Computer Science and Mechanism Design: Two Case Studies namely: auctions with revenue redistribution...

Computer Science and Mechanism Design: Two Case Studies

namely:auctions with revenue redistribution

&voting in highly anonymous environments

Vincent Conitzer

Duke University

computer science economics

computational techniques to help economists design better mechanisms

(part 1 of this talk)

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

v( ) = 2 v( ) = 5 v( ) = 7

v( ) = 1 v( ) = 4 v( ) = 3

pays 4

receives 4

Second-price (Vickrey) auction

v( ) = 2 v( ) = 4 v( ) = 3

v( ) = 2 v( ) = 4 v( ) = 3

pays 3

receives 3

Vickrey auction without a seller

v( ) = 2 v( ) = 4 v( ) = 3

pays 3(money wasted!)

Can we redistribute the payment?

v( ) = 2 v( ) = 4 v( ) = 3

pays 3receives 1receives 1

receives 1

Idea: give everyone 1/n of the payment

not strategy-proofBidding higher can increase your redistribution payment

Strategy-proof redistribution[Bailey 97, Porter et al. 04, Cavallo 06]

v( ) = 2 v( ) = 4 v( ) = 3

pays 3receives 1receives 2/3

receives 2/3

Idea: give everyone 1/n of second-highest other bid

strategy-proofYour redistribution does not depend on your bid;

incentives are the same as in Vickrey

2/3 wasted (22%)

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

R1 = V3/nR2 = V3/nR3 = V2/nR4 = V2/n...Rn-1= V2/nRn = V2/n

Can we do better?

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) * second-highest other bid, - 2/[(n-2)(n-3)] third-highest other bid

• Total redistributed:

V2-6V4/[(n-2)(n-3)] • Efficient & strategy-proof• Voluntary participation & non-

deficit (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?

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




...

Rn-1= C0+C1V1+C2V2+C3V3+...+CiVi +...+Cn-1Vn

Rn = C0+C1V1+C2V2+C3V3+...+CiVi +...+Cn-1Vn-1

Redistribution to each bidder

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 v( ) = 3

pays 2 pays 2

strategy-proofOur techniques can be generalized to this setting

Results for m+1th price auction

BC = Bailey-Cavallo

WO = Worst-case 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

v(second )=0 v(second )=4 v(second )=1

pays 5

v (first )=3 v(first )=5 v(first )=2

payment of i = total efficiency when i is not present

- total efficiency when i is present

b2

b1

Another example

v(second )=2 v(second )=3 v(second )=1

pays 2

v (first )=2 v(first )=5 v(first )=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 0example (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 % Tally 1 Elvis Presley 13.73 625045 2 Yitzhak Rabin 13.17 599473 3 Adolf Hitler 11.36 516926 4 Billy Graham 10.35 471114 5 Albert Einstein 9.78 445218 6 Martin Luther King 8.40 382159 7 Pope John Paul II 8.18 372477 8 Gordon B Hinckley 5.62 256077 9 Mohandas Gandhi 3.61 164281 10 Ronald Reagan 1.78 81368 11 John Lennon 1.41 64295 12 American GI 1.35 61836 13 Henry Ford 1.22 55696 14 Mother Teresa 1.11 50770 15 Madonna 0.85 38696 16 Winston Churchill 0.83 37930 17 Linus Torvalds 0.53 24146 18 Nelson Mandela 0.47 21640 19 Princess Diana 0.36 16481 20 Pope Paul VI 0.34 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 % Tally 1 Jesus Christ 48.36 610238

2 Adolf Hitler 14.00 1767323 Ric Flair 8.33 1051164 Prophet Mohammed 4.22 533105 John Flansburgh 3.80 479836 Mohandas Gandhi 3.30 417627 Mustafa K Ataturk 2.07 261728 Billy Graham 1.75 221099 Raven 1.51 1917810 Pope John Paul II 1.15 1452911 Ronald Reagan 0.98 1244812 Sarah McLachlan 0.85 1077413 Dr William L Pierce 0.73 933714 Ryan Aurori 0.60 767015 Winston Churchill 0.58 734116 Albert Einstein 0.56 710317 Kurt Cobain 0.32 408818 Bob Weaver 0.29 378319 Bill Gates 0.28 362920 Serdar Gokhan 0.28 3627

Ric Flair

John Flansburgh(They Might Be Giants)

Atatürk

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 false-name-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, AAMAS-04, 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!

Computer Science and Mechanism Design: Two Case Studies namely: auctions with revenue redistribution...

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