CSC2556 Lecture 2 Manipulation in Voting Credit for many visuals: - PowerPoint PPT Presentation

CSC2556 Lecture 2 Manipulation in Voting Credit for many visuals: Ariel D. Procaccia CSC2556 - Nisarg Shah 1

Recap • Voting ➢ 𝑜 voters, 𝑛 alternatives ➢ Each voter 𝑗 expresses a ranked preference ≻ 𝑗 ➢ Voting rule 𝑔 o Takes as input the collection of preferences ≻ o Returns a single alternative • A plethora of voting rule ➢ Plurality, Borda count, STV, Kemeny, Copeland, maximin, … CSC2556 - Nisarg Shah 2

Incentives • Can a voting rule incentivize voters to truthfully report their preferences? • Strategyproofness ➢ A voting rule is strategyproof if a voter cannot submit a false preference and get a more preferred alternative (under her true preference) elected, irrespective of the preferences of other voters. ➢ Formally, a voting rule 𝑔 is strategyproof if there is no ′ s.t. preference profile ≻ , voter 𝑗 , and false preference ≻ 𝑗 ′ 𝑔 ≻ −𝑗 , ≻ 𝑗 ≻ 𝑗 𝑔 ≻ CSC2556 - Nisarg Shah 3

Strategyproofness • None of the rules we saw are strategyproof! • Example: Borda Count ➢ In the true profile, 𝑐 wins ➢ Voter 3 can make 𝑏 win by pushing 𝑐 to the end 1 2 3 1 2 3 b b a b b a a a b a a c Winner Winner c c c c c d b a d d d d d b CSC2556 - Nisarg Shah 4

Borda’s Response to Critics My scheme is intended only for honest men! Random 18 th century French dude CSC2556 - Nisarg Shah 5

Strategyproofness • Are there any strategyproof rules? ➢ Sure • Dictatorial voting rule ➢ The winner is always the most Dictatorship preferred alternative of voter 𝑗 • Constant voting rule ➢ The winner is always the same • Not satisfactory (for most cases) Constant function CSC2556 - Nisarg Shah 6

Three Properties • Strategyproof: Already defined. No voter has an incentive to misreport. • Onto: Every alternative can win under some preference profile. • Nondictatorial: There is no voter 𝑗 such that 𝑔 ≻ is always the alternative most preferred by voter 𝑗 . CSC2556 - Nisarg Shah 7

Gibbard-Satterthwaite • Theorem: For 𝑛 ≥ 3 , no deterministic social choice function can be strategyproof, onto, and nondictatorial simultaneously  • Proof: We will prove this for 𝑜 = 2 voters. ➢ Step 1: Show that SP implies “strong monotonicity” [Assignment] ➢ Strong Monotonicity (SM): If 𝑔 ≻ = 𝑏 , and ≻ ′ is such that ′ 𝑦 , then 𝑔 ≻ ′ = 𝑏 . ∀𝑗 ∈ 𝑂, 𝑦 ∈ 𝐵: 𝑏 ≻ 𝑗 𝑦 ⇒ 𝑏 ≻ 𝑗 o If 𝑏 still defeats every alternative it defeated in every vote in ≻ , it should still win. CSC2556 - Nisarg Shah 8

Gibbard-Satterthwaite • Theorem: For 𝑛 ≥ 3 , no deterministic social choice function can be strategyproof, onto, and nondictatorial simultaneously  • Proof: We will prove this for 𝑜 = 2 voters. ➢ Step 2: Show that SP+onto implies “Pareto optimality” [Assignment] ➢ Pareto Optimality (PO): If 𝑏 ≻ 𝑗 𝑐 for all 𝑗 ∈ 𝑂 , then 𝑔 ≻ ≠ 𝑐 . o If there is a different alternative that everyone prefers, your choice is not Pareto optimal (PO). CSC2556 - Nisarg Shah 9

