Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions, voting power, and political instability
Andrew Gelman
Department of Statistics and Department of Political Science
Columbia University
25 Oct 2005
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Topics
I
Coalition-formation as a prisoner’s dilemma: a potential
theoretical explanation for political instability
I
Mathematical and statistical models of voting power
Both topics involve:
I
I
I
I
Open problems in mathematics (probability theory, cellular
automata)
Open problems in political science (connections to empirical
data)
Collaboration with Francis Tuerlinckx, Joe Bafumi, and
Jonathan Katz
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions and political instability
First part of the talk
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalitions in an election with 9 voters
I
Several possibilities:
I
I
I
I
I
No coalitions
Single coalition of 9 voters
Single coalition of 3 voters
Three coalitions of 3 voters each
Compute Pr (voter is decisive) for:
I
I
I
voters in a coalition
voters outside the coalition
average over all 9 voters
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
“Voting power” vs. “satisfaction”
I
Voting power: Pr (voter is decisive)
I
Satisfaction: Pr (your desired outcome wins)
Not the same:
I
I
I
Consider an election in which 90% of the voters vote for A and
10% vote for B:
Almost everyone is satisfied but voters have essentially no
power.
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
“Voting power” vs. “satisfaction”
I
Voting power: Pr (voter is decisive)
I
Satisfaction: Pr (your desired outcome wins)
Not the same:
I
I
I
Consider an election in which 90% of the voters vote for A and
10% vote for B:
Almost everyone is satisfied but voters have essentially no
power.
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
“Voting power” vs. “satisfaction”
I
Voting power: Pr (voter is decisive)
I
Satisfaction: Pr (your desired outcome wins)
Not the same:
I
I
I
Consider an election in which 90% of the voters vote for A and
10% vote for B:
Almost everyone is satisfied but voters have essentially no
power.
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
“Voting power” vs. “satisfaction”
I
Voting power: Pr (voter is decisive)
I
Satisfaction: Pr (your desired outcome wins)
Not the same:
I
I
I
Consider an election in which 90% of the voters vote for A and
10% vote for B:
Almost everyone is satisfied but voters have essentially no
power.
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
“Voting power” vs. “satisfaction”
I
Voting power: Pr (voter is decisive)
I
Satisfaction: Pr (your desired outcome wins)
Not the same:
I
I
I
Consider an election in which 90% of the voters vote for A and
10% vote for B:
Almost everyone is satisfied but voters have essentially no
power.
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
“Voting power” vs. “satisfaction”
I
Voting power: Pr (voter is decisive)
I
Satisfaction: Pr (your desired outcome wins)
Not the same:
I
I
I
Consider an election in which 90% of the voters vote for A and
10% vote for B:
Almost everyone is satisfied but voters have essentially no
power.
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a “prisoner’s dilemma”
Payoffs (in terms of your own voting power) from coalition:
Your option
Stay alone
Join a coalition
Have other voters
formed coalitions?
No
Yes
Moderate Very low
High
Low
I
Joining a coalition increases your voting power
I
But if all voters form a coalition, they all have low voting
power
I
Suboptimal outcome if all players act rationally
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a “prisoner’s dilemma”
Payoffs (in terms of your own voting power) from coalition:
Your option
Stay alone
Join a coalition
Have other voters
formed coalitions?
No
Yes
Moderate Very low
High
Low
I
Joining a coalition increases your voting power
I
But if all voters form a coalition, they all have low voting
power
I
Suboptimal outcome if all players act rationally
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a “prisoner’s dilemma”
Payoffs (in terms of your own voting power) from coalition:
Your option
Stay alone
Join a coalition
Have other voters
formed coalitions?
No
Yes
Moderate Very low
High
Low
I
Joining a coalition increases your voting power
I
But if all voters form a coalition, they all have low voting
power
I
Suboptimal outcome if all players act rationally
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a “prisoner’s dilemma”
Payoffs (in terms of your own voting power) from coalition:
Your option
Stay alone
Join a coalition
Have other voters
formed coalitions?
