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Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory Serguei Kaniovski March 26, 2013 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 1 / 20

Overview Examples of expertise and power computations based on a non-trivial 1 probability distribution over the set of all voting outcomes The property of exchangeability as a stochastic model of a representative 2 agent (voter) Known parameterizations of the joint probability distribution of n correlated 3 binary random variables The probability of at least k successes in n correlated binary trials. The 4 generalized binomial distribution The bounds on this probability when the higher-order correlations are 5 unknown Application to the Condorcet Jury Theorem and voting power in the sense of 6 Penrose - Banzhaf Concluding remarks 7 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 2 / 20

Examples of voting under simple majority rule ( n = 3) p = 0 . 5 p = 0 . 75 p = 0 . 75 p 1 = 0 . 75 v = ( v 1 , v 2 , v 3 ) c = 0 c = 0 c = 0 . 2 p 2 , 3 = 0 . 6 c = 0 . 2 1 1 1 0.125 0.422 0.506 0.357 1 1 0 0.125 0.141 0.094 0.136 1 0 1 0.125 0.141 0.094 0.136 1 0 0 0.125 0.047 0.056 0.122 0 1 1 0.125 0.141 0.094 0.051 0 1 0 0.125 0.047 0.056 0.056 0 0 1 0.125 0.047 0.056 0.056 0 0 0 0.125 0.016 0.044 0.086 Condorcet probability 0.5 0.844 0.788 0.679 Banzhaf probability 1 0.5 0.376 0.3 0.384 Banzhaf probability 2 0.5 0.376 0.3 0.365 Banzhaf probability 3 0.5 0.376 0.3 0.365 Computing the probability of a correct verdict, or the voting power as the probability of casting a decisive vote, requires a joint probability distribution on the set of all voting profiles v ∈ R 2 n . The influence of voting weights and decision rule is separate from that of the distribution. Exchangeability leads to a representative agent model, in which the independence assumption is relaxed Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 3 / 20

Voting in the U.S. Supreme Court REHNQUIST WARREN 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 0.00 0.00 0 100 200 300 400 500 0 100 200 300 400 500 Empirical evidence overwhelmingly refutes the assumption of independent votes required in the classic versions of the Condorcet Jury Theorem and the Banzhaf measure of voting power Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 4 / 20

The joint probability distribution of n binary r.v. The Bahadur parametrization � Z i = ( V i − p i ) / p i (1 − p i ) for all i = 1 , 2 , . . . , n , p i = p c i , j = E ( Z i Z j ) for all 1 ≤ i < j ≤ n , c i , j = c c i , j , k = E ( Z i Z j Z k ) for all 1 ≤ i < j < k ≤ n , c i , j , k = c 3 . . . c 1 , 2 ,..., n = E ( Z 1 Z 2 . . . Z n ) , c 1 , 2 ,..., n = c n � � � � π v = ¯ π v 1 + c i , j z i z j + c i , j , k z i z j z k + · · · + c 1 , 2 ,..., n z 1 z 2 . . . z n i < j i < j < k n i (1 − p i ) (1 − v i ) is the probability under the independence � p v i where ¯ π v = i =1 The George - Bowman parametrization for exchangeable binary r.v. i ( − 1) j C j � π i = i λ n − i + j , where λ i = P ( X 1 = 1 , X 2 = 1 , . . . , X i = 1) , λ 0 = 1 j =0 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 5 / 20

The generalized binomial distribution This probability finds wide application in reliability and decision theory The probability of at least k successes in n correlated binary trials n � � P k n ( p , C ) = P k c i , j α i α j A k n ( p , I ) + i , j ( α ) , where p h h =1 1 ≤ i < j ≤ n  0 , k = 0   � � α 2 i 1 . . . α 2 α 2 j 1 . . . α 2  i n − k − j n − k − 1 , k = 1 , . . . , n − 1   A k i , j ( α ) = i s � = i , i s � = j j s � = i , j s � = j  1 ≤ i 1 < ··· < i n − k ≤ n 1 ≤ j 1 < ··· < j n − k − 1 ≤ n    1 , k = n  � Here p is the vector of marginal probabilities, α such that α i = (1 − p i ) / p i , C = ( c ij ) the n × n correlation matrix, I the n × n identity matrix In the following we will use a simpler formula, in which the r.v. are exchangeable and all higher-order correlations vanish. In this case the distribution is completely defined by n , p and the second-order correlation coefficient c Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 6 / 20

