Iterative Combinatorial Auctions: Achieving Economic and …parkes/pubs/defense.pdf · 2010. 3....

Parkes Iterative Combinatorial Auctions 1

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Iterative Combinatorial Auctions:Achieving Economic and Computational

Efficiency

David C. Parkes

Computer and Information Science,

University of Pennsylvania.

Dissertation Defense, April 24, 2001.

Advisor: Prof. Lyle H. Ungar


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Mechanism Design

Mechanism design addresses the incentive problems in

distributed systems with:

(a) self-interested users and/or computational devices

(b) private information and goals

and a system-wide objective (e.g. efficiency, fairness,

optimality, etc.)

Computational mechanism design addresses the tension

between game-theoretic concerns and computability.


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Applications

� Distributed open networks

– routing, multi-cast problems (Feigenbaum et al., Ronen 00)

– load balancing (Ferguson et al. 95, Korilis et al. 91)

� Distributed artificial intelligence

– job-shop scheduling (Wellman et al. 00)

– task allocation (Sandholm 93, Rosenschein & Zlotkin 94)

� Electronic Commerce

– bandwidth auctions (Enron)

– surplus inventory auctions (e.g. flights)

– logistics (logistics.com)

– procurement (Ariba, CommerceOne)


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Solution Concepts

� Nash equilibrium. Every agent plays a

utility-maximizing strategy, given the utility-maximizing

strategies of other agents.

– assumes rational and well-informed agents

� Dominant strategy. Every agent has a

utility-maximizing strategy whatever the strategies of

other agents.

– stronger implementation concept, does not require

well-informed or rational agents


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Dissertation: Overview

� Chapter 2. Mechanism Design.

� Chapter 3. Linear programming & Efficient Iterative

Combinatorial Auction Design.

� Chapter 4. iBundle auction. Experimental results.

� Chapter 5. Linear programming and Iterative

Generalized Vickrey Auction Design.

� Chapter 6. iBundle Extend & Adjust. Experimental

results.

� Chapter 7. Bounded-Rational Compatible auction theory.

–Experimental Analysis of auctions with BR agents.

� Chapter 8. Train Scheduling Application.


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A Large Problem: London Bus

Routes

� 3.5 million passengers each day

� 5000 buses

� 700 routes

� Competitive tendering since 1997


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London Bus RoutesAgent Problem: Expected value of a bundle of routes

S � G for operator i:

vi(S) = �i(S)� ci(S)

for expected revenue �i(S) and operating cost ci(S) (must

meet timetable requirements, bus types, frequency, etc.)

Government objective: (allocative efficiency):

max(S1;::: ;SI)

X

i

vi(Si)

s:t: Si \ Sj = ;; 8i 6= j

This is an example of a combinatorial allocation problem.


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The Generalized Vickrey Auction(Vickrey 61, Groves 71, Clarke 73)

1. Agents submit bids, or reported values, v̂i(S) for all

sets S � G.

2. Compute allocation S� to maximize reported value:

V � = max(S1;::: ;SI)

X

i

v̂i(Si)

3. Compute best allocation without each agent i:

(V�i)� = max

(S1;::: ;SI )

X

j 6=i

v̂j(Sj)

4. Compute Payments:

pgva(i) = v̂i(S�i )� (V � � (V�i)

�)

Optimal Mechanism: allocative-efficiency,

strategy-proofness (truth-revelation is a dominant strategy),

individual-rationality.


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Special Case: A Single ItemConsider the Vickrey auction for a single item.

Let b1 denote the highest bid, and b2 the second-highest

bid. Agent 1 pays

pgva = b1 � (b1 � b2) = b2

) sell item to the highest bidder for the second-highest

bid.

This is strategy-proof because an agent’s bid defines the

range of prices it is willing to accept, but not the actual price.


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Computational ProblemsTotally centralized solution:

� Winner-determination problem, selecting bids to

maximize reported value, is NP-hard

– equivalent to the maximum weighted clique problem

� Hard problem for agents to compute value

vi(S) = �i(S)� ci(S)

for a single combination of routes, and an exponential

number of routes to consider.

