Optimal False-Name-Proof Voting Rules with Costly Voting: Study by Wagman et al., Study notes of Computers and Information technologies

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Optimal False-Name-Proof Voting

Rules with Costly Voting

Liad Wagman Vincent 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

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 tiA 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, tiA, tiB) = Pj(xA + tiA, xB + tiB) - (tiA + 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).

False-name-proof voting rule

for 2 alternatives

• FNP2: Suppose xA ≥ xB. Then

PA(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

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

QuickTime™ and a TIFF (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 a TIFF (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.

Discussion

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