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: A Formal Model n agents: I = {1, …, n} vector of weights w = ( w 1, …, w n ): –integers in binary (unless stated otherwise) threshold (quota) T a coalition J is winning if  i in J w i ≥ T Notation: (w 1, …, w n ; T)

Measuring Power: 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 

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]:  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 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: w i ≤ w j implies  i ≤  j Can change individual player’s power: – (15, 20, 25; 30):  1 = 1/3 – (15, 20, 25; 45):  1 = 0 By how much?

Choosing the Quota: Bounds on Ratio Theorem: assume w 1 ≤ … ≤ w n. 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 ≤  n ≤ 1 –lower bound: (1, …, 1, n; T) T=1: everyone is equal T=n:  1 =…=  n-1 = 0,  n = 1

Choosing the Quota: Bounds on Difference Theorem: assume w 1 ≤ … ≤ w n. 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 ≤  i ≤ 1, 0 ≤  i ≤ 1/(n-i+1) –lower bound: (1, 2, 4, …, 2 n-1 ; T) T=2 i :  1 =…=  i = 0,  i =…=  n = 1/(n-i+1)

Vulnerability to Control Given T 1, T 2 and a player i, is T 1 better for i than T 2 ? 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 w 1 ≤ … ≤ w n, player 0 of weight w Claim: there is no value of quota that makes player 0 a dummy iff  i < t w i + w ≥ w t 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 ???