CSC304 Lecture 15 Voting 2: Gibbard-Satterthwaite Theorem CSC304 - - PowerPoint PPT Presentation

CSC304 Lecture 15 Voting 2: Gibbard-Satterthwaite Theorem CSC304 - Nisarg Shah 1

Recap • We introduced a plethora of voting rules ➢ Plurality ➢ Plurality with runoff ➢ Borda ➢ Kemeny ➢ Veto ➢ Copeland ➢ 𝑙 -Approval ➢ Maximin ➢ STV • All these rules do something reasonable on a given preference profile ➢ Only makes sense if preferences are truthfully reported CSC304 - Nisarg Shah 2

Recap • Set of voters 𝑂 = {1, … , 𝑜} 1 2 3 • Set of alternatives 𝐵 , 𝐵 = 𝑛 a c b • Voter 𝑗 has a preference b a a ranking ≻ 𝑗 over the c b c alternatives • Preference profile ≻ = collection of all voter rankings • Voting rule (social choice function) 𝑔 ➢ Takes as input a preference profile ≻ ➢ Returns an alternative 𝑏 ∈ 𝐵 CSC304 - Nisarg Shah 3

Strategyproofness • Would any of these rules incentivize voters to report their preferences truthfully? • A voting rule 𝑔 is strategyproof if for every ➢ preference profiles ≻ , ➢ voter 𝑗 , and ′ = ≻ 𝑘 for all 𝑘 ≠ 𝑗 ➢ preference profile ≻ ′ such that ≻ 𝑘  it is not the case that 𝑔 ≻ ′ ≻ 𝑗 𝑔 ≻ CSC304 - Nisarg Shah 4

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 CSC304 - Nisarg Shah 5

Borda’s Response to Critics My scheme is intended only for honest men! Random 18 th century French dude CSC304 - Nisarg Shah 6

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 CSC304 - Nisarg Shah 7

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 top alternative for voter 𝑗 . CSC304 - 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 1: Show that SP is equivalent to “strong monotonicity” [HW 3?] ➢ 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. CSC304 - Nisarg Shah 9

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” [HW 3?] ➢ 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). CSC304 - Nisarg Shah 10

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 A b N A N A A Y 𝐽(𝑏, 𝑐) N N N Y A Y Y Y a N Y 𝑔 ≻ ′′ = 𝑏 ′ Say 𝑔 ≻ 1 , ≻ 2 = 𝑏 𝑔 ≻ 1 , ≻ 2 = 𝑏 ′ • PO: 𝑔 ≻ 1 , ≻ 2 ∈ {𝑏, 𝑐} • PO: 𝑔 ≻ 1 , ≻ 2 • Due to strong ∈ {a, b} ′ • SP: 𝑔 ≻ 1 , ≻ 2 ≠ 𝑐 monotonicity CSC304 - Nisarg Shah 11

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 𝐽(𝑏, 𝑐) , we have 𝐸 1, 𝑏 or 𝐸 2, 𝑐 . CSC304 - Nisarg Shah 12

Gibbard-Satterthwaite • Proof for n=2: ➢ On some 𝐽(𝑏 ∗ , 𝑐 ∗ ) , suppose 𝐸 1, 𝑏 ∗ holds. ➢ Then, we show that voter 1 is a dictator. That is, 𝐸(1, 𝑐) must hold for every other 𝑐 as well. ➢ Take 𝑐 ≠ 𝑏 . Because 𝐵 ≥ 3 , there exists 𝑑 ∈ 𝐵\{𝑏 ∗ , 𝑐} . ➢ Consider 𝐽(𝑐, 𝑑) . We either have 𝐸(1, 𝑐) or 𝐸 2, 𝑑 . ➢ But 𝐸(2, 𝑑) is incompatible with 𝐸(1, 𝑏 ∗ ) o Who would win if voter 1 puts 𝑏 ∗ first and voter 2 puts 𝑑 first? ➢ Thus, we have 𝐸(1, 𝑐) , as required. ➢ QED! CSC304 - Nisarg Shah 13

Circumventing G-S • Randomization ➢ Gibbard characterized all randomized strategyproof rules ➢ Somewhat better, but still too restrictive • Restricted preferences ➢ Median for facility location (more generally, for single- peaked preferences on a line) ➢ Will see other such settings later • Money ➢ E.g., VCG is nondictatorial, onto, and strategyproof, but charges payments to agents CSC304 - Nisarg Shah 14

Circumventing G-S • Equilibrium analysis ➢ Maybe good alternatives still win under Nash equilibria? • Lack of information ➢ Maybe voters don’t know how other voters will vote? CSC304 - Nisarg Shah 15

Circumventing G-S • Computational complexity (Bartholdi et al.) ➢ Maybe the rule is manipulable, but it is NP-hard to find a successful manipulation? ➢ Groundbreaking idea! NP-hardness can be good!! • Not NP-hard: plurality, Borda, veto, Copeland, maximin, … • NP-hard: Copeland with a peculiar tie-breaking, STV, ranked pairs, … CSC304 - Nisarg Shah 16

Circumventing G-S • Computational complexity ➢ Unfortunately, NP-hardness just says it is hard for some worst-case instances. ➢ What if it is actually easy for most practical instances? ➢ Many rules admit polynomial time manipulation algorithms for fixed #alternatives ( 𝑛 ) ➢ Many rules admit polynomial time algorithms that find a successful manipulation on almost all profiles (the fraction of profiles converges to 1) • Interesting open problem to design voting rules that are hard to manipulate on average CSC304 - Nisarg Shah 17

Social Choice • Let’s forget incentives for now. • Even if voters reveal their preferences truthfully, we do not have a “right” way to choose the winner. • Who is the right winner? ➢ On profiles where the prominent voting rules have different outputs, all answers seem reasonable [HW3]. CSC304 - Nisarg Shah 18

Axiomatic Approach • Define axiomatic properties we may want from a voting rule • We already defined some: ➢ Majority consistent ➢ Condorcet consistent ➢ Onto ➢ Strategyproof ➢ Strongly monotonic ➢ Pareto optimal CSC304 - Nisarg Shah 19

Axiomatic Approach • We will see four more: ➢ Unanimity ➢ Weak monotonicity ➢ Consistency (!) ➢ Independence of irrelevant alternatives (IIA) • Problem? ➢ Cannot satisfy many interesting combinations of properties ➢ Arrow’s impossibility result ➢ Other similar impossibility results CSC304 - Nisarg Shah 20

Other Approaches? • Statistical ➢ There exists an objectively true answer o E.g., say the question is: “ Sort the candidates by the #votes they will receive in an upcoming election .” ➢ Every voter is trying to estimate the true ranking ➢ Goal is to find the most likely ground truth given votes • Utilitarian ➢ Back to “numerical” welfare maximization, but we still ask voters to only report ranked preferences o 𝑏 ≻ 𝑗 𝑐 ≻ 𝑗 𝑑 simply means 𝑤 𝑗 𝑏 ≥ 𝑤 𝑗 𝑐 ≥ 𝑤 𝑗 𝑑 ➢ How well can we maximize welfare subject to such partial information? CSC304 - Nisarg Shah 21