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Vincent Conitzer Duke University

Vincent Conitzer Duke University. Computer Science and Mechanism Design: Two Case Studies namely: auctions with revenue redistribution & voting in highly anonymous environments. computational techniques to help economists design better mechanisms (part 1 of this talk). computer science.

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Vincent Conitzer Duke University

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  1. Vincent Conitzer Duke University Computer Science and Mechanism Design: Two Case Studiesnamely:auctions with revenue redistribution&voting in highly anonymous environments

  2. 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)

  3. Auctions with revenue distribution[Guo & C., ACM-EC 07/accepted with minor revisions to Games and Economic Behavior]

  4. First-price auction receives 4 ) = 4 v( ) = 1 ) = 3 v( v( ) = 5 v( ) = 7 v( ) = 2 v( pays 4

  5. Second-price (Vickrey) auction receives 3 ) = 4 v( ) = 2 ) = 3 v( v( ) = 4 v( ) = 3 v( ) = 2 v( pays 3

  6. Vickrey auction without a seller ) = 4 v( ) = 2 ) = 3 v( v( pays 3 (money wasted!)‏

  7. Can we redistribute the payment? Idea: give everyone 1/n of the payment ) = 4 v( ) = 2 ) = 3 v( v( receives 1 receives 1 pays 3 receives 1 not strategy-proof Bidding higher can increase your redistribution payment

  8. Strategy-proof redistribution[Bailey 97, Porter et al. 04, Cavallo 06] Idea: give everyone 1/n of second-highest other bid ) = 4 v( ) = 2 ) = 3 v( v( receives 2/3 receives 1 pays 3 receives 2/3 2/3 wasted (22%)‏ strategy-proof Your redistribution does not depend on your bid; incentives are the same as in Vickrey

  9. R1 = V3/n R2 = V3/n R3 = V2/n R4 = V2/n ... Rn-1= V2/n Rn = V2/n Bailey-Cavallo mechanism… • Bids: V1≥V2≥V3≥... ≥Vn≥0 • First run Vickrey auction • Payment is V2 • First two bidders receiveV3/n • Remaining bidders receive V2/n • Total redistributed:2V3/n+(n-2)V2/n Can we do better?

  10. 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]

  11. 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 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)‏

  12. 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)‏

  13. 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?

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. m-unit auction with unit demand:VCG (m+1th price) mechanism ) = 4 v( ) = 2 ) = 3 v( v( pays 2 pays 2 strategy-proof Our techniques can be generalized to this setting

  20. Results for m+1th price auction BC = Bailey-Cavallo WO = Worst-case Optimal

  21. 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

  22. 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

  23. 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

  24. Multi-unit auction with nonincreasing marginal values b2 )=4 v(second )=0 )=1 v(second v(second )=3 )=2 v(first )=5 v(first v (first b1 pays 5 payment of i = total efficiency when i is not present - total efficiency when i is present

  25. Another example )=3 v(second )=2 )=1 v(second v(second )=2 )=4 v(first )=5 v(first v (first pays 3 pays 2

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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)‏

  31. Voting in highly anonymous environments

  32. Time Magazine “Person of the Century” poll – “results” (January 19, 2000) 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 […] #   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   4711145   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

  33. 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 176732 3 Ric Flair 8.33 105116 4 Prophet Mohammed 4.22 53310 5 John Flansburgh 3.80 47983 6 Mohandas Gandhi 3.30 41762 7 Mustafa K Ataturk 2.07 26172 8 Billy Graham 1.75 22109 9 Raven 1.51 19178 10 Pope John Paul II 1.15 14529 11 Ronald Reagan 0.98 12448 12 Sarah McLachlan 0.85 10774 13 Dr William L Pierce 0.73 9337 14 Ryan Aurori 0.60 7670 15 Winston Churchill 0.58 734116 Albert Einstein 0.56 7103 17 Kurt Cobain 0.32 4088 18 Bob Weaver 0.29 3783 19 Bill Gates 0.28 3629 20 Serdar Gokhan 0.28 3627

  34. Ric Flair

  35. John Flansburgh(They Might Be Giants)

  36. Atatürk

  37. Raven

  38. Howard Stern

  39. Cartman

  40. Optimus Prime

  41. 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…

  42. 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

  43. 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…

  44. 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

  45. 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?

  46. 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

  47. 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

  48. 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!)

  49. 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!

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