for all d with
Tight(d) ? ?k lt ?d for all d with Not(Tight(d))
21Differential-revelation Theoretical results
- Any algortihm of the modified EBF family finds a
welfare-maximizing feasible allocation - Given an arbitrary subset of rank lattice nodes,
the set M is the same whether the original EBF or
the differential-revelation EBF is used - No additional revelation is needed to determine
the VCG payments
22Policy-independent elicitor algorithms
23Some of our elicitors query types
- 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?
24Interrogation An Example
Questions of the Auctioneer Answers of the
Agents
- a1,a2 Give me your highest ranking bundle
- a1,a2 Give me your next best bundle
- a1 Give me your valuation for AB and Aa2 Give
me your valuation for AB and B
- a1 AB, a2 AB(not feasible)
- a1 A, a2 B(feasible)
- a1 vAB8, vA4a2 vAB9, vB6
25General Algorithmic Framework for Elicitation
Algorithm Solve(Y,G) while not Done(Y,G) do o
SelectOp(Y,G) ? Choose question I
PerformOp(o,N) ? Ask bidder G Propagate(I,G) ?
Update data structures with answer Y
Candidates(Y,G) ? Curtail set of candidate
allocations
Output Y set of optimal allocations Input Y
set of candidate allocations (some may turn
out infeasible, some suboptimal) G partially
augmented order graph
26General Task of the Procedures
- Done checks if the topological structure has
been sufficiently explored to exclude
existence of better solutions - In SelectOp, a Policy determines which questions
to ask next - PerformOp asks the questions and obtains
answers - Propagate will update the augmented order graph
- Candidates will determine a new set of potential
solutions based on the update graph
27(Partially) Augmented Order Graph
8
8
8
8
8
8
Ø
B
A
AB
Agent1
A
Ø
0
0
0
0
gt
Allocations
B
B
4
0
3
6
2
6
1
9
Ø
A
B
AB
Agent2
1
1
0
0
1
6
1,1
1,2
2,1
Rank
Upper Bound
3,1
2,2
1,3
1
9
2,3
3,2
1,4
1,4
AB
6
2,4
3,3
4,2
3,4
4,3
Lower Bound
4,4
Some interesting procedures for combining
different types of info
28Storing the answer
- Interval constraint networks, 1 per agent
- Nodes store upper/lower bounds on value of
bundle - Edge (b,b) means vi(b) ? vi(b)
- At start create all nodes, add edges for free
disposal
29Constraint Network
111
1 per agent
110
101
011
100
010
001
000
30Constraint Network
0,?
111
Upper bound
0,?
0,?
0,?
110
101
011
Lower bound
0,?
0,?
0,?
100
010
001
0
000
31Constraint Propagation
0,?
111
vi(110)5
0,?
0,?
5
110
101
011
0,?
0,?
0,?
100
010
001
0
000
32Constraint Propagation
0,?
111
vi(110)5
0,?
0,?
5
110
101
011
0,?
0,?
0,?
100
010
001
0
000
33Constraint Propagation
5,?
111
vi(110)5
0,?
0,?
5
110
101
011
0,5
0,5
0,?
100
010
001
0
000
34Constraint Propagation
5,?
111
vi(110)5
0,?
0,?
5
110
101
011
0,5
0,5
0,?
100
010
001
0
000
35Constraint Propagation
5,?
111
vi(110)5
0,?
0,?
5
110
101
011
0,5
0,5
0,?
100
010
001
0 ,5
000
000
Additional edges from order queries
36Constraint propagation
- Davis87 shows propagation is
- complete for assimilation (values for UB, LB are
as tight as they can be made) - incomplete for inference (cannot always use
values to infer vi(b) ? vi(b)) - Need to use both values and network topology
during inference
37Are we done yet?
- Need to stop when enough information has been
gathered - Store list of possible allocations (candidates)
C - After each phase, eliminate allocations that
cannot be optimal v(c) ? v(c) - Stop when C 1
38We 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
39What query should the elicitor ask next ?
- Simplest answer value query
- Ask for the value of a bundle vi(b)
- How to pick b, i?
40Random elicitation
- 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?
