Manipulation, Control, and Beyond: Computational Issues in Weighted Voting Games
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Transcript of Manipulation, Control, and Beyond: Computational Issues in Weighted Voting Games
Manipulation, Control, and Beyond: Computational Issues in
Weighted Voting Games
Edith Elkind (U. of Southampton)Based on joint work with:
Y.Bachrach, G. Chalkiadakis, P. Faliszewski, N. R. Jennings, L. A. Goldberg, P. W. Goldberg,
M. Wooldridge, M. Zuckerman
Power of Coalitions
• Parties: A (45%), B (45%), C (10%)• Need 50% to pass a bill• Winning coalitions: AB, AC, BC, ABC• A, B, and C have equal power• Given a budget K, what is the right way
to distribute it between A, B, and C?– by weight: 0.45K, 0.45K, 0.1K – by power: K/3, K/3, K/3
Applications• Systems of self-interested agents
working together to achieve a goal• Agent’s contribution = weight• Suppose the goal is achieved
– agents have generated a surplus– how should they divide it?– one possible answer:
according to voting power
Weighted Voting Games: A Formal Model
• n agents: I = {1, …, n}• vector of weights w = ( w1, …, wn ):
integers in binary (unless stated otherwise)
• threshold T• a coalition J is winning if ∑ iJ wi ≥ T• value of a coalition: v(J)=1 if J wins, else v(J)=0 • imputation:
p = (p1, …, pn) s.t. pi ≥ 0, p1 + … + pn = 1
• notation: w(J) = ∑iJ wi , p(J) = ∑iJ pi
The “most fair” imputation: Shapley value
Shapley value of agent i: i = a fraction of all permutations of n agents for which i is pivotal
….. i …..< T
≥ T
Axioms:• efficiency: i i = 1• symmetry• dummy• additivity
monotonicity: wi ≥ wj implies i ≥ j
Plan of the talk
• Manipulation of Shapley value– by voters: weight-splitting– by center: changing the threshold
• Stability in weighted voting games– -core, least core, nucleolus
• Vector voting games
Plan of the talk
• Manipulation of Shapley value– by voters: weight-splitting– by center: changing the threshold
• Stability in weighted voting games– -core, least core, nucleolus
• Vector voting games
Dishonest voters (Bachrach, E., AAMAS’08)
• Can an agent increase his power by splitting his weight between two identities?
• Example: – [2, 2; 4]: 2 = 1/2 [2,1,1; 4]: 2 = 3 = 1/3
2/3 > 1/2 !• Another example:
– [2, 2; 3]: 2 = 1/2 [2,1,1; 3]: 2 = 3 = 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
• Is computational hardness a barrier to manipulation?
• Theorem: it is NP-hard to check if a beneficial split exists
• computing Shapley value is #P-hard anyway…– central authority may have more
computational resources than a single agent– an agent may want to increase his power
even if he cannot compute it
Plan of the talk
• Manipulation of Shapley value– by voters: weight-splitting– by center: changing the threshold
• Stability in weighted voting games– -core, least core, nucleolus
• Vector voting games
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 ???
Choosing the threshold: bounds on ratio
• Theorem: assume w1≤ … ≤ wn. 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 ≤ n ≤ 1– lower bound: (1, …, 1, n)
• T=1: everyone is equal• T=n: 1 =…= n-1 = 0, n = 1
Choosing the threshold: bounds on difference
• Theorem: assume w1≤ … ≤ wn. 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 ≤ i ≤ 1, 0 ≤ i ≤ 1/(n-i+1)– lower bound: (1, 2, 4, …, 2n-1)
• T=2i: 1 =…= i = 0, i =…= n = 1/(n-i+1)
Separating the players
• Suppose wi < wj
• We have i ≤ j
• Can we ensure i < j ?– yes: set T = wj
• Can we ensure i = j?– yes: set T = wi
Making a given player a dummy
• w, w1≤ … ≤ wn
• Claim: player 0 is never a dummy iff i < t wi + w ≥ wt for any t = 1, …, n
• Proof:=>: if i < t wi + w < wt for some t, set T = wt
<=: w1 w1+ w2 w1+… + wn
T
Computational complexity
• Given T1, T2 and a player i, is T1 better for i than T2?
