Computational
Complexity
of
Social Choice Procedures
DIMACS Tutorial on Social Choice and
Computer Science
May 2004
Craig A. Tovey
Georgia Tech
Part I: Who wins the
election?
Introduction
Notation
Rationality Axioms
Social Choice
HOW
and
should
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 Alternatives
Majority 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.
Notation




[m]
P([m])
||x||
A1 >i A2
1..m
set of all permutations
of [m]
Norm of x, default Euclidean
Voter i prefers A1 to A2
 Social Choice Function (SCF):
chooses a winner
 Social Welfare Ordering (SWO): chooses an ordering
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  t:
|i2 [n]: At >i Aj| > n/2
THEN the group should select Aj.
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
1
2
3
2
3
1
3
1
2
1
3
2
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.
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
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
3>4
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
2>3
1
3
5
2
4
4
2
5
3
1
4
1
5
3
2
2
3
5
1
4
Formulation of Social Choice
Problem

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

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

Alternatives Aj, j2 [m]
Voters i 2 [n]
For each i, preferences Pi 2 P([m])
Voting rule f: P[m]n a [m]
Social Welfare Ordering (SWO):
P[m]n a P[m]
SWP: permit ties in SWO
Sometimes we permit ties in P_i
Axiomatic Viewpoint
Rationality 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 alternatives
3. 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.
Maximum Likelihood
Voting
Theorem (Young & Levenglick 1978)
Kemeny is the only SWP that
simultaneously satisfies:
1. Neutral
2. Condorcet
3. 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 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 P NP no SWP simultaneously
satisfies:
1. Neutral
2. Condorcet
3. Consistent over disjoint voter set union
4. Polynomial-time computable
Maximum Likelihood Voting
Theorem [Ravi Kumar 2001] Kemeny
optimum is NP-hard for 4 voters
 Theorem [Hemaspaandra-SpakowskiVogel ~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 P[m]n ,8 i 2 [n],8 P2 P[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-valued
2. No dictator
3. Strategyproof
4.
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. Anonymous
2. Neutral
3. Condorcet winner consistent
4. 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.
ii.
iii.
iv.
v.
Plurality
Borda count
Maximin (Simpson)
Copeland
(outdegree in graph
of pairwise contests)
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. Neutral
2. No dictator
3. Condorcet winner
4. Anonymous
5. Unanimity (Pareto)
6. Polynomial-time computable
7. NP-complete to manipulate (by 1 voter)
Note: 1st result of this type
Single-Valued Version
Break ties by lexicographic order
Theorem[BTT89b]: Both second order
Copeland, and Copeland with second order
tiebreak satisfy:
1. Single-valued
2. No dictator
3. Condorcet winner
4. Anonymous
5. Unanimity (Pareto)
6. Polynomial-time computable
7. NP-complete to manipulate (by 1 voter)
Note: 1st result of this type
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
C2
X5
X6
X7
Y5
Y6
Y7
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 Copeland
2. 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.
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 are
h1,
h2, …
~1,
~2,…
fewest
next
fewest ….
hn
~n
Most supporters h_1
h1
~1
…
a few supporters
h1
h1
h1
s4
…
s7
…
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
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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-tocompute procedure
Part IV: Complexity of
Other Kinds of Manipulation
Agenda
Manipulation
Manipulating Voters
Coalitions
Agenda Control
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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*
 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).

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
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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
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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.
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]
Modern Manipulation
The Ethicist (NY TIMES 2004)
Bush supporter donates money to
Nader campaign.

Related Work

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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

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
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 singlepeaked prefs, see MOR 2003.
Spatial Model

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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
A1
A2
x3
x2
A3
A1 >A2 > A3 > A1
Condorcet’s voting paradox in Euclidean
model
x1
A1
A2
x3
x2
A3
Hyperplane normal to and bisecting line segment A1A2
permitted alternatives, no
Condorcet winner exists
A1 x1
A2
x3
x2
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-NPcomplete
Theorem: [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-NPcomplete.
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 NPhard
Computing any other solution concept
that coincides with Condorcet winner
when it exists, is NP-hard
Related Work

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
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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
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.