3-parameter generalized binomial distribution For an odd n , c 3 = c 4 = · · · = c n = 0 and ( p , c ) ∈ B n (Bahadur set) n n p t (1 − p ) n − t = I p ( k , n − k + 1) P k � C t n ( p , 0) = t = k � k − 1 � ∂ I p ( k , n − k + 1) P k n ( p , c ) = I p ( k , n − k + 1) + 0 . 5 c ( n − 1) n − 1 − p ∂ p where I x ( a , b ) is the regularized incomplete beta function Bounds on P k n , p for given n and p can be found by linear programming. Di Cecco provides bounds for given n , p and c such that ( p , c ) ∈ B n Bounds on P k n , p when all correlation coefficients are unknown � np − k + 1 � np � � ≤ P k max n − k + 1 , 0 n , p ( c , c 3 , . . . , c n ) ≤ min k , 1 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 7 / 20

The Condorcet probability and the Bahadur set The set B n contains all admissible values of c for given n and p , provided c 3 = c 4 = · · · = c n = 0 5 (p,c) Condorcet probability P 9 The upper bound on c for n=9 ( max c B 9 (p,c) ) 1.0 0.3 0.9 0.2 0.8 0.7 0.1 c=0.0 0.6 c=0.1 c=0.2 0.5 0.0 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 p ( p ≥ 0.5 ) p 1 2 B n is such that 0 < c < n − 1 for p ≈ 1 and 0 < c < n − 1 for p ≈ 0 . 5 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 8 / 20

Bounds on the probability of at least 5 successes in 9 trials All correlation coefficients are unknown Second−order correlation coefficient c=0.2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p p Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 9 / 20

Voting power In a ‘one person, one vote’ election with two alternatives a vote is decisive if it n breaks a tie. With n + 1 voters, the probability of a tie equals C n π n 2 2 For an odd n , c 3 = c 4 = · · · = c n = 0 and ( p , c ) ∈ B n n n n V k 2 2 (1 − p ) n ( p , 0) = C n p 2 � n (2 p − 1) 2 n ( p , 0) + nc n � n − 2 n − 2 V k n ( p , c ) = V k 2 2 (1 − p ) 4 C n p + 2 p (1 − p ) − 1 2 2 Bounds on V k n ( c , c 3 , . . . , c n ) when all correlation coefficients are unknown 0 ≤ V k n ( c , c 3 , . . . , c n ) ≤ 2 min { p , 1 − p } For given n , p and c , bounds can be found by linear programming Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 10 / 20

Bounds on voting power All correlation coefficients are unknown Second−order correlation coefficient c=0.2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p p Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 11 / 20

Penrose - Banzhaf and Straffin power measures A general formula for Total Criticality � � π T [ w ( T ∪ { i } ) − w ( T )] + π S [ w ( S ) − w ( S \ { i } )] S ⊆ N : i ∈ S T ⊂ N : i / ∈ T Bazhaf measure of voting power: Assumption: p = 0 . 5 and c = 0 1 π T = π S = 2 | N | Straffin measure based on Homogeneity assumption: Assumption: p ∼ U [0 , 1] (unconditionally Corr ( V i , V j ) = 1 / 3) π T = | T | !( | N | − | T | )! and π S = | S | !( | N | − | S | )! ( | N | + 1)! ( | N | + 1)! Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 12 / 20

The average Game Mean Discrepancy (GMD) n # Games # Stoch. Models Measure of Power per Game Penrose-Banzhaf Shapley-Shubik 2 5 1,0000 -0.002 -0.002 3 19 6,518 -0.048 -0.022 4 167 4,883 -0.104 -0.037 5 7,580 3,943 -0.139 -0.040 6 7,828,353 3,304 -0.154 -0.034 The negative mean GMD implies that, on average, any voting power analysis carried out using the standard techniques will apportion too much power to the players. The standard versions of the Banzhaf and Straffin measures overestimate power The Straffin measure is closer to the probability of being critical than the Banzhaf measure Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 13 / 20

The GMD for n = 6 0.20 0.20 Penrose−Banzhaf Penrose−Banzhaf Shapley−Shubik Shapley−Shubik 0.15 0.15 Relative frequency Relative frequency 0.10 0.10 0.05 0.05 0.00 0.00 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 Game Mean Discrepancy Game Mean Discrepancy The right panel shows the GMD for both measures. The left panel shows the GMD for the Straffin measure when the assumption of the binomial model hold. This is ideal for the Banzhaf measure Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications to Voting Theory 14 / 20