41Experimental setup for all graphs in this talk
- Simulations
- Draw agents valuation functions from a random
distribution where free disposal is honored - Run the auction auctioneer asks queries of
agents, agents look up answer from a file - Each point on plots is average of 10 runs
42Random elicitation
- Not much better than theoretical bound
queries
queries
2 agents
4 items
80
1000
60
Full revelation
100
Queries
40
10
20
1
9
2
2
3
4
5
6
3
4
5
6
7
8
10
agents
items
43Querying 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
44Querying 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
queries
queries
80
1000
60
Full revelation
100
40
Queries
10
20
1
2
3
4
5
6
3
4
5
6
7
8
9
2
10
agents
items
45Best value query elicitation policy so far
Focus 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.
Omniscient elicitor
Optimal elicitor implementable, but utterly
intractable.
46Worst-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 Nisans home page
L is the number of items
Proof.
47Universal revelation reducer
- Def. For a given query class, a universal
revelation reducer is an elicitor that will ask
less than everything whenever the shortest
certificate includes less than all queries - Thrm. Hudson Sandholm ACMEC-03, AAMAS-04 No
deterministic universal revelation reducer exists
for value queries - Randomized ones exists, e.g., the random elicitor
48Restricted preferences
- Even worst-case number of queries is polynomial
when agents valuation functions fall within
certain natural classes
49Read-once valuations
Returns sum of c highest-valued inputs if at
least k inputs are positive, 0 otherwise
PLUS
MAX
ALL
ALL
1000
500
400
100
200
150
- 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 agents 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
50Toolbox valuations
- Items are viewed as tools
- Agent can accomplish multiple goals
- Each goal has a value requires some subset of
tools - Agents 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
51Computational 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, s1s2)
522-wise dependent valuations
- Prop. If an agent has a 2-wise dependent
valuation function, elicitor finds it in m(m1)/2
queries - Thrm. If an agents valuation function is
approximately 2-wise dependent, elicitor finds an
approximation in m(m1)/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
53Gk 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
54Combining 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
55Combining polynomially elicitable classes
- Computational complexity?
- O(items2 items t) for union of
- Read-once valuations (with SUM and MAX gates
only) - Toolbox valuations (with t goals)
- 2-wise dependent valuations
- Toolbox-t
- INTERVAL
56In some settings, learning only a tiny part of
valuation fns suffices to allocate optimally
- Consider 2 agents
- Each has some subsets of items that he likes
- Each such subset is of size log m
- Agents 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 gt can require
super-polynomial number of queries to learn an
agents valuation fn - But Thrm. Optimal allocation can be determined
in poly(m) queries
57In 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
- Agents valuation is
- 1 for all bundles of size S, except for S
itself - 0 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.
58Power 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
59Other 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-takers non-price preferences in
(combinatorial) reverse auctions Boutilier,
Sandholm, Shields AAAI-04
60Demand queries
- If the prices (on items or some bundles) were p,
which bundle would you buy?
61Value queries vs. demand queries
- A value query can be simulated by a polynomial
number of demand queries BlumrosenNisan EC-05 - A demand query cannot be simulated in a
polynomial number of value queries
BlumrosenNisan 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
62Ascending 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 03 - To allocate optimally, enough info has to be
elicited to determine the minimal competitive
equilibrium prices Parkes Nisan-Segal 03 - Could also use descending prices
63XOR-bidding language Sandholm ICE-98, IJCAI-99
- (umbrella, 4) XOR (raincoat, 5)
XOR (umbrella,raincoat,
7) XOR - Bidders valuation is the highest-priced term, of
the terms whose bundle the bidder receives
64Power 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
2?(vm) queries are needed in the worst case - even if each agents XOR-valuation has only O(vm)
terms
65Conclusions on preference elicitation in
combinatorial auctions
- Reduces the number of local plans needed
- Capitalizes on multi-agent elicitation
- Truth-promoting push-pull mechanism
66Future 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)
67Future research on multiagent preference
elicitation
- Scalable general elicitors (in queries, CPU, RAM)
- Current run-time exp in items, poly in agents
- Current space exp in items, linear in agents
- More powerful queries, e.g. side constraints
- New query policies
- New polynomially elicitable valuation classes
- Using models of how costly it is to answer
different queries Hudson S. AMEC-02 - Decision-theoretic elicitation using priors
- Elicitors for markets beyond combinatorial
auctions - (Combinatorial) reverse auctions exchanges
- (Combinatorial) markets with side constraints
- (Combinatorial) markets with multiattribute
features -
- Other applications (e.g. voting Conitzer S.
