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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

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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

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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

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Weighted Voting Games: A Formal Model n agents: I = {1, …, n} vector of weights w = ( w 1, …, w n ): integers in binary (unless stated otherwise) threshold T a coalition J is winning if ∑ i J w i ≥ T value of a coalition: v(J)=1 if J wins, else v(J)=0 imputation: p = (p 1, …, p n ) s.t. p i ≥ 0, p 1 + … + p n = 1 notation: w(J) = ∑ i J w i, p(J) = ∑ i J p i

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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: w i ≥ w j implies i ≥ j

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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

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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

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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 …

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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

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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’ …

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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

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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

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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 ???

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Choosing the threshold: bounds on ratio Theorem: assume w 1 ≤ … ≤ w n. 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

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Choosing the threshold: bounds on difference Theorem: assume w 1 ≤ … ≤ w n. 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, …, 2 n-1 ) T=2 i : 1 =…= i = 0, i =…= n = 1/(n-i+1)

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Separating the players Suppose w i < w j We have i ≤ j Can we ensure i < j ? –yes: set T = w j Can we ensure i = j ? –yes: set T = w i

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Making a given player a dummy w, w 1 ≤ … ≤ w n Claim: player 0 is never a dummy iff i < t w i + w ≥ w t for any t = 1, …, n Proof: =>: if i < t w i + w < w t for some t, set T = w t <=: w1w1 w1+ w2w1+ w2 w 1 +… + w n T

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Computational complexity Given T 1, T 2 and a player i, is T 1 better for i than T 2 ? 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

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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

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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 p j > 0 p(J) < 1 J

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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

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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

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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

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Pseudopolynomial Algorithms Hardness reduction from Partition assumes large weights –recall: w i are given in binary, –poly-time algorithm runs in time poly (n, log max w i ) What if weights are small? –formally, the weights are given in unary –we are happy with algorithms that run in time poly (n, max w i )

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max C p 1 +…+ p n = 1 p i ≥ 0 for all i = 1, …, n ∑ i J p i ≥ 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

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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

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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 w i, 1/ ) (FPTAS) Proof idea: use FPTAS for Knapsack inside the separation oracle

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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): p 1 = (1/4, 1/4, 1/6, 1/6, 1/6) p 1 (ab) = ½, p 1 (cde) = ½, p 1 (J) > ½ for other winning coalitions p 2 = (1/3, 1/6, 1/6, 1/6, 1/6) p 2 (ab) = ½, p 2 (cde) = ½, p 2 (bcd) = … = p 2 (bde) = ½ p 2 (J) > ½ for other winning coalitions Some imputations in the least core are better than others…

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Nucleolus Given an imputation p, order all coalitions by p(J) v(J) (min to max) p(J 1 ) -v(J 1 ), p(J 2 ) -v(J 2 ), …, p(J 2 n ) - v(J 2 n ) 2 n numbers Nucleolus: the imputation x that corresponds to lexicographically maximal 2 n -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

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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

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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 w i < 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)

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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

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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 (w 1 i, …, w k i ) –vector of thresholds T 1, …, T k –J wins if w 1 (J) ≥ T 1, …, w k (J) ≥ T k Any simple game can be represented as a k-vector weighted voting game

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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) –T 1 =T 2 =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 –w 1 +w 2 ≥ T, w 3 +w 4 ≥ T –w 1 +w 3 < T, w 2 +w 4 < T contradiction!

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Equivalence k 1 -vector weighted voting game G 1 k 2 -vector weighted voting game G 2 Question: are G 1 and G 2 equivalent (have the same set of winning coalitions)? NP-hard –even if k 1 = k 2 = 1, or –even if all weights are in {0, 1}, but not both: pseudopolynomial algorithm for k < C

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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

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Minimality (2/2) k 1 -vector weighted voting game G 1 Question: is G 1 minimum, i.e., is there a k 2 -vector weighted voting game G 2 with k 2 < k 1 that is equivalent to G 1 ? NP-hard –even if k 1 = 2 pseudopolynomial algorithm for checking whether G 1 is equivalent to a weighted voting game (LP with a separation oracle)

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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

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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

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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

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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

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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

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