No
Yes
Moderate Very low
High
Low
I
Joining a coalition increases your voting power
I
But if all voters form a coalition, they all have low voting
power
I
Suboptimal outcome if all players act rationally
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Voting power is not a zero-sum game!
I
I
Consider 2 extremes
Everyone votes
I
I
I
One voter is chosen at random and gets to decide the
outcome
I
I
I
Pr (decisive vote) = Pr (election is tied)
Assuming
random voting, this probability is proportional to
√
1/ n
Pr (decisive vote) = 1 for that person and 0 for all others
Avg Pr (decisive vote) = 1/n
But both systems are “fair”!
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
n=3
n=5
n=7
n=9
n = 11
n = 13
n = 17
n = 21
0.0
0.1
Pr (decisive vote)
0.2 0.3 0.4
0.5
Voting power if you are in a coalition of size m in an
electorate of size n
0
5
10
15
coalition size, m
Andrew Gelman
20
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
n = 21
n = 31
n = 51
n = 101
n = 201
n = 501
n = 1001
0.0
Pr (decisive vote)
0.1
0.2
0.3
Voting power if you are in a coalition of size m in an
electorate of size n (for larger populations)
0
200
400
600
coalition size, m
Andrew Gelman
800
1000
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Optimal coalition size m in an electorate of size n
I
I
√
Using combinatorics, derive that optimal m is approx 1.4 n
Optimal coalition sizes:
I
I
I
I
I
I
n = 10 (faculty committee or student club): mopt = 4
n = 100 (U.S. Senate): mopt = 14
n = 435 (House of Representatives): mopt = 30
n = 5,000,000 (Pennsylvania): mopt = 3,000
n = 100,000,000 (United States): mopt = 14,000
Voting power in optimal coalition is approx 0.57n−1/4
(compared to 0.80n−1/2 if there is no coalition)
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
But . . .
I
Your voting power if there are no coalitions is 0.80n−1/2
I
Your voting power if you’re in an optimal coalition is
0.57n−1/4
I
But . . . your voting power if everyone divides into optimal
coalitions is 0.65n−1/2
I
Example of the prisoner’s dilemma of coalition formation
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
But . . .
I
Your voting power if there are no coalitions is 0.80n−1/2
I
Your voting power if you’re in an optimal coalition is
0.57n−1/4
I
But . . . your voting power if everyone divides into optimal
coalitions is 0.65n−1/2
I
Example of the prisoner’s dilemma of coalition formation
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
But . . .
I
Your voting power if there are no coalitions is 0.80n−1/2
I
Your voting power if you’re in an optimal coalition is
0.57n−1/4
I
But . . . your voting power if everyone divides into optimal
coalitions is 0.65n−1/2
I
Example of the prisoner’s dilemma of coalition formation
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
But . . .
I
Your voting power if there are no coalitions is 0.80n−1/2
I
Your voting power if you’re in an optimal coalition is
0.57n−1/4
I
But . . . your voting power if everyone divides into optimal
coalitions is 0.65n−1/2
I
Example of the prisoner’s dilemma of coalition formation
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
But . . .
I
Your voting power if there are no coalitions is 0.80n−1/2
I
Your voting power if you’re in an optimal coalition is
0.57n−1/4
I
But . . . your voting power if everyone divides into optimal
coalitions is 0.65n−1/2
I
Example of the prisoner’s dilemma of coalition formation
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Coalition-formation as a random walk in the space of trees
I
I
Picture of a coalition structure as a tree
Possible moves in tree-space:
I
I
I
I
I
Several voters forming a coalition
A coalition disbanding or dividing into sub-coalitions
A set of coalitions forming a super-coalition
A set of coalitions merging into a single coalition
Restrict to locally beneficial moves: Pr (decisive) must
increase for all voters who are involved in the decision
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
A potential explanation for political instability
I
Locally beneficial moves are nontransitive
I
Similar to cartels in economics
I
How easily can an individual voter compute ∆ Pr (decisive),
in order to decide whether a particular move is a good idea?
