Manipulation and Control
in Weighted Voting Games
Based on:
Bachrach, Elkind,
AAMAS’08
Zuckerman, Faliszewski, Bachrach, Elkind,
AAAI’08
How Do You Measure
Political Power?
• Parties:
– A: 15 voters
– B: 20 voters
– C: 25 voters
• Need 50% (30 voters) to pass a bill
• Winning coalitions: AB, AC, BC, ABC
• A, B, and C have equal power
power ≠ weight!
Weighted Voting Games:
Formal Model
• n agents: I = {1, …, n}
• vector of weights w = ( w1, …, wn ):
– integers in binary (unless stated otherwise)
• threshold (quota) T
• a coalition J is winning if
• Notation: (w1, …, wn; T)
S i in J wi ≥ T
A
Measuring Power: Shapley Value
Shapley value of agent i:
fi = a fraction of all permutations of n
agents for which i is pivotal
p: ….. i …..
<T
≥T
Axioms:
• efficiency: Si fi = 1
• symmetry
• dummy
• additivity
Shapley (and me)
Dishonest Voters
(Bachrach, E., AAMAS’08)
• Can an agent increase his power by
splitting his weight between two identities?
• Example:
– [2, 2; 4]: f2 = 1/2 [2,1,1; 4]: f2 = f3 = 1/3
2/3 > 1/2 !
• Another example:
– [2, 2; 3]: f2 = 1/2 [2,1,1; 3]: f2 = f3 = 1/6
2/6 < 1/2 …
Effects of Manipulation:
Bad Guys Gain
• Theorem:
an agent can increase his power by a
factor of 2n/(n+1), and this bound is tight
• Proof:
– lower bound:
[2, …, 2; 2n] → [2, …, 1, 1; 2n]:
1/n → 2/(n+1)
– upper bound:
careful bookkeeping of permutations
Effects of Manipulation:
Bad Guys Lose
• Theorem:
an agent can decrease his power by a
factor of (n+1)/2, and this bound is tight
• Proof:
– l.b.: [2, …, 2; 2n-1] → [2, …, 1, 1; 2n-1]:
1/n → 2(n-1)!/(n+1)!
– u.b.: careful bookkeeping of permutations:
…. i ….
… i’ i’’ …
… i’’ i’ …
Computational Aspects
• WVGs are susceptible to manipulation,
but are they vulnerable?
• Shapley value is
– #P-hard if weights are given in binary
– poly-time if weights are given in unary
• Unary weights:
– try all splits, compute Shapley value: poly-time
• Binary weights:
– Theorem: it is NP-hard to check
if a beneficial split exists
Manipulation by the Center: Why?
• Government formation:
– central authority (speaker) wants to change
relative powers of some parties:
• e.g., extreme right-wing party in the parliament:
can we reduce its power to 0?
• EU parliament:
– there is an intuitive understanding of relative
importance of different members
– can we implement this understanding?
Changing the Quota
(ZFBE, AAAI’08)
• What can the center achieve by changing
the quota?
• Can’t change the order of players:
wi ≤ wj implies fi ≤ fj
• Can change individual player’s power:
– (15, 20, 25; 30): f1 = 1/3
– (15, 20, 25; 45): f1 = 0
• By how much?
Choosing the Quota:
Bounds on Ratio
• Theorem: assume w1 ≤ … ≤ wn.
By changing T,
– can change n’s power by a factor of n,
and this bound is tight
– for players 1, …, n-1, the power can go
from > 0 to 0 (no bound on ratio)
• Proof:
– upper bound: 1/n ≤ fn ≤ 1
– lower bound: (1, …, 1, n; T)
• T=1: everyone is equal
• T=n: f1 =…= fn-1 = 0, fn = 1
Choosing the Quota:
Bounds on Difference
• Theorem: assume w1 ≤ … ≤ wn.
By changing T,
– for n: can change the power by ≤ 1-1/n,
and this bound is tight
– for i < n: can change the power by ≤ 1/(n-i+1),
and this bound is tight
• Proof:
– upper bound: 1/n ≤ fi ≤ 1, 0 ≤ fi ≤ 1/(n-i+1)
– lower bound: (1, 2, 4, …, 2n-1; T)
• T=2i: f1 =…= fi = 0, fi =…= fn = 1/(n-i+1)
Vulnerability to Control
• Given T1, T2 and a player i,
is T1 better for i than T2?
• Unary weights: poly-time
• Binary weights: PP-complete
– L is in PP if there exists an NP-machine M s.t.
x L iff M accepts w.p. ≥ ½
• Barrier to manipulation
• Pinpointed the exact complexity
Making a Given Player a Dummy
• w1≤ … ≤ wn, player 0 of weight w
• Claim:
there is no value of quota
that makes player 0 a dummy iff
S i < t wi + w ≥ wt for any t = 1, …, n
• Can check in linear time!
Banzhaf Index: a Different Way of
Measuring Power
• Banzhaf index vs. Shapley value:
– counting sets vs. counting permutations
• Popular with political scientists
• Similar results hold:
– computation: everything carries over
– tight bounds:
• control: see paper
• manipulation: private communication by Aziz,
Paterson
Single-winner elections vs.
weighted voting
Single-winner elections:
• n voters, m candidates
each voter has a
preference order
• manipulation
– cheating by voters
• control
– cheating by center
• bribery
Weighted voting:
• n weighted voters,
threshold T
• manipulation
– weight splitting/merging
• control
– changing the threshold
– ???
• bribery ???