But: any “optimal” mechanism must be outcome equivalent

with the VCG mechanism (Green & Laffont, 77)


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Tractable Winner Determination

� Restricted bidding languages

(Rothfkopf et al. 98, Vohra & de Vries 00)

– limited bid prices, e.g. sub-modular

– limited bundle types, e.g. consecutive items, etc.

� Implement approximate solutions, and compute

approximate Vickrey payments and/or change the

mechanism.

(KfirDahav et al. 98, Lehmann et al. 00)

� Distribute computation to agents; perhaps issue

“challenges”.

(Brewer 99, Nisan & Ronen 00)


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Tractable Agent Problems

� Introduce bidding programs ; allow the auctioneer to

compute the value of a particular bundle “on-the-fly”

(Nisan & Ronen 00)

– helps if formulation easier than solution

– but, shifts computation to the center and introduces

issues of trust and privacy.

� Use dynamic methods ; ask for some information,

perform intermediate computation, ask for some more

information, etc.

– compute the GVA outcome without complete

enumeration of vi(S) from every agent

– might also simplify the auctioneer’s problem


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Complete Info. is Not

Necessary!

� Single item: values v1 = 4; v2 = 8; v3 = 12.

– info. v1 � 8; v2 = 8; v3 � 8.

� Combinatorial auction: agents have non-overlapping

optimal bundles.

1

23

4

: : : can compute and verify the efficient allocation in both

cases.


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Competitive Equilibrium PricesAt competitive equilibrium prices the efficient allocation S�:

(1) allocates a bundle to maximize the utility (value -

price) of every agent

(2) maximizes revenue to auctioneer at the prices.

i.e. solves the CAP with only “best-response” information

from each agent.

Note: prices may need to be non-linear and

non-anonymous, but will always exist (Bikchandani & Ostroy 98)

Important questions:

– what information is required to compute CE prices?

– how much computation must agents perform to provide

that information?

– can agents manipulate the outcome?


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Untangling Best-Response

) look for a dynamic procedure to compute CE prices, and

study computational and strategic properties.


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Contributions

� Connection between linear programming and iterative

auction design for the CAP.

� iBundle, the first efficient ascending-price auction for a

reasonable agent bidding strategy.

� Primal-dual method, VICKAUCTION, to compute both

Vickrey payments and the efficient allocation with only

best-response information from agents.

� Experimental design, iBundle Extend&Adjust, for an

iterative Generalized Vickrey Auction.

– key innovation is a two phase design, with prices

adjusted to Vickrey payments after termination.


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Important Previous Work

year author(s) assum result

1981 Crawford & Knoer linear-additive Vick

1986 Demange et al. assignment problem Vick

1997 Ausubel weakly diminishing returns Vick

homog. items

2000 Gul & Stacchetti gross-substitutes min CE

2000 Ausubel gross-substitutes Vick

Interesting methods with no provable results include: Banks

et al., Rassenti et al., etc., De Martini et al, Wurman &

Wellman, : : :

General problem was largely open.


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Rest of the talk.

� Iterative Best-Response information is sufficient.

(assume myopic best-response)

– primal-dual method COMBAUCTION

– iBundle auction

– experimental & theoretical analysis

� Iterative mechanisms solve problems without

complete information

– discuss the computational requirements on agents

� Vickrey payments prevent strategic-manipulation ,

and justify myopic best-response

– primal-dual method VICKAUCTION

– iBundle, Extend&Adjust

– experimental & theoretical analysis


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A Primal-Dual Formulation(Bikchandani & Ostroy 98)

Variables

Primal allocation of items to agents

Dual non-linear, possibly non-anonymous prices

LP1

STRONGERFORMULATION

AB

AC AB,1

BC B,2 A,2

LP2 LP3

single-item conflict general conflict item & agent conflict

AB A

AC

prices on items bundle prices discriminatorybundle prices


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Strong DualityFeasible primal S� = (S1; : : : ; SI) & dual pi(�) solutions

are optimal iff complementary-slackness conditions hold.