• NP-hard (reduction from “is i a dummy?”)• 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
Plan of the talk
• Manipulation of Shapley value– by voters: weight-splitting– by center: changing the threshold
• Stability in weighted voting games– -core, least core, nucleolus
• Vector voting games
Good imputations: other criteria
• Fairness: Shapley value, Banzhaf power index
• Stability: core– p is in the core if for any J we have p(J) ≥ v(J)– core can be empty
Lemma: the core is empty <=> no player belongs to all winning coalitions pj > 0
p(J) < 1J
Relaxing the Notion of the Core• - core: p is in the - core iff for any J
p(J) ≥ v(J) - – each winning coalition gets at least 1- – nonempty for large enough (e.g., = 1)
• least core: smallest non-empty -core• if least core = - core
– there is a p s.t. p(J) ≥ 1 - for any winning J– for any ’ there is no p
s.t. p(J) ≥ 1 - ’ for any winning J
Computational Issues:Our Results (E., Goldberg, Goldberg, Wooldridge, AAAI’07)
• Is the core non-empty? – poly-time: use the lemma
• Is the -core non-empty?
• Is a given imputation p in the -core?
• Is a given imputation p in the least core?
• Construct an imputation in the least core. – p
Given a weighted voting game (I; w; T)
reductions from Partition
Computational Issues:Our Results (E., Goldberg, Goldberg, Wooldridge, AAAI’07)
• Is the core non-empty? – poly-time: use the lemma
• Is the -core non-empty? – coNP-hard
• Is a given imputation p in the -core? – coNP-hard
• Is a given imputation p in the least core? – NP-hard
• Construct an imputation in the least core. – NP-hard
Given a weighted voting game (I; w; T)
reductions from Partition
Pseudopolynomial Algorithms
• Hardness reduction from Partition assumes large weights– recall: wi are given in binary, – poly-time algorithm <=>
runs in time poly (n, log max wi)
• What if weights are small?– formally, the weights are given in unary– we are happy with algorithms
that run in time poly (n, max wi)
max Cp1+…+ pn = 1
pi ≥ 0 for all i = 1, …, n
∑ iJ pi ≥ C for any J s.t. w(J) ≥ T
linear program exponentially many ineqs
Claim: least core = (1 - C)-core
Algorithm For the Least Core
LPs and Separation Oracles
• Separation oracle: – input: (p, C)– output: “yes” if (p, C) satisfies the LP,
violated constraint otherwise• Claim: LPs with poly-time separation oracles
can be solved in poly-time.• Our case: given (p, C),
is there a J with w(J) ≥ T, p(J) < C?– reduces to Knapsack => solvable in pseudopoly time
• Works for other problems listed above
Approximation Algorithms
• Back to large weights…• Theorem: suppose least core = -core.
Then for any we can compute ’ s.t. ’ and ’-core is non-empty in time poly (n, log max wi, 1/)
(FPTAS) • Proof idea: use FPTAS for Knapsack inside
the separation oracle
Refining the Notion of Least Core
I = {a, b, c, d, e}, w = {3, 3, 2, 2, 2}, T = 6– 2 disjoint winning coalitions => ≥ ½– ½-core not empty (hence least core = ½-core):
• p1 = (1/4, 1/4, 1/6, 1/6, 1/6)
p1(ab) = ½, p1(cde) = ½ ,
p1(J) > ½ for other winning coalitions
• p2 = (1/3, 1/6, 1/6, 1/6, 1/6)
p2(ab) = ½, p2(cde) = ½ , p2(bcd) = … = p2(bde) = ½
p2(J) > ½ for other winning coalitionsSome imputations in the least core are better than others…
Nucleolus
• Given an imputation p, order all coalitions by p(J) v(J) (min to max)
p(J1) -v(J1), p(J2) -v(J2), …, p(J2n) - v(J2n) 2n numbers
• Nucleolus: the imputation x that corresponds to lexicographically maximal 2n-vector – always in the least core
winning coalitions:v(J) = 1, p(J) v(J) ≤ 0
losing coalitions:v(J) = 0, p(J) v(J) ≥ 0
Computing Nucleolus
• Can be computed by solving n sequential exp-size linear programs (similar to LP for the least core)
• Our result: NP-hard to compute– not clear if it is in NP
• Pseudopolynomial algorithm for the least core does not seem to generalize
• Approximation algorithm for the least core does not seem to generalize
Computing Nucleolus: Positive Results
• Can we approximate nucleolus payoffs by (normalized) weights?
