Computational Complexity of Social Choice Procedures

1. Computational Complexityof Social Choice Procedures DIMACS Tutorial on Social Choice and Computer Science May 2004 Craig A. Tovey Georgia Tech

2. Part I: Who wins the election? Introduction Notation Rationality Axioms

3. Social Choice HOW should and does (normative) (descriptive) a group of individuals make a collective decision? Typical Voting Problem: select a decision from a finite set given conflicting ordinal preferences of set of agents. No T.U., no transferable good.

4. Case of 2 AlternativesMajority Rule n voters, 2 alternatives Theorem (Condorcet) If each voter’s judgment is independent and equally good (and not worse than random), then majority rule maximizes the probability of the better alternative being chosen.

5. Notation • [m] 1..m • P([m]) set of all permutations of [m] • ||x|| Norm of x, default Euclidean • A1 >i A2 Voter i prefers A1 to A2 • Social Choice Function (SCF): chooses a winner • Social Welfare Ordering (SWO): chooses an ordering

6. Social Choice What if there are ¸ 3 alternatives? Plurality can elect one that would lose to every other (Borda). Alternatives A1,…,Am Condorcet Principle (Condorcet Winner) IF an alternative is pairwise preferred to each other alternative by a majority 9 t2 [m] s.t. 8 j2 [m], j ¹ t: |i2 [n]: At >i Aj| > n/2 THEN the group should select Aj.

7. Condorcet’s Voting Paradox Condorcet winner may fail to exist Example: choosing a restaurant Craig prefers Indian to Japanese to Korean John prefers Korean to Indian to Japanese Mike prefers Japanese to Korean to Indian Each alternative loses to another by 2/3 vote

8. 1 1 2 3 2 3 1 3 1 2 2 3

9. Pairwise Relationships 8 directed graphs G=(V,E) 9 a population of O(|V|) voters with preferences on |V| alternatives whose pairwise majority preferences are represented by G. Proof: Cover edges of K|V| with O(|V|) ham paths Create 2 voters for each path, each direction

10. Now the tournament graph has no edges. Assign to each ordered pair (i,j) a voter with preference ordering {…j,i,…}. Don’t re-use! Flip i and j to create any desired edge. 1 2 3 4 5 5 4 3 2 1 1 3 5 2 4 4 2 5 3 1 4 1 5 3 2 2 3 5 1 4 1 2 3 4 5 5 4 3 2 1 1 3 5 2 4 4 2 5 3 1 4 1 5 3 2 2 3 5 1 4

11. Now the tournament graph has no edges. Assign to each ordered pair (i,j) a voter with preference ordering {…j,i,…}. Don’t re-use! Flip i and j to create any desired edge. 1 2 3 4 5 5 3 4 2 1 1 3 5 2 4 4 2 5 3 1 4 1 5 3 2 2 3 5 1 4 1 2 3 4 5 5 4 2 3 1 1 3 5 2 4 4 2 5 3 1 4 1 5 3 2 2 3 5 1 4 3 > 4 2> 3

12. Formulation of Social Choice Problem • Alternatives Aj, j2 [m] • Voters i 2 [n] • For each i, preferences Pi2P([m]) • Voting rule f: P[m]na [m] • Social Welfare Ordering (SWO): P[m]naP[m] • SWP: permit ties in SWO • Sometimes we permit ties in P_i

13. Axiomatic ViewpointRationality CriteriaProperties • Anonymous: symmetric on [n] • Neutral: symmetric on Aj, j2 [m] • monotone: if Aj is selected, and voter i elevates Aj in Pi (no other change), then Aj will still be selected. • strict monotone: ties permitted, but an elevation changes a tie to unique selection.

14. Axiomatic justification of Majority Rule • Theorem (May, 1952) Let m=2. Majority rule is the unique method that is anonymous, neutral, and strictly monotone. (Note for m =2 monotonicity ) strategyproof.)

15. So, what if there are ¸ 3 alternatives and there is no Condorcet winner? some (Cond. consistent) SCFs • Copeland: outdegree – indegree in tournament graph. • Simpson: min # votes mustered against any opponent • Dodgson: minimize the # of pairwise adjacent swaps in voter preferences to make alternative a Condorcet winner • Multistage elimination tree (Shepsle & Weingast)

16. So, what if there are ¸ 3 alternatives and there is no Condorcet winner? some (Condorcet consistent) SCOs • Copeland, Simpson, Dodgson • no scoring method (Fishburn 73) • MLE Kemeny (1959), Young (1985), Condorcet?!: Let d(P,P’)= # pairwise disagreements between P,P’. Choose P to

17. Arrow’s (im)possibility theorem Arrow(1951, 1963) Let m ¸ 3. No SWP simultaneously satisfies: • Unanimity (Pareto) • IIA: indep. of irrelevant alternatives • No dictator, no i2 [n] s.t. f(P[n])=Pi original proof uses sets of voters similar to what we’ve seen many combinations of properties are inconsistent Main point: No fully satisfactory aggregation of social preferences exists.