CS1 Bundle S�i maximizes agent i’s utility, solves:

maxS

vi(S)� pi(S)

(or S�i = ;, if maxS vi(S)� pi(S) � 0).

CS2 Allocation S� maximizes the auctioneer’s revenue,

solves:

max(S1;::: ;SI)

X

i2I

pi(Si)

Auction interpretation: with myopic best-response bids:

(CS1) every agent that bids receives a bundle, and (CS2)

the allocation maximizes revenue for the auctioneer.


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Primal-Dual Methodology

STOP

START

(solve restricted primal Winner Determination

problem, achieve CS2 and CS1 for as many agents as possible)

Increase prices

(test CS conditions) Termination?

Myopic Best-response (get CS conditions)

(progress towards an optimal dual solution)


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The iBundle Auction

� Agents bid on bundles, repeat winning bids.

– exclusive-or bidding language

� Maintain a provisional allocation that maximizes

revenue, given bids received.

� Increase price by � > 0 on bundles that receive

unsuccessful bids.

– simple rule to introduce non-anonymous prices

dynamically

� Terminate when all bidding agents receive a bundle.

Standard auctions (e.g. the English auction, the

simultaneous ascending-price auction) are special cases.


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iBundle: Simple ExampleA B C AB BC AC ABC

Agent 1 60 50 0 200� 50 60 220

Agent 2 0 60 50 60 200 50 220

Agent 3 50 0 75� 50 75 200 220

Optimal Value: V � = 275

1 3 5 7 9 11 13 150

50

100

150

200

250

300

Round

Revenue

Value

(� = 25)


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Theoretical Analysisc.f. Bertsekas’ AUCTION algorithm.

Theorem. (Optimality) iBundle computes allocations with

value within 3minfjGj; jIjg� of optimal with myopic

best-response agent strategies.

jGj items, jIj agents, � bid increment.

Worst-case number of rounds / 1=�, allowing

efficiency-computation tradeoffs.

Proof outline: maintain CS2 and CS1 for agents in prov.

allocation, terminate with CS1 for every agent.

Note: can also prove special cases with anonymous prices

(e.g. “core” + optional, single partition, superadditive values,

etc.)


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Experimental Analysis

� Implement winner-determination as a branch-and-bound

search in bid-space, with carefully selected heuristics

(Sandholm, 1999)

� Test on multiple problem distributions (Banks et al. 89;

Sandholm 99; more...)

� Compare computation in GVA and iBundle for the same

winner-determination algorithm.

– approximations

– average-case information revelation

– effect of price-discrimination


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Approximations:

Adjusting the Bid IncrementProblem sets: 50 items, 50–400 bids, 5–40 agents.

5 10 15 20 25 30 35 4010

0

101

102

103

104

105

106

Number of Agents

Auc

tione

er C

PU

Tim

e (s

)

80%

85%

Truthful

95%99%

GVA

Order-of-magnitude speed-up for small loss in allocative

efficiency.

Note: also experimented with greedy local WD algorithms,

and cache/hot-start techniques and heuristics.


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Information Revelation(averaged over a set of problems)

Auction E� Inf

GVA 100 % 100 %

iBundle+ 99 % 71 %

iBundle+� 99.1 % 71 %

Sim. Asc. Price 83 % –

Naive 70 % –

+ terminates after approximately 18 rounds

� with price-discrimination

Information metric:

Inf =

PS max pbid;i(S)P

S vi(S)


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Agent Computation

� Characterize conditions in which BR is polynomial while

complete revelation is exponential:

– e.g. hard but approximable single-bundle valuation

problem, polynomial number of interesting bundles

� Bounded Rational Compatible (BRC) auctions:

– characterize auction designs in which agents can

follow an equilibrium strategy with approximate

valuations.

p(S )

v(S )

u(S )

1

1

1

p(S )

v(S )

u(S )

2

2

2

(dominates)


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Metadeliberation in Iterative

AuctionsAgents can adjust bids and compute incremental values in

response to bids of other agents.

