## Statistical Science

- Statist. Sci.
- Volume 17, Issue 4 (2002), 420-435.

### The mathematics and statistics of voting power

Andrew Gelman, Jonathan N. Katz, and Francis Tuerlinckx

#### Abstract

In an election, *voting power*---the probability that a single
vote is decisive---is affected by the rule for aggregating votes into
a single outcome. Voting power is important for studying political
representation, fairness and strategy, and has been much discussed in
political science. Although power indexes are often considered as
mathematical definitions, they ultimately depend on statistical models
of voting. Mathematical calculations of voting power usually have
been performed under the model that votes are decided by coin flips.
This simple model has interesting implications for weighted elections,
two-stage elections (such as the U.S. Electoral College) and
coalition structures.
We discuss empirical failings of the coin-flip model
of voting and consider, first, the implications for voting power and,
second, ways in which votes could be modeled more realistically.
Under the random voting model, the standard deviation of the average of
*n* votes
is proportional to $1/\sqrt{n}$, but under more general models, this variance
can have the form $cn^{-\alpha}$ or $\sqrt{a-b\log n}$.
Voting power calculations under more
realistic models present research challenges in modeling and
computation.

#### Article information

**Source**

Statist. Sci., Volume 17, Issue 4 (2002), 420-435.

**Dates**

First available in Project Euclid: 10 April 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1049993201

**Digital Object Identifier**

doi:10.1214/ss/1049993201

**Mathematical Reviews number (MathSciNet)**

MR1977137

**Zentralblatt MATH identifier**

1062.91019

**Keywords**

Banzhaf index decisive vote elections electoral college Ising model political science random walk trees

#### Citation

Gelman, Andrew; Katz, Jonathan N.; Tuerlinckx, Francis. The mathematics and statistics of voting power. Statist. Sci. 17 (2002), no. 4, 420--435. doi:10.1214/ss/1049993201. https://projecteuclid.org/euclid.ss/1049993201