18. Maximum Likelihood Voting Theorem (Young & Levenglick 1978) Kemeny is the only SWP that simultaneously satisfies: • Neutral • Condorcet • Consistent over disjoint voter set union “The only drawback … is the difficulty in computing it ….” [Moulin 1988]

19. Part II: Who won the election? Procedures that are hard to execute

20. Maximum Likelihood Voting Theorem: [Bartholdi Tovey Trick 89a]: Kemeny score (or winner) is NP-hard. Proof: Use the tournament construction and reduce from feedback arc set. Note: 1st archival result of this type (together w/Dodgson score thm). Found earlier in Orlin letter 81; Wakabayashi thesis 86. Corollary: If P¹ NP no SWP simultaneously satisfies: • Neutral • Condorcet • Consistent over disjoint voter set union • Polynomial-time computable

21. Maximum Likelihood Voting • Theorem [Ravi Kumar 2001] Kemeny optimum is NP-hard for 4 voters • Theorem [Hemaspaandra-Spakowski-Vogel ~2001]: Kemeny Winner is complete for P||NP • Theorem [Kumar 2004] “Median rank aggregation” is a O(1)-factor approximation to Kemeny optimum. note: approximation may lose all rationality properties --- an example of differing tastes in social choice and computer science. additional note: there is some work on “approximate” adherence to axioms,e.g. Nisan&Segal 2002 for almost Pareto.

22. Dodgson Score Theorem: [Bartholdi Tovey Trick 89a] Dodgson score is NP-hard. Proof: reduction from X3C. Remark: polynomial for fixed m or fixed n. Sharper result by Hemaspaandra2-Rothe [JACM 97] Theorem: Dodgson Winner is complete for P||NP

23. Significance • Computational complexity of computation should be one of the criteria by which voting procedures are evaluated • In different recent work, Segal [2004] finds the minimally informative messages verifying that an alternative is in the Pareto choice set – communication complexity [e.g.Kushilevitz & Nisan 97]

24. Part III: Strategic Voting Manipulation by Individual Voters

25. Strategic voting • As early as Borda, theorists noted the “nuisance of dishonest voting” • Very common in plurality voting • Majority voting is strategyproof when m=2 • How about m¸ 3? Answer is closely related to Arrow’s Theorem [see also Blair and Muller 1983].

26. Strategyproof • A voting rule is strategyproof if 8 u 2P[m]n ,8 i 2 [n],8 P2P[m]: f(u) ¸i f(Pi,u-i). Equivalently, for all possible profiles of preferences, “everyone votes sincerely” is a Nash equilibrium. If everyone else is sincere, no voter benefits by being insincere.

27. Gibbard-Satterthwaite Theorem (1973, 1975) Let m¸ 3. No voting rule simultaneously satisfies: • Single-valued • No dictator • Strategyproof • 8 j2 [m] 9 voter population profile that elects j Proof: similar to proof of, or uses, Arrow’s theorem.

28. Gardenfors’s Theorem Let m ¸ 3. No SWP simultaneously satisfies: • Anonymous • Neutral • Condorcet winner consistent • Strategyproof

29. Works for voting procedures represented as polynomial time computable candidate scoring functions s.t. responsive (high score wins) “monotone-iia” Plurality Borda count Maximin (Simpson) Copeland (outdegree in graph of pairwise contests) Monotone increasing functions of above Greedy Manipulation Algorithm [BTT89b]1st inquiry into computational difficulty of manipulation

30. Definition Second order Copeland: sum of Copeland scores of alternatives you defeat Once used by NFL as tie-breaker. Used by FIDE and USCF in round-robin chess tournaments (the graph is the set of results)

31. A New “Good” Use of Complexity: resisting manipulation Theorem[BTT89b]: Both second order Copeland, and Copeland with second order tiebreak satisfy: • Neutral • No dictator • Condorcet winner • Anonymous • Unanimity (Pareto) • Polynomial-time computable • NP-complete to manipulate (by 1 voter) Note: 1st result of this type

32. Single-Valued Version Break ties by lexicographic order Theorem[BTT89b]: Both second order Copeland, and Copeland with second order tiebreak satisfy: • Single-valued • No dictator • Condorcet winner • Anonymous • Unanimity (Pareto) • Polynomial-time computable • NP-complete to manipulate (by 1 voter) Note: 1st result of this type

33. Proof Ideas Last-round-tournament-manipulation is NP-Complete w.r.t. 2nd order Copeland. 3,4-SAT (To84) Special candidate C0, clause candidates Cj Literal candidates Xi,Yi Y5 X5 C2 X6 Y6 Y7 X7

34. Proof Ideas All arcs in graph are fixed except those between each literal and its complement Clause candidate loses to all literals except the three it contains To stop each clause from gaining 3 more 2nd order Copeland points, must pick one losing (= True) literal for each clause

35. Proof Ideas Pad so each clause candidate is • tied with C_0 in 1st order Copeland • 3 behind C_0 in 2nd order Copeland This proves last round tourn manip hard. Then use arbitrary graph construction to make all other contests decided by 2 votes, so one voter can’t affect other edges.

