PHIL 308S: Voting Theory and Fair Division
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1 PHIL 308S: Voting Theory and Fair Division Lecture 12 Eric Pacuit Department of Philosophy University of Maryland, College Park ai.stanford.edu/ epacuit October 18, 2012 PHIL 308S: Voting Theory and Fair Division 1/32
2 { axed&vi"at 1&i'e Cax: Do About Manipulation It) PHIL 308S: Voting Theory and Fair Division 2/32
3 Agenda manipulation Misrepresenting preferences Sophisticated voting What is wrong with manipulation? PHIL 308S: Voting Theory and Fair Division 3/32
4 A voting rule V is manipulable provided there are two profiles P and P and a voter i such that PHIL 308S: Voting Theory and Fair Division 4/32
5 A voting rule V is manipulable provided there are two profiles P and P and a voter i such that P j = P j for all j i, and PHIL 308S: Voting Theory and Fair Division 4/32
6 A voting rule V is manipulable provided there are two profiles P and P and a voter i such that P j = P j for all j i, and Voter i prefers V ( P ) to V ( P). PHIL 308S: Voting Theory and Fair Division 4/32
7 A voting rule V is manipulable provided there are two profiles P and P and a voter i such that P j = P j for all j i, and Voter i prefers V ( P ) to V ( P). Intuition: P i is voter i s true preference. PHIL 308S: Voting Theory and Fair Division 4/32
8 A voting rule V is manipulable provided there are two profiles P and P and a voter i such that P j = P j for all j i, and Voter i prefers V ( P ) to V ( P). Intuition: P i is voter i s true preference. If V ( P) and V ( P ) are singletons, then i prefers V ( P ) to V ( P) means V ( P )P i V ( P) PHIL 308S: Voting Theory and Fair Division 4/32
9 What if V ( P) and V ( P ) are not singletons? PHIL 308S: Voting Theory and Fair Division 5/32
10 Preference Lifting, I Given a preference ordering over a set of objects X, we want to lift this to an ordering ˆ over (X ). Given, what reasonable properties can we infer about ˆ? S. Barberá, W. Bossert, and P.K. Pattanaik. Ranking sets of objects. In Handbook of Utility Theory, volume 2. Kluwer Academic Publishers, PHIL 308S: Voting Theory and Fair Division 6/32
11 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? PHIL 308S: Voting Theory and Fair Division 7/32
12 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? PHIL 308S: Voting Theory and Fair Division 7/32
13 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? You know that w x y z Can you infer that {w, x, y} ˆ {w, y, z}? PHIL 308S: Voting Theory and Fair Division 7/32
14 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? You know that w x y z Can you infer that {w, x, y} ˆ {w, y, z}? You know that w x y z Can you infer that {w, x} ˆ {y, z}? PHIL 308S: Voting Theory and Fair Division 7/32
15 Preference Lifting, III There are different interpretations of X ˆ Y : You will get one of the elements, but cannot control which. You can choose one of the elements. You will get the full set. PHIL 308S: Voting Theory and Fair Division 8/32
16 Preference Lifting, IV Kelly Principle (EXT) {x} ˆ {y} provided x y (MAX) A ˆ Max(A) (MIN) Min(A) ˆ A J.S. Kelly. Strategy-Proofness and Social Choice Functions without Single- Valuedness. Econometrica, 45(2), pp , PHIL 308S: Voting Theory and Fair Division 9/32
17 Preference Lifting, IV Gärdenfors Principle (G1) A ˆ A {x} if a x for all a A (G2) A {x} ˆ A if x a for all a A P. Gärdenfors. Manipulation of Social Choice Functions. Journal of Economic Theory. 13:2, , PHIL 308S: Voting Theory and Fair Division 10/32
18 Preference Lifting, IV Gärdenfors Principle (G1) A ˆ A {x} if a x for all a A (G2) A {x} ˆ A if x a for all a A P. Gärdenfors. Manipulation of Social Choice Functions. Journal of Economic Theory. 13:2, , Independence (IND) A {x} ˆ B {x} if A ˆ B and x A B PHIL 308S: Voting Theory and Fair Division 10/32
