Computer Science and Mechanism
Design: Two Case Studies
namely:
auctions with revenue redistribution
&
voting in highly anonymous environments
Vincent Conitzer
Duke University
computational techniques to help
economists design better mechanisms
(part 1 of this talk)
computer science
economics
techniques from economics to help
computer scientists deal with new
settings
(part 2 of this talk)
Auctions with revenue distribution
[Guo & C.,
ACM-EC 07/accepted with minor revisions to Games and
Economic Behavior]
First-price auction
receives 4
v(
)=1
v(
)=2
v(
v(
)=4
)=5
pays 4
)=3
v(
v(
)=7
Second-price (Vickrey) auction
receives 3
v(
)=2
v(
)=2
v(
v(
)=4
)=4
pays 3
)=3
v(
v(
)=3
Vickrey auction without a seller
v(
)=2
v(
)=4
pays 3
(money wasted!)
v(
)=3
Can we redistribute the payment?
Idea: give everyone 1/n
of the payment
v(
)=2
receives 1
v(
)=4
pays 3
receives 1
v(
)=3
receives 1
not strategy-proof
Bidding higher can increase your redistribution payment
Strategy-proof redistribution
[Bailey 97, Porter et al. 04, Cavallo 06]
Idea: give everyone 1/n of
second-highest other bid
v(
)=2
receives 1
v(
)=4
pays 3
receives 2/3
)=3
v(
receives 2/3
2/3 wasted (22%)
strategy-proof
Your redistribution does not depend on your bid;
incentives are the same as in Vickrey
Bailey-Cavallo mechanism…
•
•
•
•
•
Bids: V1≥V2≥V3≥... ≥Vn≥0
First run Vickrey auction
Payment is V2
First two bidders receive V3/n
Remaining bidders receive
V2/n
• Total redistributed:
2V3/n+(n-2)V2/n
Can we do better?
R1 = V3/n
R2 = V3/n
R3 = V2/n
R4 = V2/n
...
Rn-1= V2/n
Rn = V2/n
Desirable properties
Strategy-proofness
 Voluntary participation: bidder’s utility always
nonnegative
 Efficiency: bidder with highest valuation gets item
 Non-deficit: sum of payments is nonnegative



i.e. total Vickrey payment ≥ total redistribution
(Strong) budget balance: sum of payments is zero

i.e. total Vickrey payment = total redistribution
Impossible to get all
 We sacrifice budget balance


Try to get approximate budget balance
Other work sacrifices: incentive compatibility [Parkes
01], efficiency [Faltings 04], non-deficit [Bailey 97], budget
balance [Cavallo 06]

Another redistribution mechanism
• Bids: V1≥V2≥V3≥V4≥... ≥Vn≥0
• First run Vickrey
• Redistribution:
Receive 1/(n-2) * secondhighest other bid,
- 2/[(n-2)(n-3)] third-highest
other bid
• Total redistributed:
V2-6V4/[(n-2)(n-3)]
• Efficient & strategy-proof
• Voluntary participation & nondeficit (for large enough n)
R1 = V3/(n-2) - 2/[(n-2)(n-3)]V4
R2 = V3/(n-2) - 2/[(n-2)(n-3)]V4
R3 = V2/(n-2) - 2/[(n-2)(n-3)]V4
R4 = V2/(n-2) - 2/[(n-2)(n-3)]V3
...
Rn-1= V2/(n-2) - 2/[(n-2)(n-3)]V3
Rn = V2/(n-2) - 2/[(n-2)(n-3)]V3
Comparing redistributions
•
•
•
•
•
•
•
Bailey-Cavallo: ∑Ri =2V3/n+(n-2)V2/n
Second mechanism: ∑Ri =V2-6V4/[(n-2)(n-3)]
Sometimes the first mechanism redistributes more
Sometimes the second redistributes more
Both redistribute 100% in some cases
What about the worst case?
