Mechanism Design without Money

1. Mechanism Design without Money Lecture 13

2. Condorcet winner • Voting rules are trying to generalize the concept of “majority” in different ways • Condorcet proposed another way, which relies on pairwise elections

3. Condorcet winner Does it define a new voting rule? Recall that a beats b in pairwise elections, if more voters prefer a over b a is a Condorcet winnerif a beats any other candidate in a pairwise election. 3

4. The Condorcet Paradox c a b c there is no Condorcet winner! a b b a c

5. The Condorcet property • As we saw, there might not be a Condorcet winner in some case • However, if there is one, then it is unique • We say that a voting rule has the Condorcet property, if the Condorcet winner is always selected (when there is one)

6. Voting rules: Borda • Given m candidates • ith ranked candidate score m-i • Candidate with greatest sum of scores wins • Example • 42 votes: A>B>C>D • 26 votes: B>C>D>A • 15 votes: C>D>B>A • 17 votes: D>C>B>A • B wins Jean Charles de Borda, 1733-1799

7. a a b b b c c c d d d Manipulation • If the Borda rule is used, then a will win • a has 8 points, while b only has 7 a

8. d b a a b b c c c a d d Manipulation Manipulation • But if voter 3 lies about his preferences… • Now a only has 6 points, and b wins! • What would happen if we used Plurality?

9. Manipulation (2) • For each voting rule we saw, there is a voting profile, in which one voter can manipulate the outcome by lying. • Does it matter? • Are there rules which are immune to such strategic behavior? It depends… NO (at least not good rules)

10. Manipulation (3) Definition: a voting rule f is strategy-proof (SP), if no (single) voter can ever benefit from lying about his preferences. Formally: Claim: If |A|=2 (i.e. there are two candidates), then Plurality is Strategy-Proof • In this case all the rules we saw are also SP

11. The Gibbard-Satterthwaite Theorem Definition: A voting rule f is dictatorial if there is an individual (the dictator) whose most preferred candidate is always chosen by f. formally: Definition: A voting rule f is onto if it is possible for any of the candidates to win (given the right preference profile):

12. The Gibbard-Satterthwaite Theorem contradiction! (SP) The Gibbard-Satterthwaite theorem: If there are at least three candidates, any voting rule that is strategy-proof and onto is dictatorial. • This is a powerful negative result (no dictator) (onto) ( if |A|≥3 )

13. Manipulating Borda • Suppose you want to add k manipulators for Borda who would promote p • Clearly, p should be ranked first in all of them • If we could rank no one else, we would be happy • But we have to give points to other candidates • So the goal is to give points to bad candidates who won’t win

14. How hard can that be? • Manipulating Borda with just two extra voters is NPC. • Reduction from a cousin of subset sum • But we know that subset sum (and family) are not really hard..

15. Greedy algorithm for k manipulators • We want to promote p • Each manipulator puts p on top • Problem – we need to give points to other candidates.. • Set the manipulators one by one • At each manipulator be greedy • Thm: If exists a manipulation with k-1 manipulators this will succeed with k manipulators

16. Probability that a manipulation exists • What is the probability that a manipulation by a single voter exists? • Pluarality: requires a tie on top, probability is about 1/n1/2 • Other systems – inverse polynomial • Done by proving that condorcet’s cycles exist with decent probability • Nathan Keller (math) works on these questions

17. Computational hardness as a barrier to manipulation • A (successful) manipulation is a way of misreporting one’s preferences that leads to a better result for oneself • Gibbard-Satterthwaite only tells us that for some instances, successful manipulations exist • It does not say that these manipulations are always easy to find • Do voting rules exist for which manipulations are computationally hard to find?