Agent

, , ,v v2 v31

Bids

Iterative

Feedback

^ ^ ^

vs. sealed-bid “shooting in the dark.”

Experimental analysis to compare agent computation for a

simple model of agent bounded-rationality.

Show iterative auctions:

(a) reduce computation

(b) increase allocative-efficiency.


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Free-riding in iBundleWhat if agents do not follow myopic BR?

– quite easy for an informed agent to manipulate if other

agents follow myopic best-response

Example:

A B AB

Agent 1 0 10� 10

Agent 2 10� 0 10

Agent 3 0 0 15

� Optimal Allocation.

Myopic best-response:

– agent 1 (B; 8), agent 2 (A; 7)

Agent 1 can wait, bid when agent 3 out-bids agent 2:

– agent 1 (B; 5), agent 2 (A; 10).


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Solution: Vickrey Payments

Theorem. (Gul & Stacchetti 00) Computing Vickrey

payments in an iterative auction with myopic BR makes

myopic BR a Bayes-Nash equilibrium of the auction.

) incentive-compatibility.

Methodology:

– primal-dual formulation of Vickrey payments

– extend iBundle to collect just enough additional

best-response information to adjust agent prices

to Vickrey payments after termination.


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Minimal Dual PricesVDLP =

X

i

u�i (pi) + maxS

X

i

pi(Si)

) optimal dual solution is not unique. In fact, the minimal

dual price to each agent i computes its Vickrey payment.

Theorem. Vickrey payment is min CE price:

pgva(i) =minp

pi(S�

i )

s:t: CompSlack(p;S�)

Derive the ADJUST* procedure, to compute Vickrey

payments from:

(a) suitable competitive equilibrium prices

(b) the primal solution, S�.


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Intuition: ADJUST*Discount prices to agent i by �(i):

�(i) = P � �MAXREV(P�i)

where MAXREV(P�i) is the revenue-maximizing

allocation without agent i, and P � is final revenue.

This maintains (CS1) and (CS2).

) the new price pi(S�

i )��(i) is also a CE price,

and we characterize conditions for the new price to

equal the Vickrey payment.


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Extend & Adjust

CS1

p(2)

p(1)price(1)

CS2CS1

pric

e(2)

I

II

Phase II extends iBundle just long enough to achieve the

conditions required to compute Vickrey payments with

ADJUST*.


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Vickrey Primal-Dual Method

VICKAUCTION = COMBAUCTION � PHASE2 � ADJUST*

Theorem. VICKAUCTION is a primal-dual algorithm to

compute the efficient allocation and the Vickrey payments in

CAP.

Proof sketch. Retain the efficient allocation from the end of

Phase I. In Phase II maintain CS1 & CS2, and increase

prices until conditions are met for Adjust* to compute

min CE prices, and therefore Vickrey payments.

note: the agent “interface” assumed in VICKAUCTION

remains myopic best-response to ascending-prices.


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iBundle, Extend&Adjust

� Phase I: iBundle. The allocation at the end of Phase I is

finally implemented as the efficient allocation.

� Phase II: Extend&Adjust. Introduce “dummy” agents

dynamically to mimic continued bidding from real agents

as they drop out, and drive up prices to agents in the

efficient allocation.

Finally, take the allocation from the end of Phase I and the

discounted prices based on Phase II.


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Worked Example

A B AB

Agent 1 0 10� 10

Agent 2 10� 0 10

Agent 3 0 0 15

Phase I: S� = (B;A; ;), P � = 15, pbid = (8; 7; 0).

�init(1) = �init(2) = 15� 15 = 0. Agents 1 and 2 are

dependent agents.

Phase II: Dummy agent: v4(AB) = 15 + L.