AAAI-02)
68Tradeoffs between
- Agents evaluation complexity
- Amount revealed to the auctioneer (crypto)
- Amount revealed to other agents (vs. to elicitor)
- Bits communicated
- Elicitors computational complexity (knowing when
to terminate, what to ask next) - Elicitors memory usage (e.g., implicit candidate
list) - Designers objective
- Designing for specific prior eliciting using
the prior - Terminating before optimal allocation,
69Thank you for your attention!
- Papers with additional results at
www.cs.cmu.edu/sandholm
70Revelation principle
- Thrm. Anything that can be accomplished with some
mechanism x can also be accomplished via a
mechanism where agents reveal their preferences
truthfully in a single step
71Sometimes a non-truthpromoting mechanism is
preferable
- Thrm. There are settings where
- Executing the social welfare maximizing
truth-promoting mechanism is NP-complete - There is a non-truthpromoting mechanism, where
- The mediator only carries out polynomial
computation - Finding a beneficial insincere revelation is
NP-complete for each agent - If an agent manages to find a beneficial
insincere revelation, the mechanism is as good as
the truth-promoting one - Otherwise, the non-truthpromoting mechanism
yields greater social welfare - Are there practical settings where
non-truthpromoting mechanisms yield a significant
benefit? - What would such mechanisms look like?
- Are there principles for designing them?
- Can they be designed automatically?
- What about multi-step non-truthpromoting
mechanisms?
72Mechanisms that take into account agents limited
computing
- Mechanisms that are average-case hard to
manipulate, or where every instance is hard - Modeling deliberation actions as part of the game
Larson Sandholm AAAI-00,
AGENTS-01 WS on Agent-based Approaches to B2B,
TARK-01, AIJ-01, AAMAS-02, Draft-03 - E.g. bidding agents that determine valuations of
items in auctions where local planning is
intractable - Anytime algorithms normative deliberation
control method - Strategic computing
- Deliberation equilibrium
- Thrm. Even for 1-item auctions, if computing is
costly, there is no mechanism that motivates
truthful bidding and avoids strategic computing - What would good mechanisms for such agents look
like? Design principles? Automated design?
73Order queries
- Order query agent i, is bundle b worth more to
you than bundle b ? - Motivation Often easier to answer than value
queries - Order queries are insufficient for determining
welfare maximizing allocations - How to interleave order, value queries?
- How to choose i, b, b ?
74Value and order queries
- Interleave
- 1 value query (of random allocatable agent-bundle
pair) - 1 order query (pick arbitrary allocatable i, b,
b ) - To evaluate, in the graphs we have
- value query costs 1
- order query costs 0.1
75Value and order queries
- Elicitation cost reduced compared to value
queries only - Cost reduction depends on relative costs of order
value queries
76Bound-approximation queries
- Often bidders can determine their valuations more
precisely by allocating more time to deliberation
S. AAAI-93, ICMAS-95, ICMAS-96, IJEC-00 Larson
S. TARK-01, AGENTS-01 workshop, SITE-02 Parkes
IJCAI workshop-99 - Get better bounds UBi(b) and LBi(b) with more
time spent deliberating - Idea dont ask for exact info if it is not
necessary - Query agent i, hint spend t time units
tightening the upper (lower) bound on b - How to choose i, b, t, UB or LB ?
- For simplicity, in the experiment graph, fix t
0.2 time units (1 unit gives exact)
77Bound-approx query policy
This slide is hidden later, it should replace
the next slide.
- For simplicity, fix t 0.2 units (1 unit gives
exact) - Can choose randomly.
- More complicated policy does slightly better
- Choose query that will change the bounds on
allocatable bundles the most - Dont know how much bounds will change
- Will try 3 policies
- Compute expectation (assume uniform distribution)
- Be optimistic assume most possible change
- Be pessimistic assume least possible change
78Bound-approximation query policy
- Could choose the query randomly
- More sophisticated policy does slightly better
- Choose query that will change the bounds on
allocatable bundles the most - Dont know exactly how much bounds will change
- Assume all legal answers equiprobable, sample to
get expectation
79Bound-approximation queries
- This policy does quite well
- Future work try other related policies
queries
queries
160
1000
Full revelation
120
100
Query cost
80
10
40
1
9
2
2
3
4
5
6
3
4
5
6
7
8
10
agents
items
80Bound-approximation a note
- To choose which query to ask, we calculated the
expected change it makes - But what is change from ? ?
- Policy actually is ask everyone for an UB on the
grand bundle first - After that, we neednt worry about ?
