Optimal False-Name-Proof Voting Rules with Costly Voting Liad WagmanVincent Conitzer Duke University Malvika Rao CS 286r Class Presentation Harvard University

Overview Introduction Definitions False-name-proof voting rule for 2 alternatives Group false-name-proofness False-name-proof voting rule for 3 alternatives Discussion

Introduction Introducing costs… Previous rules without costs unresponsive to agent preferences. Idea: no one ever benefits by voting additional times. Because we now have costs we are tying utility to money. So people’s utility function becomes comparable.

Definitions (2 alternatives) Definition 1 (State): A state consists of a pair (x A, x B ), where x j ≥ 0 is the # of votes for j in {A, B}. Definition 2 (Voting Rule): A voting rule is a mapping from the set of states to the set of probability distributions over outcomes. The probability that alternative j in {A, B} is selected in state (x A, x B ) is denoted by P j (x A, x B ). Definition 3 (Neutrality): A voting rule is neutral if P A (x, y) = P B (y, x).

Definitions (2 alternatives) Let t i A and t i B be the # of times agent i votes for A and B. If i prefers alternative j then i’s expected utility u i (x A, x B, t i A, t i B ) = P j (x A + t i A, x B + t i B ) - (t i A + t i B - 1)c. Definition 4 (Voluntary Participation): A voting rule satisfies voluntary participation if for an agent i who prefers A, for all (x A, x B ), u i (x A, x B, 1, 0) ≥ u i (x A, x B, 0, 0). Definition 5 (Strategy-proofness): A voting rule is strategy- proof if for an agent i who prefers A, for all (x A, x B ), u i (x A, x B, 1, 0) ≥ u i (x A, x B, 0, 1).

Definitions (2 alternatives) Definition 6 (False-name-proofness): A voting rule is false- name-proof (with costs) if for an agent i who prefers A, for all (x A, x B ), for all t i A ≥ 1 and t i B, u i (x A, x B, 1, 0) ≥ u i (x A, x B, t i A, t i B ). Definition 7 (Strong optimality): A neutral false-name-proof voting rule P that satisfies voluntary participation is strongly optimal if for any other such rule P´, for any state (x A, x B ) where x A ≥ x B, we have P A (x A, x B ) ≥ P´ A (x A, x B ).

False-name-proof voting rule for 2 alternatives FNP2: Suppose x A ≥ x B. Then P A (x A, x B ) = 1 if x A > x B = 0, P A (x A, x B ) = min{1, 1/2 + c(x A - x B )} if x A ≥ x B > 0 or x A = x B = 0. Theorem: FNP2 is the unique strongly optimal neutral false-name-proof voting rule with 2 alternatives that satisfies voluntary participation.

False-name-proof voting rule for 2 alternatives Proof: FNP2 is strongly optimal By neutrality for any x ≥ 0 P´ A (x, x) = 1/2. By false-name-proofness for any x > 0 P´ A (x+1, x) - P´ A (x, x) ≤ c. So P´ A (x+1, x) ≤ 1/2 + c. Similarly P´ A (x+2, x) ≤ P´ A (x+1, x) + c ≤ 1/2 + 2c. For any t > 0 P´ A (x+t, x) ≤ 1/2 + tc. Since P´ A (x+t, x) ≤ 1, P´ A (x+t, x) ≤ min{1, 1/2 + tc}. But P A (x+t, x) = min{1, 1/2 + tc}.

FNP2 Responsiveness Example: c = x B / x A

FNP2 Responsiveness Convergence to majority winner as n --> ∞.

FNP2 Responsiveness Average probability that FNP2 and majority rule disagree as a function of c.

FNP2 Responsiveness Average probability that FNP2 and majority rule disagree as a function of p (probability agent prefers A).

Group false-name-proof voting rule for 2 alternatives FNP2 is not group false-name-proof. Consider the example: c = 0.15, x A = x B = 2. If the 2 agents that prefer A each cast an additional vote then A now wins with probability 0.8. Each agent is = 0.15 better off. A rule is group false-name-proof (with costs and transfers) if for all k ≥ 1, for all (x A, x B ), for all t A ≥ k and t B, P A (x A + k, x B ) ≥ P A (x A + t A, x B + t B ) - c(t A + t B - k)/k.

Group false-name-proof voting rule for 2 alternatives Strongly optimal GFNP2: Suppose x A ≥ x B. Then P A (x A, x B ) = 1 if x A > x B = 0, P A (x A, x B ) = 1/2 if x A = x B = 0, P A (x A, x B ) = min{1, 1/2 + ∑ k (c/k) for k = x B to x A -1} if x A ≥ x B > 0. As n --> ∞ GFNP2 yields the opposite result from the majority rule at least 40% of the time. There is no finite c such that GFNP2 coincides with the majority rule.

False-name-proof voting rule for 3 alternatives Strong optimality: Voting rule P is strongly optimal if for any other rule P´, for any (x A, x B, x C ) where x A ≥ x B ≥ x C ≥ 1, either P A (x A, x B, x C ) > P´ A (x A, x B, x C ); or P A (x A, x B, x C ) = P´ A (x A, x B, x C ) and P B (x A, x B, x C ) ≥ P´ B (x A, x B, x C ). FNP3: Suppose x A ≥ x B ≥ x C ≥ 1. Then P A (x A, x B, x C ) = min{1, 1/2 + c(x A - x B ) - 1/2 max{0, 1/3 - c(x B - x C )}} P C (x A, x B, x C ) = max{0, 1/3 - c((x A + x B )/2 - x C )} P B (x A, x B, x C ) = 1 - P A (x A, x B, x C ) - P C (x A, x B, x C )

Discussion 4+ alternatives… How can we improve group false-name-proofness? GFNP3? Continuous preferences Bayes-Nash