36. Another resistant procedure • Theorem (BO:SCW 91) Single Transferable Vote is NP-hard to manipulate (by a single voter) for a single seat. • Corollary: Non-monotonicity is NP-hard to detect in STV. • Used in elections for Parliament in Ireland, Tasmania; Senate in Australia, South Africa, N. Ireland; local authorities in Ireland, Canada, Australia; school board in NYC.

37. Proof ideas Candidates with fewest votes are h1, h2, … hn ~1, ~2,… ~n Most supporters h_1 a few supporters next fewest fewest …. h1 ~1 … h1 s4 … h1 s9 … h1 s7 … where (s4,s7,s9) is from a X3Cover instance

38. Proof ideas • Placing ~1 first forces h1 to be eliminated first (and vice-versa) • Choose ~i or hi for each i2 [n] • Must distribute new votes for s candidates evenly so no s_j beats your favored candidate Simplified but has main ideas

39. Conitzer and Sandholm’s Universal Preround Complexifier • Give up neutrality • Add a pre-round of b m/2 c pairwise contests. If m is odd, one candidate gets a “bye”. The SCF is performed on the d m/2 e survivors. • Modified procedure is NP-hard, #P-hard, and PSPACE-hard respectively to manipulate by 1 voter, depending on whether pairing is ex ante, ex post, or interleaved with the voting.

40. Works for Plurality, Borda, Simpson, STV. • Tweak or Tstrong?

41. Implications • Gibbard-Satterthwaite, Gardenfors, other such theorems open door to strategic voting. Makes voting a richer phenomenon. • Both practically and theoretically, complexity can partly close door. • Plurality voting is still widely used. Voting theory penetrates slowly into politics. • One might consider using a hard-to-compute procedure

42. Part IV: Complexity of Other Kinds of Manipulation Agenda Manipulation Manipulating Voters Coalitions

43. Agenda Control • Add small # of “spoiler” candidates (alternatives) • Disqualify small # of candidates • Partition candidates and use 2-stage sequential election • Partition candidates and use run-off election • Dates back to Roman times, at least!

44. Complexity of Agenda Control • Theorem [BTT 92]: Preceding types of agenda control are NP-hard for plurality voting • Theorem [IBID] Preceding types of agenda control are polynomially solvable for Condorcet voting (note: impossible for adding candidates). 1st inquiry into computational difficulty of election manipulation

45. Election Control: Manipulating Voters • Add small # of voters Chicago voting* • Remove small # of voters Detroit voting** • Partition voters into two groups. Each group votes to nominate a candidate; then the voters as a whole decide between the candidates (if different).

46. Complexity of Election Control by Manipulating Voters • Theorem [BTT 92]: Preceding types of election control are NP-hard for Condorcet voting • Theorem [IBID] Preceding types of agenda control are polynomially solvable for plurality voting.

47. Main point: different voting procedures have different levels of computational resistance or vulnerability to various types of manipulation. Note: agenda manipulation by adding/deleting candidates relates to IIA in Arrow’s theorem, but I think that computational complexity is not a circumvention because that rationality criterion is not principally about agenda manipulation.

48. Coalitions • Coalition members may coordinate their votes • A winning coalition can force the outcome of the SCF. • Core: no coalition of voters has a safe and profitable deviation. Core is set of undominated candidates (undominated: no winning coalition unanimously prefers another candidate). Example: if SCF is Condorcet, core is Condorcet winner (if exists) or empty. • Thm [BNT 91] “Is an alternative dominated?” is NP-complete in the Euclidean model.

49. Coalitions • Core Stable: SCF has nonempty core for all preference profiles. • Theorem [Nakamura 1979]: SCF is core stable iff Nakamura number > m (minimal # winning coalitions with empty intersection). • Theorem [BNT 91] Nakamura number · m is strongly NP-complete in weighted voting games. • Theorem[Conitzer & Sandholm 2003] Core non-empty is NP-complete for non-TU and TU cooperative games.

50. Coalitions Setup: Borda voting, but voter i has weight wi on her vote. Question: Can a given coalition C strategically coordinate its votes to get a given candidate j to win, if all other voters are sincere? (an atypical question from voting or game theory viewpoints) Theorem [CS 2002] NP-complete for 3 candidates. Proof: put j first, then partition wi: i2 C between other 2 for 2nd place. Similar results for STV, Copeland,Simpson.[IBID]