18. A formal computational problem • The simplest version of the manipulation problem: • CONSTRUCTIVE-MANIPULATION: • We are given a voting rule r, the (unweighted) votes of the other voters, and an alternative p. • We are asked if we can cast our (single) vote to make p win. • E.g., for the Borda rule: • Voter 1 votes A > B > C • Voter 2 votes B > A > C • Voter 3 votes C > A > B • Borda scores are now: A: 4, B: 3, C: 2 • Can we make B win? • Answer: YES. Vote B > C > A (Borda scores: A: 4, B: 5, C: 3)

19. “Tweaking” voting rules • It would be nice to be able to tweak rules: • Change the rule slightly so that • Hardness of manipulation is increased (significantly) • Many of the original rule’s properties still hold • It would also be nice to have a single, universal tweak for all (or many) rules • One such tweak: add a preround[Conitzer & Sandholm IJCAI 03]

20. Adding a preround [Conitzer & Sandholm IJCAI-03] • A preround proceeds as follows: • Pair the alternatives • Each alternative faces its opponent in a pairwise election • The winners proceed to the original rule • Makes many rules hard to manipulate

21. Preround example (with Borda) • STEP 1: • A. Collect votes and • B. Match alternatives • (no order required) • Match A with B • Match C with F • Match D with E Voter 1: A>B>C>D>E>F Voter 2: D>E>F>A>B>C Voter 3: F>D>B>E>C>A • A vs B: A ranked higher by 1,2 • C vs F: F ranked higher by 2,3 • D vs E: D ranked higher by all • STEP 2: • Determine winners of preround • Voter 1: A>D>F • Voter 2: D>F>A • Voter 3: F>D>A • STEP 3: • Infer votes on remaining alternatives • STEP 4: • Execute original rule • (Borda) • A gets 2 points • F gets 3 points • D gets 4 points and wins!

22. Matching first, or vote collection first? • Match, then collect “A vs C, B vs D.” “A vs C, B vs D.” “D > C > B > A” • Collect, then match (randomly) “A vs C, B vs D.” “A > C > D > B”

23. Could also interleave… • Elicitor alternates between: • (Randomly) announcing part of the matching • Eliciting part of each voter’s vote “A vs F” “B vs E” “C > D” “A > E” … …

24. How hard is manipulation when a preround is added? • Manipulation hardness differs depending on the order/interleaving of preround matching and vote collection: • Theorem.NP-hard if preround matching is done first • Theorem.#P-hard if vote collection is done first • Theorem.PSPACE-hard if the two are interleaved (for a complicated interleaving protocol) • In each case, the tweak introduces the hardness for any rule satisfying certain sufficient conditions • All of Plurality, Borda, Maximin, satisfy the conditions in all cases, so they are hard to manipulate with the preround

25. Bribery • We are allowed to change the votes of a set of voters. • Stronger than manipulation – we are deleting bad voters and replacing them with good ones

26. Reducing Energy Consumption

27. The Voting Process a3 > a4 > a1 > a2 a4 > a2 > a1 > a3 a2 > a3 > a1 > a4 a1 > a2 > a3 > a4

28. The Voting Process a3 > a4 > a1 > a2 a4 > a2 > a1 > a3 a2 > a3 > a1 > a4 a1 > a2 a1 > a2 > a3 > a4

29. The Voting Process a3 > a4 > a1 > a2 a3 > a1 > a4 > a2 ? a4 > a2 > a1 > a3 a2 > a3 > a1 > a4 a1 > a2 a1 > a2 > a3 > a4

30. Persuasion Problems • Given: • A set of alternatives • A set of voters with their preferences • A preferences list of the sender • Is there a “good” set of suggestions? • K-Persuasion: send at most k suggestions

31. Add Safety Requirement • What if not all the voters accept the suggestions? • Safe-Persuasion • Is there a “good” and safe set of suggestions? • r-Safe-persuasion: send at most r suggestions

32. Persuasion ≠ Manipulation or Bribery • In coalitional manipulation and bribery • The manipulators always obey their suggestions • There is no requirement that they will benefit from it • How the manipulators attain full knowledge? • In persuasion • Voters can accept or decline the sender’s suggestions • Send suggestion only to voters that will benefit from it, and we add safety requirement • The sender is the election organizer

33. Some Complexity Results

34. Some Intuition • In Plurality • The score of the current winner cannot be decreased • K-Approval and Bucklin • Persuasion is in P from UCM • K-Persuasion is hard from X3C • In Borda • Augmenting the hardness proof of [Betzler et al., 2011]

35. Hardness of is safe for k approval

36. Hardness for k approval

37. Test • Four out of five • Mix of easy and moderate questions – no one is supposed to get a bad grade, not trivial to get a 100 • You need to know the slides, and need to be able to solve simple exercises (e.g. find a Nash equilibrium of a game) • If there are several sections, scoring is not identical • Test from last year’s course is online • Not exactly the same material, but same style of questions