I
Approximate calculations (similar to “expected utility”
calculations for economic actors)
I
Also, as with economics, can imagine reaching a global
optimum using side payments
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
A potential explanation for political instability
I
Locally beneficial moves are nontransitive
I
Similar to cartels in economics
I
How easily can an individual voter compute ∆ Pr (decisive),
in order to decide whether a particular move is a good idea?
I
Approximate calculations (similar to “expected utility”
calculations for economic actors)
I
Also, as with economics, can imagine reaching a global
optimum using side payments
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
A potential explanation for political instability
I
Locally beneficial moves are nontransitive
I
Similar to cartels in economics
I
How easily can an individual voter compute ∆ Pr (decisive),
in order to decide whether a particular move is a good idea?
I
Approximate calculations (similar to “expected utility”
calculations for economic actors)
I
Also, as with economics, can imagine reaching a global
optimum using side payments
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
A potential explanation for political instability
I
Locally beneficial moves are nontransitive
I
Similar to cartels in economics
I
How easily can an individual voter compute ∆ Pr (decisive),
in order to decide whether a particular move is a good idea?
I
Approximate calculations (similar to “expected utility”
calculations for economic actors)
I
Also, as with economics, can imagine reaching a global
optimum using side payments
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
A potential explanation for political instability
I
Locally beneficial moves are nontransitive
I
Similar to cartels in economics
I
How easily can an individual voter compute ∆ Pr (decisive),
in order to decide whether a particular move is a good idea?
I
Approximate calculations (similar to “expected utility”
calculations for economic actors)
I
Also, as with economics, can imagine reaching a global
optimum using side payments
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
A potential explanation for political instability
I
Locally beneficial moves are nontransitive
I
Similar to cartels in economics
I
How easily can an individual voter compute ∆ Pr (decisive),
in order to decide whether a particular move is a good idea?
I
Approximate calculations (similar to “expected utility”
calculations for economic actors)
I
Also, as with economics, can imagine reaching a global
optimum using side payments
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Coalitions and voting power
Optimal coalition sizes
Political instability
Open problems
Open problems
I
Conjecture: the coalition-formation process is inherently
unstable for n > 3
I
Local calculations of changes in voting power
Generalizing the model:
I
I
I
I
I
I
Weighted voting
Network structure of voters
Unequal probabilities pi for individual voters
Different preferences for different issues
Fitting to real data (legislative committees, courts, political
parties, . . . )
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Mathematical and statistical models of voting power
Second part of the talk
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power in a 2-level electoral system
I
I
U.S. Electoral College, or E.U. Council of Ministers
Electoral college:
I
I
I
nj voters in state j
ej electoral votes contribute to U.S. total
Your vote is decisive if:
I
I
Your state is tied, and
Your state’s electoral votes are decisive, if your state is tied
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Your voting power in a presidential election
I
Voting power = Pr (your vote is decisive):
I
I
I
Pr (your state is tied) ×
Pr (your state’s electoral votes are decisive, if your state is tied)
Simple calculation based on random voting model:
I
I
√
Pr (your state is tied) ∝ 1/ nj
Pr (your state’s electoral votes are decisive) ∝ ej
I
√
Combining: voting power ∝ ej / nj
I
According to this model, higher voting power in large states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Random voting model
I
I
I
I
Voters are flipping coins
√
Implies that voting power ∝ ej / nj
Focus on voting power as a function of nj
Key prediction of random voting model:
I
I
Large elections should be much closer (in percentage terms)
than small elections
“Law of large numbers”
I
Classical voting power calculations can also be defined
combinatorially
(Mathematically, same as the random voting model)
I
But does the model describe reality?
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
Checking the random voting model
.
.
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.
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lowess fit
alpha = -0.16
alpha = -0.5
2*10^6
6*10^6
10^7
Total vote for the two leading candidates
Andrew Gelman
Coalitions, voting power, and political instability
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
More checks of the random voting model
.
.
.
.