Bailey-Cavallo worst case: V3=0
– fraction redistributed: 1-2/n
• Second mechanism worst case: V2=V4
– fraction redistributed: 1-6/[(n-2)(n-3)]
• For large enough n, 1-6/[(n-2)(n-3)]≥1-2/n, so
second is better (in the worst case)
Generalization: linear
redistribution mechanisms
• Run Vickrey
• Amount redistributed to bidder:
C0 + C1 V-i,1 + C2 V-i,2 + ... + Cn-1 V-i,n-1
where V-i,j is the j-th highest other bid for bidder i
• Bailey-Cavallo: C2 = 1/n
• Second mechanism: C2 = 1/(n-2), C3 = - 2/[(n-2)(n-3)]
• Bidder’s redistribution does not depend on own bid, so
strategy-proof
• Efficient
• Other properties?
Redistribution to each bidder
Recall: R=C0 + C1 V-i,1 + C2 V-i,2 + ... + Cn-1 V-i,n-1
R1 = C0+C1V2+C2V3+C3V4+...+CiVi+1+...+Cn-1Vn
R2 = C0+C1V1+C2V3+C3V4+...+CiVi+1+...+Cn-1Vn
R3 = C0+C1V1+C2V2+C3V4+...+CiVi+1+...+Cn-1Vn
R4 = C0+C1V1+C2V2+C3V3+...+CiVi+1+...+Cn-1Vn
...
Rn-1= C0+C1V1+C2V2+C3V3+...+CiVi +...+Cn-1Vn
Rn = C0+C1V1+C2V2+C3V3+...+CiVi +...+Cn-1Vn-1
Individual rationality & non-deficit
• Voluntary participation:
equivalent to
Rn=C0+C1V1+C2V2+C3V3+...+CiVi+...+Cn-1Vn-1
≥0
for all V1≥V2≥V3≥... ≥Vn-1≥0
• Non-deficit:
∑Ri≤V2 for all V1≥V2≥V3≥... ≥Vn-1≥Vn≥0
Worst-case optimal (linear)
redistribution
Try to maximize worst-case redistribution %
Variables: Ci, K
Maximize K
subject to:
Rn≥0 for all V1≥V2≥V3≥... ≥Vn-1≥0
∑Ri≤V2 for all V1≥V2≥V3≥... ≥Vn≥0
∑Ri≥KV2 for all V1≥V2≥V3≥... ≥Vn≥0
Ri as defined in previous slides
Transformation into linear program
• Claim: C0=0
• Lemma: Q1X1+Q2X2+Q3X3+...+QkXk≥0 for
all X1≥X2≥...≥Xk≥0
is equivalent to
Q1+Q2+...+Qi≥0 for i=1 to k
• Using this lemma, can write all constraints
as linear inequalities over the Ci
Worst-case optimal remaining %
n=5: 27% (40%)
n=6: 16% (33%)
n=7: 9.5% (29%)
n=8: 5.5% (25%)
n=9: 3.1% (22%)
n=10: 1.8% (20%)
n=15: 0.085% (13%)
n=20: 3.6 e-5 (10%)
n=30: 5.4 e-8 (7%)
the data in the parenthesis are for the Bailey-Cavallo mechanism
m-unit auction with unit demand:
VCG (m+1th price) mechanism
v(
)=2
v(
)=4
pays 2
v(
)=3
pays 2
strategy-proof
Our techniques can be generalized to this setting
Results for m+1th price auction
BC = BaileyCavallo
WO = Worstcase Optimal
Analytical characterization of WO
mechanism
• Unique optimum
• Can show: for fixed m, as n goes to infinity, worst-case
redistribution percentage approaches 100% linearly
• Rate of convergence 1/2
Worst-case optimality outside the
linear family
• Theorem: The worst-case optimal linear
redistribution mechanism is also worst-case optimal
among all VCG redistribution mechanisms that are
–
–
–
–
–
deterministic,
anonymous,
strategy-proof,
efficient,
non-deficit
• Voluntary participation is not mentioned
– Sacrificing voluntary participation does not help
• Not uniquely worst-case optimal
Remarks
• Moulin's working paper “Efficient, strategy-proof and
almost budget-balanced assignment”
pursues different worst-case objective (minimize
waste/efficiency)
– results in same mechanism in the unit-demand
setting (!)