Gibbard-Satterthwaite • Proof for n=2: Consider problem instance 𝐽(𝑏, 𝑐) ′ ≻ 𝟐 ≻ 𝟑 ≻ 𝟐 ≻ 𝟑 ′′ ′′ ≻ 𝟐 ≻ 𝟑 a b a b a b a A b A N 𝐽(𝑏, 𝑐) N Y Y a 𝑔 ≻ 1 , ≻ 2 ∈ {𝑏, 𝑐} 𝑔 ≻ ′′ = 𝑏 ′ 𝑔 ≻ 1 , ≻ 2 = 𝑏 ➢ PO ′ • PO: 𝑔 ≻ 1 , ≻ 2 • Due to strong ∈ {a, b} ′ • SP: 𝑔 ≻ 1 , ≻ 2 Say 𝑔 ≻ 1 , ≻ 2 = 𝑏 ≠ 𝑐 monotonicity CSC2556 - Nisarg Shah 10

Gibbard-Satterthwaite • Proof for n=2: ➢ If 𝑔 outputs 𝑏 on instance 𝐽(𝑏, 𝑐) , voter 1 can get 𝑏 elected whenever she puts 𝑏 first. o In other words, voter 1 becomes dictatorial for 𝑏 . o Denote this by 𝐸(1, 𝑏) . ➢ If 𝑔 outputs 𝑐 on 𝐽(𝑏, 𝑐) o Voter 2 becomes dictatorial for 𝑐 , i.e., we have 𝐸(2, 𝑐) . • For every (𝑏, 𝑐) , we have either 𝐸 1, 𝑏 or 𝐸 2, 𝑐 . CSC2556 - Nisarg Shah 11

Gibbard-Satterthwaite • Proof for n=2: ➢ Fix 𝑏 ∗ and 𝑐 ∗ . Suppose 𝐸 1, 𝑏 ∗ holds. ➢ Then, we show that voter 1 is a dictator. o That is, 𝐸(1, 𝑑) holds for every 𝑑 ≠ 𝑏 ∗ as well. ➢ Take 𝑑 ≠ 𝑏 ∗ . Because 𝐵 ≥ 3 , there exists 𝑒 ∈ 𝐵\{𝑏 ∗ , 𝑑} . ➢ Consider 𝐽(𝑑, 𝑒) . We either have 𝐸(1, 𝑑) or 𝐸 2, 𝑒 . ➢ But 𝐸(2, 𝑒) is incompatible with 𝐸(1, 𝑏 ∗ ) o Who would win if voter 1 puts 𝑏 ∗ first and voter 2 puts 𝑒 first? ➢ Thus, we have 𝐸(1, 𝑑) , as required. ➢ QED! CSC2556 - Nisarg Shah 12

Circumventing G-S • Restricted preferences (later in the course) ➢ Not allowing all possible preference profiles ➢ Example: single-peaked preferences o Alternatives are on a line (say 1D political spectrum) o Voters are also on the same line o Voters prefer alternatives that are closer to them • Use of money (later in the course) ➢ Require payments from voters that depend on the preferences they submit ➢ Prevalent in auctions CSC2556 - Nisarg Shah 13

Circumventing G-S • Randomization (later in this lecture) • Equilibrium analysis ➢ How will strategic voters act under a voting rule that is not strategyproof? ➢ Will they reach an “equilibrium” where each voter is happy with the (possibly false) preference she is submitting? • Restricting information ➢ Can voters successfully manipulate if they don’t know the votes of the other voters? CSC2556 - Nisarg Shah 14

Circumventing G-S • Computational complexity ➢ So we need to use a rule that is the rule is manipulable. ➢ Can we make it NP-hard for voters to manipulate? [Bartholdi et al., SC&W 1989] ➢ NP-hardness can be a good thing! • 𝑔 -M ANIPULATION problem (for a given voting rule 𝑔 ): ➢ Input: Manipulator 𝑗 , alternative 𝑞 , votes of other voters (non-manipulators) ➢ Output: Can the manipulator cast a vote that makes 𝑞 uniquely win under 𝑔 ? CSC2556 - Nisarg Shah 15

Example: Borda • Can voter 3 make 𝑏 win? 1 2 3 1 2 3 b b b b a a a a a c c c c c d d d d d b CSC2556 - Nisarg Shah 16