.
alpha = -0.23
lowess fit
alpha = -0.5
0
4*10^6
8*10^6
1.2*10^7
Total vote for the two leading candidates
Proportional vote differential
0.2
0.4
0.6
0.8
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.
..
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lowess fit
.
0.0
alpha = -0.5
Proportional vote differential
0.2
0.4
0.6
0.8
alpha = 0.06
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lowess fit
alpha = -0.11
. .
alpha = -0.5
.
0 50000
150000
250000
Total vote for the two leading candidates
0
200000
400000
600000
Total vote for the two leading candidates
U.S. statewide offices
European national elections
U.S. Senate elections
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alpha = -0.01
.
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lowess fit
= -0.5
Proportional vote differential
0.2
0.4
0.6
0.8
0
20000
60000
100000
Total vote for the two leading candidates
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. . alpha
0
4*10^6
8*10^6
1.4*10^7
Total vote for the two leading candidates
Andrew Gelman
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
U.S. state senate elections
0.0
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
U.S. state house elections
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alpha = -0.07
.
. . ..
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. ..
. . .. ..
lowess fit
.
..
.
.. . . .
alpha = -0.5
0
10^7
3*10^7
Total vote for the two leading candidates
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Voting power by state, 2000 (math and stat models)
empirical model
Pr (individual voter is decisive)
0.00004
0.00008
0.00012
random voting model
CA
Pr (individual voter is decisive)
4*10^-8
10^-7
NY
TX
PA FL
OHIL
MI
WV
AR
AZ
VTNV
NH
IA
ME
DE
0
10
20
30
40
Number of electoral votes for the state
50
OH
TN
OR
FL
PA
MO
WI
MI
MN
LA
WA
VA
CO
HI
SD
GA
NC
MS
SC
KYAL
ND
AK
MT
IL
CA
IN
NJ
CT MD
WYRI
KS OK
NE
ID
DC UT
0
0.0
NJ
NC
INGA
VA
TN
MA
MD
MO
OKAL
MS
DC
WI
AZLA WA
CT
AKRINM SC
WY
KY MN
HIWV
ARIACO
NE
OR
VTIDUTKS
ND
NH
SD
DE
NV
ME
MT
NM
0
MA
TXNY
10
20
30
40
Number of electoral votes for the state
Voters in large states do not have an electoral college advantage!
Andrew Gelman
Coalitions, voting power, and political instability
50
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic models using forecasting
I
A voter in Ohio or Florida has more voting power (is more
likely to cast a decisive vote) than a voter in Utah or Texas
I
Estimate Pr (decisive vote) in each state and each year using
hierarchical time-series cross-sectional forecasting model
I
Empirical comparisons of voting power in large and small
states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic models using forecasting
I
A voter in Ohio or Florida has more voting power (is more
likely to cast a decisive vote) than a voter in Utah or Texas
I
Estimate Pr (decisive vote) in each state and each year using
hierarchical time-series cross-sectional forecasting model
I
Empirical comparisons of voting power in large and small
states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic models using forecasting
I
A voter in Ohio or Florida has more voting power (is more
likely to cast a decisive vote) than a voter in Utah or Texas
I
Estimate Pr (decisive vote) in each state and each year using
hierarchical time-series cross-sectional forecasting model
I
Empirical comparisons of voting power in large and small
states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic models using forecasting
I
A voter in Ohio or Florida has more voting power (is more
likely to cast a decisive vote) than a voter in Utah or Texas
I
Estimate Pr (decisive vote) in each state and each year using
hierarchical time-series cross-sectional forecasting model
I
Empirical comparisons of voting power in large and small
states
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic probability models
I
Simplest model: votes are independent with probability p
instead of 1/2
I
I
This model is useless: it still predicts a standard deviation that
is tiny when n is large
We must take the next step and allow votes to be correlated
I
Probability models on trees (voters within neighborhoods
within cities within states within regions within a country)
I
Two tree models: Ising model and latent Gaussian model
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic probability models
I
Simplest model: votes are independent with probability p
instead of 1/2
I
I
This model is useless: it still predicts a standard deviation that
is tiny when n is large
We must take the next step and allow votes to be correlated
I
Probability models on trees (voters within neighborhoods
within cities within states within regions within a country)
I
Two tree models: Ising model and latent Gaussian model
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic probability models
I
Simplest model: votes are independent with probability p
instead of 1/2
I
I
