Complexity Applied to

Social Choice

Manipulation

&

Spatial Equilibria

John Bartholdi, Michael Trick, Lakshmi Narasimhan, Craig Tovey

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

EnlightenmentCondorcet (1785), de Borda (1781)

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.

Theorem (May, 1952) Majority rule is the unique method that is anonymous, neutral, and strictly monotone. (Note monotonicity ) strategyproof.)

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

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

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

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 seenmany combinations of properties are inconsistent

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

Condorcet-Young-Kemeny Maximum Likelihood Voting

Theorem (Kemeny 59, Young Levenglick 78, Bartholdi Tovey Trick 89; Wakabayashi 86).

No SWP simultaneously satisfies:

1. Neutral

2. Condorcet

3. Consistent over disjoint voter set union

4. Polynomial-time computable

Strategic voting

As early as Borda theorists noted the “nuisance of dishonest voting”

Very common in plurality votingMajority voting is strategyproof when m=2How about m¸ 3? Answer is closely related

to Arrow’s Theorem [see also Blair and Muller 1983].

Gibbard-Satterthwaite Theorem

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

1. Single-valued2. No dictator3. Strategyproof (non-manipulable)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

4. Strategyproof

Greedy Manipulation Algorithm

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

1. responsive (high score wins)

2. “monotone-iia”

i. Plurality

ii. Borda count

iii. 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 “Good” Use of Complexity

Theorem: 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

Break ties by lexicographic order

Theorem: 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

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

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 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.

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.

Related Work• Voting Schemes for which It Can Be Difficult to Tell Who

Won the Election, Social Choice and Welfare 1989. Bartholdi, Tovey, Trick

• Aggregation of binary relations: algorithmic and polyhedral investigations, 1986, Univerisity of Augsburg Ph.D. dissertation. Y. Wakabayashi

• The densest hemisphere problem, Theor. Comp. Sci, 1978. Johnson, Preparata

• The Computational Difficulty of Manipulating an Election, SCW 1989. Bartholdi, Tovey, Trick

• Limiting median lines do not suffice to determine the yolk, SCW 1992. Stone, Tovey

Related Work• Single Transferable Vote Resists Strategic Voting, SCW

1991. Bartholdi, Orlin• 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