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
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