Computational Complexity of Social Choice Procedures

1
Computational Complexity of 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 voters 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
Condorcets 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 KV 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,. Dont 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,. Dont 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 Pi 2 P(m)
• Voting rule f Pmn a m
• Social Welfare Ordering (SWO)
• Pmn a Pm
• 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 m2. 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
Arrows (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(Pn)Pi
• original proof uses sets of voters similar to
  what weve 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 PNP
• 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.
  NisanSegal 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 PNP

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 m2
• How about m 3? Answer is closely related to
  Arrows Theorem see also Blair and Muller 1983.

26
Strategyproof

• A voting rule is strategyproof if
• 8 u 2 Pmn ,8 i 2 n,8 P2 Pm
• 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, Arrows
  theorem.

28
Gardenforss Theorem

• Let m 3. No SWP simultaneously satisfies
• Anonymous
• Neutral
• Condorcet winner consistent
• Strategyproof

29
Greedy Manipulation Algorithm BTT89b1st
inquiry into computational difficulty of
manipulation

• 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

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

• TheoremBTT89b 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
• TheoremBTT89b 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 cant affect other edges.

36
Another resistant procedure

• Theorem (BOSCW 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 Sandholms 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 Arrows 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.
• TheoremConitzer 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

51
Modern Manipulation

• The Ethicist (NY TIMES 2004)
• Bush supporter donates money to Nader campaign.

52
Related Work

• Voting Schemes for which It Can Be Difficult to
  Tell Who Won the Election, Social Choice and
  Welfare 1989. Bartholdi, Tovey, Trick BTT89a
• Aggregation of binary relations algorithmic and
  polyhedral investigations, 1986, Univerisity of
  Augsburg Ph.D. dissertation. Y. Wakabayashi
• The Computational Difficulty of Manipulating an
  Election, SCW 1989. Bartholdi, Tovey, Trick
  BTT89b

53
Related Work

• Single Transferable Vote Resists Strategic
  Voting, SCW 1991. Bartholdi, Orlin
• Universal Voting Protocol Tweaks to Make
  Manipulation Hard. Conitzer, Sandholm.

54
PART V

• SPATIAL (EUCLIDEAN) MODEL

55
Definition of Spatial Model

• Voter i has ideal (bliss) point xi 2
• Each alternative is represented by a point in
• A1 i A2 iff xi-A1 xi A2
• Can use norms other than Euclidean e.g.
  ellipsoidal indifference curves

56
1D spatial model

• Informally used by U.S. press and many others
• Shockingly effective predictively in current U.S.
  politics. See Keith Pooles website, e.g.
  Supreme Court.
• Similar to single-peaked preferences (a little
  more restrictive). For polyhedral explanation of
  nice behavior of single-peaked prefs, see MOR
  2003.

57
Spatial Model

• Largely descriptive role rather than normative
• The workhorse of empirical studies in political
  science
• k1,2 are the most popular of dimensions
• In U.S. k2 gives high accuracy (90) , k1 also
  very accurate since 1980s, and 1850s to early
  20th century.

58
(No Transcript)
59
What do the dimensions mean?

• Different schools of thought
• Use expert domain knowledge or contextual
  information to define dimensions and/or place
  alternatives
• Fit data (e.g. roll call) to achieve best fit
• Maximize data fit in 1st dimension, then 2nd
• Impute meaning to fitted model

60
2D is qualitatively richer than 1D
x1
A1
A2
x2
x3
A3
A1 A2 A3 A1
61
Condorcets voting paradox in Euclidean model
x1
A1
A2
x2
x3
A3
Hyperplane normal to and bisecting line segment
A1A2
62
Even if all points in alternatives, no Condorcet winner exists
x1
A1
A2
x2
x3
63
Chaos theorems

• McKelvey 1979, Schofield 83.
• Majority vote can take the agenda anywhere.
• (not precisely the meaning of chaos in system
  dynamics)

64
Major Question Conditions for Existence of
Stable Point (Undominated, Condorcet Winner)

• Plott (67) For case all xi distinct
• Slutsky(79) General case, not finite
• Davis, DeGroot, Hinich (72) Every hyperplane
  through x is median, i.e. each closed halfspace
  contains at least half the voter ideal points.
• McKelvey, Schofield (87) More general, finite,
  but exponential.
• Are there better conditions?

65
Recognizing a Stable (Undominated) Point is
co-NP-complete

• Theorem BNT 91Given x1xn and x0 in determining whether x0 is dominated is
  NP-complete.
• Proof use Johnson Preparata 1978.
• Algorithm BNT 91 In O(kn) given x_1x_n can
  find x_0 which is undominated if any point is.
• Corollary Majority-rule stability is
  co-NP-complete.

66
Implications

• Puts to rest efforts to find simpler necessary
  and sufficient conditions. In this case
  complexity theory provides insight.
• Computing the radius of the yolk is NP-hard
• Computing any other solution concept that
  coincides with Condorcet winner when it exists,
  is NP-hard

67
Related Work

• The densest hemisphere problem, Theor. Comp. Sci,
  1978. Johnson, Preparata
• Limiting median lines do not suffice to determine
  the yolk, SCW 1992. Stone, Tovey
• A polynomial time algorithm for computing the
  yolk in fixed dimension, Math Prog 1992. Tovey
• Dynamical Convergence in the Spatial Model, in
  Social Choice, Welfare and Ethics, eds. Barnett,
  Moulin, Salles, Schofield, Cambridge 1995. Tovey
• Some foundations for empirical study in the
  Euclidean spatial model of social choice, in
  Political Economy, eds. Barnett, Hinich,
  Schofield, Cambridge 1993. Tovey

68
Part VI Discussion

• What can we learn from each other?
• Benefits of multidisciplinary meetings.

69
Possible Benefits

• Idea to use for real problem faced in your field.
  
• New area to generate papers in your field.
  (Lets be honest).
• Opportunity to help solve a problem in another
  field.
• Acquire idea or info from another field which
  alters a basic question in your field.