This model is useless: it still predicts a standard deviation that
is tiny when n is large
We must take the next step and allow votes to be correlated
I
Probability models on trees (voters within neighborhoods
within cities within states within regions within a country)
I
Two tree models: Ising model and latent Gaussian model
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic probability models
I
Simplest model: votes are independent with probability p
instead of 1/2
I
I
This model is useless: it still predicts a standard deviation that
is tiny when n is large
We must take the next step and allow votes to be correlated
I
Probability models on trees (voters within neighborhoods
within cities within states within regions within a country)
I
Two tree models: Ising model and latent Gaussian model
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic probability models
I
Simplest model: votes are independent with probability p
instead of 1/2
I
I
This model is useless: it still predicts a standard deviation that
is tiny when n is large
We must take the next step and allow votes to be correlated
I
Probability models on trees (voters within neighborhoods
within cities within states within regions within a country)
I
Two tree models: Ising model and latent Gaussian model
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
More realistic probability models
I
Simplest model: votes are independent with probability p
instead of 1/2
I
I
This model is useless: it still predicts a standard deviation that
is tiny when n is large
We must take the next step and allow votes to be correlated
I
Probability models on trees (voters within neighborhoods
within cities within states within regions within a country)
I
Two tree models: Ising model and latent Gaussian model
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Ising model on a tree of voters
I
Put a 1 or −1 at each node of the tree; correlations along
branches
I
Look at the implied probability distribution for V n , the
average of n voters
I
Standard deviation of V n has the form cn−α
I
Random voting model implies α = 0.5; actual data fit α = 0.1
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Ising model on a tree of voters
I
Put a 1 or −1 at each node of the tree; correlations along
branches
I
Look at the implied probability distribution for V n , the
average of n voters
I
Standard deviation of V n has the form cn−α
I
Random voting model implies α = 0.5; actual data fit α = 0.1
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Ising model on a tree of voters
I
Put a 1 or −1 at each node of the tree; correlations along
branches
I
Look at the implied probability distribution for V n , the
average of n voters
I
Standard deviation of V n has the form cn−α
I
Random voting model implies α = 0.5; actual data fit α = 0.1
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Ising model on a tree of voters
I
Put a 1 or −1 at each node of the tree; correlations along
branches
I
Look at the implied probability distribution for V n , the
average of n voters
I
Standard deviation of V n has the form cn−α
I
Random voting model implies α = 0.5; actual data fit α = 0.1
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Ising model on a tree of voters
I
Put a 1 or −1 at each node of the tree; correlations along
branches
I
Look at the implied probability distribution for V n , the
average of n voters
I
Standard deviation of V n has the form cn−α
I
Random voting model implies α = 0.5; actual data fit α = 0.1
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
Estimating the parameter α
.
.
.
.
.
.
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0
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lowess fit
alpha = -0.16
alpha = -0.5
2*10^6
6*10^6
10^7
Total vote for the two leading candidates
Andrew Gelman
Coalitions, voting power, and political instability
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Estimating the parameter α
.
.
.
.
.
alpha = -0.23
lowess fit
alpha = -0.5
0
4*10^6
8*10^6
1.2*10^7
Total vote for the two leading candidates
Proportional vote differential
0.2
0.4
0.6
0.8
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0.0
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0.2
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0.0
.
..
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lowess fit
.
0.0
alpha = -0.5
Proportional vote differential
0.2
0.4
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alpha = 0.06
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lowess fit
alpha = -0.11
. .
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.
0 50000
150000
250000
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0
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400000
600000
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U.S. statewide offices
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.
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alpha = -0.01
.