– different mechanism results after removing
voluntary participation requirement
Multi-unit auction with
nonincreasing marginal values
b2
v(second
v (first
)=0
v(second
)=3
v(first
)=4
)=5
v(second
v(first
b1
pays 5
payment of i = total efficiency when i is not present
- total efficiency when i is present
)=1
)=2
Another example
v(second
v (first
)=2
v(second
)=2
v(first
)=3
)=5
pays 2
v(second
v(first
)=1
)=4
pays 3
Approach
• We construct a mechanism that has the same
worst-case performance as the earlier WO
mechanism
• Multi-unit auction with unit demand is a special
case of multi-unit auction with nonincreasing
marginal value
• The new mechanism is optimal in the worst
case
Gadgets
• Let S be a set of bidders. Define function R
recursively:
• R(S,0)=VCG(S)
– total VCG payment from selling all units (using
VCG mechanism) to the set of bidders S
• R(S,i) is defined as
– remove 1 bidder from the first m+i bidders of S
(order by initial marginal value)
– denote the new set by S'
– average over all R(S', i-1) (m+i choices)
– domain: i ≤ |S|-m
Mechanism construction
• The set of all bidders: A={a1,a2,...,an}
– ai is the bidder with the ith highest initial marginal
value
– the set of bidders other than ai: A-i = A - {ai}
• We redistribute to bidder i
1/m ∑j=m+1..n-1 Cj R(A-i , j-m-1)
– The Ci are the same as in unit demand setting
– The mechanism is strategy-proof: redistribution is
independent of your own bid
• This mechanism is worst-case optimal
More general settings?
• If marginal values are not required to be
nonincreasing, the worst-case
redistribution percentage is at most 0
example (nothing can be redistributed, details omitted)
one bidder bids 1 on m units
one bidder bids 1 on 1 unit
the other bidders bid 0
The original VCG mechanism is already worst-case
optimal
• Similar example for general combinatorial
auction with single-minded bidders
Other results
• Undominated redistribution mechanisms
[Guo & C. AAMAS08a]
• Optimal-in-expectation mechanisms [Guo &
C. AAMAS08b]
• Inefficient allocation
– burning units for higher welfare
• Auctions of heterogeneous objects
– submodular valuations (e.g. unit demand)
Voting in highly anonymous
environments
Time Magazine “Person of the Century”
poll – “results” (January 19, 2000)
# Person
1 Elvis Presley
2 Yitzhak Rabin
3 Adolf Hitler
4 Billy Graham
5 Albert Einstein
6 Martin Luther King
7 Pope John Paul II
8 Gordon B Hinckley
9 Mohandas Gandhi
10 Ronald Reagan
11 John Lennon
12 American GI
13 Henry Ford
14 Mother Teresa
15 Madonna
16 Winston Churchill
17 Linus Torvalds
18 Nelson Mandela
19 Princess Diana
20 Pope Paul VI
%
13.73
13.17
11.36
10.35
9.78
8.40
8.18
5.62
3.61
1.78
1.41
1.35
1.22
1.11
0.85
0.83
0.53
0.47
0.36
0.34
Tally
625045
599473
516926
471114
445218
382159
372477
256077
164281
81368
64295
61836
55696
50770
38696
37930
24146
21640
16481
15812
To Our Readers
We understand that some of our users are
upset that the votes for Jesus Christ as
"Person of the Century" were removed from
our online poll.
[…]
TIME's Man of the Year, instituted in the
1930s, has always been based on the impact
of living persons on events around the world.
[…]
When we removed the votes for Jesus we also
removed the votes for the Prophet
Mohammed. We did not wish to see this poll
turned into a mockery, because in our
experience it is quite possible that supporters
of the leading figures might have turned to
computer robots to churn out hundreds of
thousands of "phony" votes for their
champions.