19 Preference Lifting, V Theorem (Kannai and Peleg). If X 6, then no weak order satisfies both the Gärdenfors principle and independence. Y. Kannai and B. Peleg. A Note on the Extension of an Order on a Set to the Power Set. Journal of Economic Theory, 32(1), pp , PHIL 308S: Voting Theory and Fair Division 11/32
20 Suppose that V ( P) and V ( P ) are not singletons X is weakly dominates Y for i provided x X y Y xr i y and x X y Y xp i y PHIL 308S: Voting Theory and Fair Division 12/32
21 Suppose that V ( P) and V ( P ) are not singletons X is weakly dominates Y for i provided x X y Y xr i y and x X y Y xp i y X is preferred by an optimist to Y : max i (X, P)P i max i (Y, P) PHIL 308S: Voting Theory and Fair Division 12/32
22 Suppose that V ( P) and V ( P ) are not singletons X is weakly dominates Y for i provided x X y Y xr i y and x X y Y xp i y X is preferred by an optimist to Y : max i (X, P)P i max i (Y, P) X is preferred by a pessimist to Y : min i (X, P)P i min i (Y, P) PHIL 308S: Voting Theory and Fair Division 12/32
23 Suppose that V ( P) and V ( P ) are not singletons X is weakly dominates Y for i provided x X y Y xr i y and x X y Y xp i y X is preferred by an optimist to Y : max i (X, P)P i max i (Y, P) X is preferred by a pessimist to Y : min i (X, P)P i min i (Y, P) X has higher expected utility : There exists a utility function representing P i such that, if p(x) = 1 X and p(y) = 1 Y, then p(x) u(x) > p(y) u(y) y Y x X PHIL 308S: Voting Theory and Fair Division 12/32
24 The Gibbard-Satterthwaite Theorem A social choice function is strategy-proof if for no individual i there exists a profile R and a linear order R i such that V ( R i, R i ) is ranked above V ( R) according to i. Theorem. Any social choice function for three or more alternatives that is Pareto and strategy-proof must be a dictatorship. M. A. Satterthwaite. Strategy-proofness and Arrow s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10(2): , A. Gibbard. Manipulation of voting schemes: A general result. Econometrica, 41(4): , PHIL 308S: Voting Theory and Fair Division 13/32
25 Agenda manipulation Misrepresenting preferences Sophisticated voting What is wrong with manipulation? PHIL 308S: Voting Theory and Fair Division 14/32
26 Example I The following example is due to [Brams & Fishburn] P A = o 1 > o 3 > o 2 P B = o 2 > o 3 > o 1 P C = o 3 > o 1 > o 2 Size Group I II 4 A o 1 o 1 3 B o 2 o 2 2 C o 3 o 1 If the current winner is o, then agent i will switch its vote to some candidate o provided 1. o is one of the top two candidates as indicated by a poll 2. o is preferred to the other top candidate PHIL 308S: Voting Theory and Fair Division 15/32
27 Example I The following example is due to [Brams & Fishburn] P A = o 1 > o 3 > o 2 P B = o 2 > o 3 > o 1 P C = o 3 > o 1 > o 2 Size Group I II 4 A o 1 o 1 3 B o 2 o 2 2 C o 3 o 1 If the current winner is o, then agent i will switch its vote to some candidate o provided 1. o is one of the top two candidates as indicated by a poll 2. o is preferred to the other top candidate PHIL 308S: Voting Theory and Fair Division 16/32
28 Example II PA = (o 1, o 4, o 2, o 3 ) PB = (o 2, o 1, o 3, o 4 ) PC = (o 3, o 2, o 4, o 1 ) PD = (o 4, o 1, o 2, o 3 ) PE = (o 3, o 1, o 2, o 4 ) Size Group I II III IV 40 A o 1 o 1 o 4 o 1 30 B o 2 o 2 o 2 o 2 15 C o 3 o 2 o 2 o 2 8 D o 4 o 4 o 1 o 4 7 E o 3 o 3 o 1 o 1 If the current winner is o, then agent i will switch its vote to some candidate o provided 1. i prefers o to o, and 2. the current total for o plus agent i s votes for o is greater than the current total for o. PHIL 308S: Voting Theory and Fair Division 17/32