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lowess fit
= -0.5
Proportional vote differential
0.2
0.4
0.6
0.8
0
20000
60000
100000
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. . alpha
0
4*10^6
8*10^6
1.4*10^7
Total vote for the two leading candidates
Andrew Gelman
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
U.S. state senate elections
0.0
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
U.S. state house elections
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.
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lowess fit
.
..
.
.. . . .
alpha = -0.5
0
10^7
3*10^7
Total vote for the two leading candidates
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Latent Gaussian model on a tree of voters
I
Put a continuous value at each node of the tree; random
variation added as you go down the tree
I
I
Look at the implied probability distribution for V n , the
average of n voters
√
Standard deviation of V n has the form a − b log n
I
Also fits the data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Latent Gaussian model on a tree of voters
I
Put a continuous value at each node of the tree; random
variation added as you go down the tree
I
I
Look at the implied probability distribution for V n , the
average of n voters
√
Standard deviation of V n has the form a − b log n
I
Also fits the data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Latent Gaussian model on a tree of voters
I
Put a continuous value at each node of the tree; random
variation added as you go down the tree
I
I
Look at the implied probability distribution for V n , the
average of n voters
√
Standard deviation of V n has the form a − b log n
I
Also fits the data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Latent Gaussian model on a tree of voters
I
Put a continuous value at each node of the tree; random
variation added as you go down the tree
I
I
Look at the implied probability distribution for V n , the
average of n voters
√
Standard deviation of V n has the form a − b log n
I
Also fits the data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Latent Gaussian model on a tree of voters
I
Put a continuous value at each node of the tree; random
variation added as you go down the tree
I
I
Look at the implied probability distribution for V n , the
average of n voters
√
Standard deviation of V n has the form a − b log n
I
Also fits the data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
√
0.0
Proportional vote differential
0.2
0.4
0.6
0.8
Comparing the models cn−α and
a − b log n
.
.
.
.
.
.
. ..
.
.
..
.
.
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.
.............. ... .......... ........... . ... . .. .. . . . .. .. .... . .
.. .
. . .. . .
. . . .
0
.
.
.
.
.
.
.
alpha = -0.16
b = 0.01
.
2*10^6
6*10^6
10^7
Total vote for the two leading candidates
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Open problems
I
General statements about voting power (for example, under
what conditions are proportional voting systems roughly
“fair”?)
I
Fitting more realistic tree models to electoral data
I
Fitting similar models to other data (for example, 5-level
survey of smoking in India)
I
Network models
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Open problems
I
General statements about voting power (for example, under
what conditions are proportional voting systems roughly
“fair”?)
I
Fitting more realistic tree models to electoral data
I
Fitting similar models to other data (for example, 5-level
survey of smoking in India)
I
Network models
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Open problems
I
General statements about voting power (for example, under
what conditions are proportional voting systems roughly
“fair”?)
I
Fitting more realistic tree models to electoral data
I
Fitting similar models to other data (for example, 5-level
survey of smoking in India)
I
Network models
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Open problems
I
General statements about voting power (for example, under
what conditions are proportional voting systems roughly
“fair”?)
I
Fitting more realistic tree models to electoral data
I
Fitting similar models to other data (for example, 5-level
survey of smoking in India)
I
Network models
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Standard voting power measures
Data from real elections
More complex mathematical/statistical modeling
Open problems
Open problems
I
General statements about voting power (for example, under
what conditions are proportional voting systems roughly
“fair”?)
I
Fitting more realistic tree models to electoral data
I
Fitting similar models to other data (for example, 5-level
survey of smoking in India)
I
Network models
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability
Coalitions and political instability
Mathematical and statistical models of voting power
Conclusions
Conclusions
I
Coalitions
I
I
I
Voting power
I
I
I
Forming coalitions can help you but hurt others
Potential explanation for political instability (even in the
absence of “real” disputes)
The electoral college favors voters in small states
“Voting power” calculations that claim otherwise are based on
the falsifiable (and falsified) rule that elections will be
extremely close in large states
Mathematical models can give insights but must be looped
back to real data
Andrew Gelman
Coalitions, voting power, and political instability