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Theorems for Exchangeable Binary Random Variables with Applications Serguei Kaniovski November 19, 2010 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications

Overview Examples of expertise and power computations based on a 1 non-trivial probability distribution over the set of all voting outcomes The property of exchangeability as a stochastic model of a 2 representative agent (voter) Known parameterizations of the joint probability distribution of n 3 correlated binary random variables The probability of at least k successes in n correlated binary trials. 4 Some results for exchangeable random variables with vanishing higher-order correlations The bounds on this probability when the correlations are unknown 5 Application to the Condorcet Jury Theorem and voting power in the 6 sense of Penrose - Banzhaf Concluding remarks 7 Where to find my work 8 Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications

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

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 Penrose - Banzhaf measure of voting power Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications

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

The probability of at least k successes in n trials This probability finds wide application in reliability and decision theory 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

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 c=0.1 0.6 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

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

Voting power In a ‘one person, one vote’ election with two alternatives a vote is decisive n if it breaks a tie. With n + 1 voters, the probability of a tie equals C 2 n π n 2 For an odd n , c 3 = c 4 = · · · = c n = 0 and ( p , c ) ∈ B n n n n V k n ( p , 0) = C 2 2 (1 − p ) 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 ) + 2 p (1 − p ) − 1 4 C n p 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

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

Generating a joint probability distribution This is helpful in developing probabilistic voting scenarios The following regularization yields a non-exchangeable distribution � π v ≥ 0 , π v = 1 for all v ∈ V v ∈ V � π v = p i for all i = 1 , 2 , . . . , n v ∈ V ( i ) � � π v = p i p j + c i , j p i (1 − p i ) p j (1 − p j ) for all 1 ≤ i < j ≤ n v ∈ V ( i , j ) π v ) 2 � min π v 0 . 5 ( π v − ¯ The objective function v V ( i ) is the set of all v such that v i = 1, and V ( i , j ) = V ( i ) ∩ V ( j ) the set of all v such that v i = v j = 1. The quadratic optimization problem has 2 n variables and 1 + 2 n + C 2 n constraints. But the Condorcet probability does not depend on c i , j for this distribution (invariance) Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications

Conclusions: Condorcet Jury The effect of correlation on the jury’s competence is negative for voting rules close to simple majority and positive for voting rules close to unanimity If the individual competence is low, it may be better to hire one expert rather than several. In all other cases simple majority rule is the optimal decision rule. A jury operating under simple majority rule will not necessarily benefit from an enlargement, unless the enlargement is substantial. The higher the individual competence, the sooner an enlargement will be beneficial Correlation-robust voting rules minimizes the effect of correlation on collective competence by making it as close as possible to that of a jury of independent jurors. The optimal correlation-robust voting rule should be preferred to simple majority rule if mitigating the effect of correlation is more important than maximizing the accuracy of the collective decision For a given competence, compute the bounds to a jurys competence as the minimum and maximum probability of a jury being correct Serguei Kaniovski Theorems for Exchangeable Binary Random Variables with Applications