Optimal False-Name-Proof Voting Rules with Costly Voting

Liad Wagman Vincent ConitzerDuke University

Malvika RaoCS 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 (xA, xB), 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 (xA, xB) is denoted by Pj(xA, xB).

• Definition 3 (Neutrality): A voting rule is neutral if PA(x, y) = PB(y, x) .

Definitions (2 alternatives)• Let ti

A and tiB be the # of times agent i votes for A and B.

If i prefers alternative j then i’s expected utility ui(xA, xB, ti

A, tiB) = Pj(xA + ti

A, xB + tiB) - (ti

A + tiB - 1)c.

• Definition 4 (Voluntary Participation): A voting rule satisfies voluntary participation if for an agent i who prefers A, for all (xA, xB), ui(xA, xB, 1, 0) ≥ ui(xA, xB, 0, 0) .

• Definition 5 (Strategy-proofness): A voting rule is strategy-proof if for an agent i who prefers A, for all (xA, xB), ui(xA, xB, 1, 0) ≥ ui(xA, xB, 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 (xA, xB), for all ti

A ≥ 1 and tiB,

ui(xA, xB, 1, 0) ≥ ui(xA, xB, tiA, 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 (xA, xB) where xA ≥ xB, we have PA(xA, xB) ≥ P´A(xA, xB).

False-name-proof voting rule for 2 alternatives

• FNP2: Suppose xA ≥ xB. ThenPA(xA, xB) = 1 if xA > xB = 0, PA(xA, xB) = min{1, 1/2 + c(xA - xB)} if xA ≥ xB > 0 or xA = xB = 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 PA(x+t, x) = min{1, 1/2 + tc}.

FNP2 Responsiveness

• Example: c = 0.15.

5 0 0 0.05 0.2 0.35 0.54 0 0.05 0.2 0.35 0.5 0.653 0 0.2 0.35 0.5 0.65 0.82 0 0.35 0.5 0.65 0.8 0.951 0 0.5 0.65 0.8 0.95 10 0.5 1 1 1 1 1xB / xA 0 1 2 3 4 5

FNP2 Responsiveness

• Convergence to majority winner as n --> ∞.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

FNP2 Responsiveness

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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

FNP2 Responsiveness

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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Group false-name-proof voting rule for 2 alternatives

• FNP2 is not group false-name-proof. Consider the example: c = 0.15, xA = xB = 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 (xA, xB), for all tA ≥ k and tB,

PA(xA + k, xB) ≥ PA(xA + tA, xB + tB) - c(tA + tB - k)/k .

Group false-name-proof voting rule for 2 alternatives

• Strongly optimal GFNP2: Suppose xA ≥ xB. Then PA(xA, xB) = 1 if xA > xB = 0, PA(xA, xB) = 1/2 if xA = xB = 0, PA(xA, xB) = min{1, 1/2 + ∑k (c/k) for k = xB to xA-1}if xA ≥ xB > 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 (xA, xB , xC) where xA ≥ xB ≥ xC ≥ 1, either PA (xA, xB , xC) > P´A (xA, xB , xC); or PA (xA, xB , xC) = P´A

(xA, xB , xC) and PB (xA, xB , xC) ≥ P´B (xA, xB , xC) .

• FNP3: Suppose xA ≥ xB ≥ xC ≥ 1. Then PA (xA, xB , xC) = min{1, 1/2 + c(xA - xB) - 1/2 max{0, 1/3 - c(xB - xC)}} PC (xA, xB , xC) = max{0, 1/3 - c((xA + xB)/2 - xC)} PB (xA, xB , xC) = 1 - PA (xA, xB , xC) - PC (xA, xB , xC)

Discussion

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