Computational Complexity

of Social Choice Procedures

DIMACS Tutorial on Social Choice and Computer Science

May 2004Craig A. ToveyGeorgia Tech

Part I: Who wins the election?

IntroductionNotationRationality Axioms

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.

Case of 2 AlternativesMajority Rule

n voters, 2 alternativesTheorem (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.

Notation [m] 1..m ([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

Social ChoiceWhat 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 t:

|i2 [n]: At >i Aj| > n/2

THEN the group should select Aj.

Condorcet’s Voting Paradox

Condorcet winner may fail to existExample: choosing a restaurantCraig prefers Indian to Japanese to KoreanJohn prefers Korean to Indian to JapaneseMike prefers Japanese to Korean to Indian

Each alternative loses to another by 2/3 vote

1

2

3

2

3

1

3

1

2

1

23

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

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.

12345

54321

13524

42531

41532

23514

12345

54321

13524

42531

41532

23514

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.

12345

53421

13524

42531

41532

23514

12345

54231

13524

42531

41532

23514

3 > 4 2 > 3

Formulation of Social Choice Problem

Alternatives Aj, j2 [m] Voters i 2 [n] For each i, preferences Pi 2 ([m]) Voting rule f: [m]n [m] Social Welfare Ordering (SWO):

[m]n [m] SWP: permit ties in SWO Sometimes we permit ties in P_i

Axiomatic ViewpointRationality Criteria

Properties

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.

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.)

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)

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

Arrow’s (im)possibility theorem

Arrow(1951, 1963) Let m ¸ 3. No SWP simultaneously satisfies:

1. Unanimity (Pareto)2. IIA: indep. of irrelevant alternatives3. No dictator, no i2 [n] s.t. f(P[n])=Pioriginal proof uses sets of voters similar to what we’ve

seenmany combinations of properties are inconsistent

Main point: No fully satisfactory aggregation of social preferences exists.

Maximum Likelihood Voting

Theorem (Young & Levenglick 1978)Kemeny is the only SWP that

simultaneously satisfies:1. Neutral2. Condorcet3. Consistent over disjoint voter

set union“The only drawback … is the difficulty in

computing it ….” [Moulin 1988]

Part II: Who won the election?

Procedures that are hard to execute

Maximum Likelihood VotingTheorem: [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 P NP no SWP simultaneously satisfies:

1. Neutral2. Condorcet3. Consistent over disjoint voter set union4. Polynomial-time computable

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.

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

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]

Part III: Strategic Voting

Manipulation by Individual Voters

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].

Strategyproof

A voting rule is strategyproof if 8 u 2 [m]n ,8 i 2 [n],8 P2 [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.

Gibbard-Satterthwaite Theorem

(1973, 1975) Let m¸ 3. No voting rule simultaneously satisfies:

1. Single-valued2. No dictator3. Strategyproof4. 8 j2 [m] 9 voter population profile that

elects j Proof: similar to proof of, or uses, Arrow’s theorem.

Gardenfors’s Theorem

Let m ¸ 3. No SWP simultaneously satisfies:

1. Anonymous2. Neutral 3. Condorcet winner consistent4. Strategyproof

Greedy Manipulation Algorithm [BTT89b]

1st inquiry into computational difficulty of manipulation

Works for voting procedures represented as polynomial time computable candidate scoring functions s.t.

1. responsive (high score wins)

2. “monotone-iia”

i. Pluralityii. Borda countiii. Maximin (Simpson)iv. Copeland

(outdegree in graph of pairwise contests)

v. Monotone increasing functions of above

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)

A New “Good” Use of Complexity: resisting manipulation

Theorem[BTT89b]: Both second order Copeland, and Copeland with second order tiebreak satisfy:

1. Neutral2. No dictator3. Condorcet winner4. Anonymous5. Unanimity (Pareto)6. Polynomial-time computable7. NP-complete to manipulate (by 1 voter)Note: 1st result of this type

Single-Valued VersionBreak ties by lexicographic order

Theorem[BTT89b]: Both second order Copeland, and Copeland with second order tiebreak satisfy:

1. Single-valued2. No dictator3. Condorcet winner4. Anonymous5. Unanimity (Pareto)6. Polynomial-time computable7. NP-complete to manipulate (by 1 voter)Note: 1st result of this type

Proof IdeasLast-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

C2

X5

X6

Y5

Y6

X7Y7

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

Proof Ideas

Pad so each clause candidate is 1. tied with C_0 in 1st order Copeland2. 3 behind C_0 in 2nd order CopelandThis proves last round tourn manip hard.Then use arbitrary graph construction to

makeall other contests decided by 2 votes, so

one voter can’t affect other edges.

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.

Proof ideas

Candidates with fewest votes areh1, h2, … hn

~1, ~2,… ~n

Most supporters h_1 a few supporters

fewestfewestnextnext

fewestfewest……..

h1

~1

…

h1

s4

…

h1

s7

…

h1

s9

…where (s4,s7,s9) is from a X3Cover instance

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

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.

Works for Plurality, Borda, Simpson, STV.

Tweak or Tstrong?

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

Part IV: Complexity of Other Kinds of Manipulation

Agenda ManipulationManipulating VotersCoalitions

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!

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

Election Control: Manipulating Voters

Add small # of voters Chicago voting* Chicago voting*

Remove small # of voters Detroit voting**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).

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.

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.

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.

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.

CoalitionsSetup: 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]

Modern Manipulation The Ethicist (NY TIMES 2004) Bush supporter donates money to

Nader campaign.

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]

Related Work Single Transferable Vote Resists

Strategic Voting, SCW 1991. Bartholdi, Orlin

Universal Voting Protocol Tweaks to Make Manipulation Hard. Conitzer, Sandholm.

PART V

SPATIAL (EUCLIDEAN) MODEL

Definition of Spatial Model

Voter i has ideal (bliss) point xi 2 <k

Each alternative is represented by a point in <k

A1 ¸i A2 iff ||xi-A1|| · || xi – A2|| Can use norms other than Euclidean

e.g. ellipsoidal indifference curves

1D spatial model

Informally used by U.S. press and many others

Shockingly effective predictively in current U.S. politics. See Keith Poole’s 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.

Spatial Model Largely descriptive role rather than

normative The workhorse of empirical studies in

political science k=1,2 are the most popular # of

dimensions In U.S. k=2 gives high accuracy (~90%) ,

k=1 also very accurate since 1980s, and 1850s to early 20th century.

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

2D is qualitatively richer than 1D

x1

x2x3

A1

A2

A3

A1 >A2 > A3 > A1

Condorcet’s voting paradox in Euclidean model

x1

x2x3

A1

A2

A3

Hyperplane normal to and bisecting line segment A1A2

Even if all points in <2 are permitted alternatives, no Condorcet winner exists

x1

x2x3

A1

A2

Chaos theorems McKelvey [1979], Schofield [83]. Majority vote can take the agenda

anywhere.(not precisely the meaning of chaos in

system dynamics)

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?

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

completeTheorem: [BNT 91]Given x1…xn and x0 in

<k, determining whether x0 is dominated is NP-complete.

Proof: use Johnson & Preparata 1978.Algorithm [BNT 91]: In O(kn) given x_1…

x_n can find x_0 which is undominated if any point is.

Corollary: Majority-rule stability is co-NP-complete.

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

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

Part VI: Discussion

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

Possible Benefits Idea to use for real problem faced in your

field. New area to generate papers in your field.

(Let’s 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.