A Greedy Algorithm • Goal: The manipulator wants to make alternative 𝑞 win uniquely • Algorithm: ➢ Rank 𝑞 in the first place ➢ While there are unranked alternatives: o If there is an alternative that can be placed in the next spot without preventing 𝑞 from winning, place this alternative. o Otherwise, return false. CSC2556 - Nisarg Shah 17

Example: Borda 1 2 3 1 2 3 1 2 3 b b a b b a b b a a a a a b a a c c c c c c c d d d d d d 1 2 3 1 2 3 1 2 3 b b a b b a b b a a a c a a c a a c c c b c c d c c d d d d d d d b CSC2556 - Nisarg Shah 18

Example: Copeland 1 2 3 4 5 a b c d e a a b e e a - 2 3 5 3 b b a c c 3 - 2 4 2 c c d b b 2 2 - 3 1 d d e a a 0 0 1 - 2 e e c d d 2 2 3 2 - Preference profile Pairwise elections CSC2556 - Nisarg Shah 19

Example: Copeland 1 2 3 4 5 a b c d e a a b e e a - 2 3 5 3 b b a c c c 3 - 2 4 2 c c d b b 2 3 - 4 2 d d e a a 0 0 1 - 2 e e c d d 2 2 3 2 - Preference profile Pairwise elections CSC2556 - Nisarg Shah 20

Example: Copeland 1 2 3 4 5 a b c d e a a b e e a - 2 3 5 3 b b a c c c 3 - 2 4 2 c c d b b d 2 3 - 4 2 d d e a a 0 1 1 - 3 e e c d d 2 2 3 2 - Preference profile Pairwise elections CSC2556 - Nisarg Shah 21

Example: Copeland 1 2 3 4 5 a b c d e a a b e e a - 2 3 5 3 b b a c c c 3 - 2 4 2 c c d b b d 2 3 - 4 2 d d e a a e 0 1 1 - 3 e e c d d 2 3 3 2 - Preference profile Pairwise elections CSC2556 - Nisarg Shah 22

Example: Copeland 1 2 3 4 5 a b c d e a a b e e a - 2 3 5 3 b b a c c c 3 - 2 4 2 c c d b b d 2 3 - 4 2 d d e a a e 0 1 1 - 3 e e c d d b 2 3 3 2 - Preference profile Pairwise elections CSC2556 - Nisarg Shah 23

When does this work? • Theorem [Bartholdi et al., SCW 89]: Fix voter 𝑗 and votes of other voters. Let 𝑔 be a rule for which ∃ function 𝑡(≻ 𝑗 , 𝑦) such that: 1. For every ≻ 𝑗 , 𝑔 chooses a candidate 𝑦 that uniquely maximizes 𝑡(≻ 𝑗 , 𝑦) . ′ 𝑧 ⇒ 𝑡 ≻ 𝑗 , 𝑦 ≤ 𝑡 ≻ 𝑗 ′ , 𝑦 2. 𝑧 ∶ 𝑦 ≻ 𝑗 𝑧 ⊆ 𝑧 ∶ 𝑦 ≻ 𝑗 Then the greedy algorithm solves 𝑔 -M ANIPULATION correctly. • Question: What is the function 𝑡 for plurality? CSC2556 - Nisarg Shah 24

Proof of the Theorem ≻ 𝑗 • Say the algorithm creates a partial 𝑞 ranking ≻ 𝑗 and then fails, i.e., every Output of 𝑐 algo next choice prevents 𝑞 from winning 𝑒 ′ • Suppose for contradiction that ≻ 𝑗 𝑣 𝑏 could make 𝑞 uniquely win ′ ≻ 𝑗 𝑑 • 𝑉 ← alternatives not ranked in ≻ 𝑗 𝑐 • 𝑣 ← highest ranked alternative in 𝑉 𝑞 ′ according to ≻ 𝑗 𝑏 𝑒 𝑉 = {𝑏, 𝑑} • Complete ≻ 𝑗 by adding 𝑣 next, and 𝑑 then other alternatives arbitrarily CSC2556 - Nisarg Shah 25