• NO for individual players: x(i) / w(i) can be arbitrarily small or arbitrarily large.
• YES for coalitions: Theorem: Suppose wi < T for all i, T ≥ ½.
Then 1/2 ≤ x(J) / w(J) ≤ 2 for all coalitions J. Also 1/2 ≤ x(J) / w(J) ≤ 3 for a larger class of games
(see the paper)
Plan of the talk
• Manipulation of Shapley value– by voters: weight-splitting– by center: changing the threshold
• Stability in weighted voting games– -core, least core, nucleolus
• Vector voting games
k-vector weighted voting games
• EU: to pass a bill, need– a certain number of countries AND– a certain fraction of EU population
• Formal model:– voter i has a vector of weights (w1
i, …, wki)
– vector of thresholds T1, …, Tk
– J wins if w1(J) ≥ T1, …, wk(J) ≥ Tk
• Any simple game can be represented as a k-vector weighted voting game
k-vector weighted voting games vs weighted voting games
• 2-vector weighted voting game G– 4 players (9, 1), (1, 9), (5, 5), (5, 5)– T1=T2=10– {1, 2} and {3, 4} are winning coalitions– {1, 3} and {2, 4} are losing coalitions
• G not equivalent to any weighted voting game– w1+w2 ≥ T, w3+w4 ≥ T– w1+w3 < T, w2+w4 < T
contradiction!
Equivalence
• k1-vector weighted voting game G1
• k2-vector weighted voting game G2
• Question: are G1 and G2 equivalent (have the same set of winning coalitions)?
• NP-hard– even if k1 = k2 = 1, or– even if all weights are in {0, 1}, but not both:
• pseudopolynomial algorithm for k < C
Minimality (1/2)
• k-vector weighted voting game G • Question: is G minimal, i.e.,
are all coordinates necessary?• NP-hard
– even if k = 2, or– even if all weights are in {0, 1}, but not both:
• pseudopolynomial algorithm for k < C
Minimality (2/2)• k1-vector weighted voting game G1 • Question: is G1 minimum, i.e., is there
a k2-vector weighted voting game G2 with k2 < k1 that is equivalent to G1?
• NP-hard– even if k1 = 2
• pseudopolynomial algorithm for checking whether G1 is equivalent to a weighted voting game (LP with a separation oracle)
Coalition Structures in Weighted Voting Games
• The work so far assumed that the grand coalition will form– Shapley value: dividing a unit of profit– core: stability of the grand coalition
• What if several winning coalitions can form simultaneously?– plausible if T < w(I)/2
• Coalition structures: partitions of I
CS in WVGs: definitions
• coalition structire: CS = {C1, …, Ck}• imputation for CS: p s.t. p(Ci) = 1 if
w(Ci) ≥ T, p(Ci) = 0 if w(Ci) < T– can be p(I)>1
• outcome: pair (CS, p)• CS-core: (CS, p) is in the CS-core if
– p is an imputation for CS– for any S w(S) ≥ T implies p(S) ≥ 1
Results
• coNP-complete to check if an outcome is in the CS-core– pseudopolynomial algorithm:
reduction to KNAPSACK• NP-hard to check if the CS-core is non-empty
– pseudopolynomial algorithm to check if, for given CS, there is a p s.t. (CS, p) is in the CS-core
• linear program with KNAPSACK-based separation oracle
– can enumerate all coalitions, check if there is an appropriate imputation
• heuristics to speed up this process
Conclusions• Weighted voting games:
a model for multiagent systems• Shapley value:
well understood, but can be manipulated • Other solution concepts (-core, least core,
nucleolus): progress on understanding computational complexity
• k-vector weighted voting games– complexity of equivalence and minimality
Extensions and open problems
• False-name voting– split into more than two identities– Manipulation by merging
• how do you distribute gains from manipulation?– Banzhaf power index
• Bribery in weighted voting games• Computational complexity of the nucleolus