[…]
Ours is a poll that attempts to judge the works
of mere men, the acts in which men render
unto Caesar
[…]
Time Magazine “Person of the Century”
poll – partial results (November 20, 1999)
# Person
%
1 Jesus Christ
48.36
2 Adolf Hitler
14.00
3 Ric Flair
8.33
4 Prophet Mohammed 4.22
5 John Flansburgh
3.80
6 Mohandas Gandhi 3.30
7 Mustafa K Ataturk 2.07
8 Billy Graham
1.75
9 Raven
1.51
10 Pope John Paul II 1.15
11 Ronald Reagan
0.98
12 Sarah McLachlan 0.85
13 Dr William L Pierce 0.73
14 Ryan Aurori
0.60
15 Winston Churchill 0.58
16 Albert Einstein
0.56
17 Kurt Cobain
0.32
18 Bob Weaver
0.29
19 Bill Gates
0.28
20 Serdar Gokhan
0.28
Tally
610238
176732
105116
53310
47983
41762
26172
22109
19178
14529
12448
10774
9337
7670
7341
7103
4088
3783
3629
3627
Ric Flair
John Flansburgh
(They Might Be Giants)
Atatürk
Raven
Howard Stern
Cartman
Optimus Prime
New 7 wonders of the world
• Vote by phone or over Internet
– Latter requires e-mail address
• Can buy more votes
– … or get another e-mail address…
Voting/social choice
• Set of alternatives
• Set of voters
• Each voter ranks all the alternatives
– E.g. b > a > d > c
– Ranking is called a vote
• Deterministic voting rule: maps any multiset of
votes to an alternative
• Randomized voting rule: maps any multiset of votes
to a probability distribution over alternatives
• Neutral rule treats alternatives symmetrically
• Anonymous rule treats votes symmetrically
Example rules
• Plurality: alternative gets 1 point for being
ranked first
• Borda: alternative gets m-1 points for being
ranked first, m-2 for being ranked second, …,
0 for being ranked last
• Single transferable vote/instant runoff:
alternative ranked first by fewest is removed
from all votes; repeat until one left
• Many others…
Strategy-proof voting rules
• Sometimes ranking alternatives differently
from true preferences makes one better off
• A voting rule is strategy-proof if no voter
ever benefits from casting a vote that does
not reflect her true preferences
• Gibbard [Econometrica 77] characterizes
strategy-proof randomized rules
Anonymity-proof voting rules
• A voting rule is false-name-proof if no voter
ever benefits from casting additional
(potentially different) votes
• A voting rule satisfies voluntary
participation if it never hurts a voter to cast
her vote
• A voting rule is anonymity-proof if it is falsename-proof & satisfies voluntary participation
• Can we characterize (neutral, anonymous,
randomized) anonymity-proof rules?
Characterization
• Theorem [C., working paper]:
• Any anonymity-proof voting rule f can be
described by a single number kf in [0,1]
• With probability kf, the rule chooses an
alternative uniformly at random
• With probability 1- kf, the rule draws two
alternatives uniformly at random;
– If all votes rank the same alternative higher among
the two, that alternative is chosen
– Otherwise, a coin is flipped to decide between the
two alternatives
Group strategy-proofness
• Group strategy-proofness: no coalition can
benefit by deviating together
• Theorem [C., working paper]:
• If group-strategyproofness is required in
addition to anonymity-proofness, then:
– for two alternatives, the characterization does not
change
– for three or more alternatives, the only remaining
rule is the one that picks an alternative uniformly at
random
What can we do?
• Make it costly to participate multiple times [Wagman &
C,, working paper]
– Allows for better anonymity-proof mechanisms
– 2 alternatives: probability of choosing majority winner goes
to 1 as number of voters grows
• Verify some identities after the fact, discard
preference reports that fail verification [C., TARK 07]
– Optimal verification is computationally hard (but there are
good approximation algorithms)
• Prevent multiple account signups by creating a
memory test that is easy to pass once, difficult to
pass twice [C., AMEC 08]
– Does not work very well (yet!)
Conclusions
• Designed auctions that redistribute revenue
optimally (in various senses)
– Computational techniques (LP) were key tool
– More general agenda: automated mechanism
design [C. & Sandholm UAI-02, ICEC-03, ACM-EC-04, AAMAS04, IJCAI-07; Jurca & Faltings ACM-EC-06, ACM-EC-07; Hyafil &
Boutilier AAAI-07, IJCAI-07; …]
• Designing voting rules for highly anonymous
environments
– Straightforward approach leads to very negative
results
– More creative approaches are needed
Thank you for your attention!