Optimal FalseNameProof Voting Rules with Costly Voting Liad

Optimal False-Name-Proof Voting Rules with Costly Voting Liad Wagman Vincent Conitzer Duke University Malvika Rao CS 286 r 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 xj ≥ 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 Pj(x. A, x. B). • Definition 3 (Neutrality): A voting rule is neutral if PA(x, y) = PB(y, x).

Definitions (2 alternatives) • Let ti. A and ti. B be the # of times agent i votes for A and B. If i prefers alternative j then i’s expected utility ui(x. A, x. B, ti. A, ti. B) = Pj(x. A + ti. A, x. B + ti. B) - (ti. A + ti. 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), ui(x. A, x. B, 1, 0) ≥ ui(x. A, x. B, 0, 0). • Definition 5 (Strategy-proofness): A voting rule is strategyproof if for an agent i who prefers A, for all (x. A, x. B), ui(x. A, x. B, 1, 0) ≥ ui(x. A, x. B, 0, 1).

Definitions (2 alternatives) • Definition 6 (False-name-proofness): A voting rule is falsename-proof (with costs) if for an agent i who prefers A, for all (x. A, x. B), for all ti. A ≥ 1 and ti. B, ui(x. A, x. B, 1, 0) ≥ ui(x. A, x. B, ti. A, ti. 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 PA(x. A, x. B) ≥ P´A(x. A, x. B).

False-name-proof voting rule for 2 alternatives • FNP 2: Suppose x. A ≥ x. B. Then PA(x. A, x. B) = 1 if x. A > x. B = 0, PA(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: FNP 2 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: FNP 2 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 + 2 c. • 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 PA(x+t, x) = min{1, 1/2 + tc}.

FNP 2 Responsiveness • Example: c = 0. 15. 5 4 3 2 1 0 x. B / x. A 0 0 0. 5 0 0 0. 05 0. 2 0. 35 0. 5 1 1 0. 05 0. 2 0. 35 0. 65 1 2 0. 35 0. 65 0. 8 1 3 0. 35 0. 65 0. 8 0. 95 1 4 0. 5 0. 65 0. 8 0. 95 1 1 5

FNP 2 Responsiveness • Convergence to majority winner as n --> ∞.

FNP 2 Responsiveness • Average probability that FNP 2 and majority rule disagree as a function of c.

FNP 2 Responsiveness • Average probability that FNP 2 and majority rule disagree as a function of p (probability agent prefers A).

Group false-name-proof voting rule for 2 alternatives • FNP 2 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. 3 - 0. 15 = 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, PA(x. A + k, x. B) ≥ PA(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 GFNP 2: Suppose x. A ≥ x. B. Then PA(x. A, x. B) = 1 if x. A > x. B = 0, PA(x. A, x. B) = 1/2 if x. A = x. B = 0, PA(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 --> ∞ GFNP 2 yields the opposite result from the majority rule at least 40% of the time. There is no finite c such that GFNP 2 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 PA (x. A, x. B , x. C) > P´A (x. A, x. B , x. C); or PA (x. A, x. B , x. C) = P´A (x. A, x. B , x. C) and PB (x. A, x. B , x. C) ≥ P´B (x. A, x. B , x. C). • FNP 3: Suppose x. A ≥ x. B ≥ x. C ≥ 1. Then PA (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)}} PC (x. A, x. B , x. C) = max{0, 1/3 - c((x. A + x. B)/2 - x. C)} PB (x. A, x. B , x. C) = 1 - PA (x. A, x. B , x. C) - PC (x. A, x. B , x. C)

Discussion • • • 4+ alternatives… How can we improve group false-name-proofness? GFNP 3? Continuous preferences Bayes-Nash