29 Example II PA = (o 1, o 4, o 2, o 3 ) PB = (o 2, o 1, o 3, o 4 ) PC = (o 3, o 2, o 4, o 1 ) PD = (o 4, o 1, o 2, o 3 ) PE = (o 3, o 1, o 2, o 4 ) Size Group I II III IV 40 A o 1 o 1 o 4 o 1 30 B o 2 o 2 o 2 o 2 15 C o 3 o 2 o 2 o 2 8 D o 4 o 4 o 1 o 4 7 E o 3 o 3 o 1 o 1 If the current winner is o, then agent i will switch its vote to some candidate o provided 1. i prefers o to o, and 2. the current total for o plus agent i s votes for o is greater than the current total for o. PHIL 308S: Voting Theory and Fair Division 18/32
30 Example II PA = (o 1, o 4, o 2, o 3 ) PB = (o 2, o 1, o 3, o 4 ) PC = (o 3, o 2, o 4, o 1 ) PD = (o 4, o 1, o 2, o 3 ) PE = (o 3, o 1, o 2, o 4 ) Size Group I II III IV 40 A o 1 o 1 o 4 o 1 30 B o 2 o 2 o 2 o 2 15 C o 3 o 2 o 2 o 2 8 D o 4 o 4 o 1 o 4 7 E o 3 o 3 o 1 o 1 If the current winner is o, then agent i will switch its vote to some candidate o provided 1. i prefers o to o, and 2. the current total for o plus agent i s votes for o is greater than the current total for o. PHIL 308S: Voting Theory and Fair Division 19/32
31 Example II PA = (o 1, o 4, o 2, o 3 ) PB = (o 2, o 1, o 3, o 4 ) PC = (o 3, o 2, o 4, o 1 ) PD = (o 4, o 1, o 2, o 3 ) PE = (o 3, o 1, o 2, o 4 ) Size Group I II III IV 40 A o 1 o 1 o 4 o 1 30 B o 2 o 2 o 2 o 2 15 C o 3 o 2 o 2 o 2 8 D o 4 o 4 o 1 o 4 7 E o 3 o 3 o 1 o 1 If the current winner is o, then agent i will switch its vote to some candidate o provided 1. i prefers o to o, and 2. the current total for o plus agent i s votes for o is greater than the current total for o. PHIL 308S: Voting Theory and Fair Division 20/32
32 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 21/32
33 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 22/32
34 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 23/32
35 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 24/32
36 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 25/32
37 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 26/32
38 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 27/32
39 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 28/32
40 Example III P A = (o 1, o 2, o 3 ) P B = (o 2, o 3, o 1 ) P C = (o 3, o 1, o 2 ) Size Group I II III IV V VI VII 40 A o 1 o 1 o 2 o 2 o 2 o 1 o 1 o 1 30 B o 2 o 3 o 3 o 2 o 2 o 2 o 3 o 3 30 C o 3 o 3 o 3 o 3 o 1 o 1 o 1 o 3 PHIL 308S: Voting Theory and Fair Division 29/32
41 Sophisticated Voting Consider a legislator voting on a pay raise. (pass and vote nay) P i (pass and vote yea) P i (fail and vote nay) P i (fail and vote yea) PHIL 308S: Voting Theory and Fair Division 30/32
42 Sophisticated Voting Consider a legislator voting on a pay raise. (pass and vote nay) P i (pass and vote yea) P i (fail and vote nay) P i (fail and vote yea) If there are three voters who voter in turn, what will the first legislator choose? PHIL 308S: Voting Theory and Fair Division 30/32
43 (P & N) P i (P & Y ) P i (F & N) P i (F & Y ) 1 Y N 2 2 Y N Y N Y N Y N Y N Y N P P P F P F F F PHIL 308S: Voting Theory and Fair Division 31/32
44 (P & N) P i (P & Y ) P i (F & N) P i (F & Y ) 1 Y N 2 2 Y N Y N Y N Y N Y N Y N P P P F P F F F PHIL 308S: Voting Theory and Fair Division 31/32
45 (P & N) P i (P & Y ) P i (F & N) P i (F & Y ) 1 Y N 2 2 Y N Y N Y N Y N Y N Y N P P P F P F F F PHIL 308S: Voting Theory and Fair Division 31/32
46 (P & N) P i (P & Y ) P i (F & N) P i (F & Y ) 1 Y N 2 2 Y N Y N Y N Y N Y N Y N P P P F P F F F PHIL 308S: Voting Theory and Fair Division 31/32
47 (P & N) P i (P & Y ) P i (F & N) P i (F & Y ) 1 Y N 2 2 Y N Y N Y N Y N Y N Y N P P P F P F F F PHIL 308S: Voting Theory and Fair Division 31/32
48 What does it mean to vote strategically? Voting as a game vs. voting as an act of communication K. Dowding and M. van Hees. In Praise of Manipulation. British Journal of Political Science, 38 : pp 1-15, PHIL 308S: Voting Theory and Fair Division 32/32
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