Agent 1 bids to 10, and discount ��nal(2) = 10� 8 = 2.

Agent 2 bids to 10, and discount ��nal(1) = 10� 7 = 3.

Outcome: Allocation S� = (B;A; ;) with payments (5; 5; 0),

equal to Vickrey payments.


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Theoretical Results

Theorem. iBundle Extend&Adjust terminates with Vickrey

payments whenever Vickrey payments are supported in

competitive equilibrium.

– linear-additive, homogeneous items & subadditive

valuation functions; assignment problem; two-agents etc.

Conjecture. iBundle Extend&Adjust is an iterative

Generalized Vickrey Auction.

) leads to efficient and IC auction that avoids complete

information revelation.

( must show termination to complete the proof)


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Experimental Results

0.2 0.4 0.6 0.8 10

5

10

15

20

Fraction of Allocations Correct

L 2 Dis

tanc

e to

GV

A P

aym

ents

Phase I Initial AdjustPhase II Min CE

(a) Uniform.

0.2 0.4 0.6 0.8 10

1

2

3

4


L 2 Dis

tanc

e to

GV

A P

aym

ents

(b) Decay.

0.75 0.8 0.85 0.9 0.95 10

5

10

15

20

25


L 2 Dis

tanc

e to

GV

A P

aym

ents

(c) Random.

0.2 0.4 0.6 0.8 10

1

2

3

4

5


L 2 Dis

tanc

e to

GV

A P

aym

ents

(d) Weighted Random


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Discussion: Phase I to Phase IIAn agent’s bids in Phase II have no effect on:

(i) its allocation or

(ii) its payment

(its bids only change the payment made by other agents).

The transition must be hard for agents to detect:

– check that Phase II is not slower than Phase I

– check that the “competitive effect” of dummy agents is

indistinguishable from real agents.

– hide as much information from agents as possible


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Phase I–II: Comput. Analysis

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

Round

Eff Rev Adj Rev

(a) Performance.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

Round

CP

U ti

me

(s/r

ound

)

(b) CPU time.

0 20 40 60 80 1000

2

4

6

8

10

Round

Num

ber

of a

gent

s

dummy regular

(c) Agents.

25 goods, 10 agents, 150 bids.

.

Note: an approximate method Adj-Pivot* , which uses earlier

provisional allocations to adjust prices, looks very promising.


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Proxy Agents (Restrict

Strategies)(see also Ausubel 96, “safe bidding terminal”)

Theorem. Truthful best-response BR(v) is a dominant

strategy in the space of all static best-resp. strat. BR(v̂).

) introduce proxy bidding agents to restrict agent

strategies, convert the auction into an iterative

direct-revelation mechanism .

Proxy 2

Proxy 1

Proxy n

Agent 2

new prices, best-response bids

Auctioneer

incrementalvalue information


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Future iBundle Work

� Methods to speed-up solution of sequential

winner-determination problems

� Approximation algorithms for winner-determination

– consider effect on agent strategies and alloc.

efficiency.

� Exploit structure: develop tractable special-cases

– hierarchical “tree” preference structures

– constraint-based bids, etc.

� Experimental study in a distributed optimization

problem

� Reduce price-discrimination in Extend&Adjust, also

investigate speed-ups from ADJUST-PIVOT*

� Agent meta-deliberation problem in an iterative

combinatorial auction:

– structural complexity, optimal policies, etc.


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Conclusions

� Increasing number of distributed systems with

self-interested agents, present interesting decentralized

optimization problems.

� Computationally tractable mechanisms require new

approaches

– e.g. primal dual methods to construct iterative

strategy-proof auctions

� Exciting research agenda at the interface between

computer science and game theory; with beautiful

theoretical problems and important applications.

Iterative Combinatorial Auctions: Achieving Economic and …parkes/pubs/defense.pdf · 2010. 3....

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Transcript of Iterative Combinatorial Auctions: Achieving Economic and …parkes/pubs/defense.pdf · 2010. 3....