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Complexity Applied to Social Choice Manipulation & Spatial Equilibria John Bartholdi, Michael Trick, Lakshmi Narasimhan, Craig Tovey

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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.

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Case of 2 Alternatives Majority Rule Enlightenment Condorcet (1785), de Borda (1781) 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. Theorem (May, 1952) Majority rule is the unique method that is anonymous, neutral, and strictly monotone. (Note monotonicity ) strategyproof.)

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Notation [m] 1..m ([m]) set of all permutations of [m] ||x|| Norm of x, default Euclidean A 1 > i A 2 Voter i prefers A 1 to A 2

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Social Choice What if there are ¸ 3 alternatives? Plurality can elect one that would lose to every other (Borda). Alternatives A 1,…,A m Condorcet Principle (Condorcet Winner) IF an alternative is pairwise preferred to each other alternative by a majority 9 t 2 [m] s.t. 8 j 2 [m], j t: |i 2 [n]: A t > i A j | > n/2 THEN the group should select A j.

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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

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123123 231231 312312 1 2 3

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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

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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. 1234512345 5432154321 1352413524 4253142531 4153241532 2351423514 1234512345 5432154321 1352413524 4253142531 4153241532 2351423514

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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. 1234512345 5342153421 1352413524 4253142531 4153241532 2351423514 1234512345 5423154231 1352413524 4253142531 4153241532 2351423514 3 > 42 > 3

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Formulation of Social Choice Problem Alternatives A j, j 2 [m] Voters i 2 [n] For each i, preferences P i 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

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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 i 2 [n] s.t. f(P [n] )=P i 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.

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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

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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].

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Gibbard-Satterthwaite Theorem (1973, 1975) Let m ¸ 3. No voting rule simultaneously satisfies: 1. Single-valued 2. No dictator 3. Strategyproof (non-manipulable) 4. 8 j 2 [m] 9 voter population profile that elects j Proof: similar to proof of, or uses, Arrow’s theorem.

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Gardenfors’s Theorem Let m ¸ 3. No SWP simultaneously satisfies: 1. Anonymous 2. Neutral 3. Condorcet winner 4. Strategyproof

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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

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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)

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A “Good” Use of Complexity Theorem: 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

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Break ties by lexicographic order Theorem: 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

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Proof Ideas Last-round-tournament-manipulation is NP- Complete w.r.t. 2 nd order Copeland. 3,4-SAT (To84) Special candidate C 0, clause candidates C j Literal candidates X i,Y i C2C2 X5X5 X6X6 Y5Y5 Y6Y6 X7X7 Y7Y7

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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 2 nd order Copeland points, must pick one losing (= True) literal for each clause

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Proof Ideas Pad so each clause candidate is 1. tied with C_0 in 1 st order Copeland 2. 3 behind C_0 in 2 nd 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.

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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.

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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

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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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