- Thrm. Upper bound on value of grand bundle is
needed for all but one agent - Thrm. With more than one bidder, eliciting the
grand bundle from every agent cannot increase the
length of the shortest elicitation certificate
81Supplementing bound-approximation queries with
order queries
- Integrated as before
- Computationally more expensive
queries
queries
160
1000
Full revelation
120
Total cost
100
80
Order cost
10
Value cost
40
1
2
3
4
5
6
3
4
5
6
7
8
9
2
10
agents
items
82A potentially better policy
- Assume auctioneer has an oracle that says which
allocation is optimal. How to verify? - To prove optimality, need to
- Prove sufficiently tight LB on optimal
- Prove sufficiently tight UB on all others
- Indicates a strategy when oracle is missing
- Usually ask queries that reduce UB
- But, need to sometimes raise LB
83Incentive compatibility
- Elicitors questions leak information about
others preferences - Can be made ex post incentive compatible
- Ask enough questions to determine VCG prices
- Worst approach bidders1 elicitors
- Could interleave these extra questions with
real questions - To avoid lazyness Not necessary from an
incentive perspective - Agents dont have to answer the questions may
answer questions that were not asked - Unlike in price feedback (tatonnement)
mechanisms Bikhchandani-Ostroy, Parkes-Ungar,
Wurman-Wellman, Ausubel-Milgrom,
Bikhchandani-deVries-Schummer-Vohra, - Push-pull mechanism
84Incentive compatibility of the different
approaches
- Classic single-shot full revelation mechanims
(e.g., Vickrey-Clarke-Groves, dAGVA) - Can be made dominant strategy incentive
compatible - (Ascending) mechanisms with price feedback (e.g.,
iBundle, akBa) - Can be made incentive compatible in ex post
equilibrium - Our new approach an elicitor agent
- Elicitors questions leak information about
others preferences - Can be made incentive compatible in ex post
equilibrium - Ask enough questions to determine VCG prices
- Could interleave these extra questions with
real questions - To avoid lazyness Not necessary from an
incentive perspective - Bidders can pass on questions answer questions
that were not asked
85Elicitation where worst-case number of queries is
polynomial in items
86Ascending combinatorial auctions
87Demand queries
- If these were the prices, which bundle would you
buy? - A value query can be simulated by a polynomial
number of demand queries - A demand query cannot be simulated in a
polynomial number of value queries Nisan
88Ascending combinatorial auctions
- Increase prices until each item is demanded only
once - Item prices vs. bundle prices
- E.g. where there exist no appropriate item prices
- Discriminatory vs. nondiscriminatory prices
Bundle Bidder 1s valuation Bidder 2s valuation
1 0 2
2 0 2
1,2 3 2
89Competitive equilibrium
- Def. Competitive equilibrium (CE)
- For each bidder, payoff max vi(S) pi(S), 0
- Sellers payoff maxS ? Feasibles ?i pi(S)
- Prices can be on bundles and discriminatory
- Thrm. Allocation S is supported in CE iff it is
an efficient allocation - Thrm Parkes 02 NisanSegal 03. In a
combinatorial auction, the information implied by
best-responses to some set of CE prices is
necessary and sufficient as a certificate for the
optimal allocation
90Communication complexity of ascending auctions
- Exponential in items in the general case
- (like any other preference elicitation scheme)
- If items are substitutes (for each agent), then a
Walrasian equilibrium exists, - i.e., nondiscriminatory per-item prices suffice
for agents to self-select the right items - Number of queries needed to find such prices is
polynomial in items Nisan Segal 03
91Conclusions on preference elicitation in
combinatorial auctions
- Combinatorial auctions are desirable winner
determination algorithms now scale to the large - Another problem The Revelation Problem
- Valuation computation / revelation /
communication - Introduced an elicitor that focuses revelation
- Provably finds the welfare maximizing (or Pareto
efficient) allocations - Policy dependent search algorithms for
elicitation - Based on topological observations
- Optimally effective among admissibly equipped
elicitors - Eliciting value differences suffices
- Policy independent general elicitation framework
- Uses value, order rank queries (etc)
- Bound-approximation queries takes incremental
revelation further - Several algorithms, data structures query
policies in the paper - Only elicits a vanishingly small fraction of the
valuations - Presented a way to make the elicitor incentive
compatible - Yields a push-pull partial-revelation mechanism