Median voter theory
Encyclopedia
The median voter theory, also known as the median voter theorem or Black's theorem, is a famous voting theorem. It posits that in a majority
election
, if voter policy preferences can be represented as a point along a single dimension
, if all voters vote deterministically for the politician who commits to a policy position closest to their own preference, and if there are only two politicians, then a politician maximizes their number of votes by committing to the policy position preferred by the median
voter. This strategy is a Nash equilibrium
. It results in voters being indifferent between the candidates and casting their votes for either candidate with equal probability. Hence, in expectation each politician receives half of the votes. If either candidate deviates to commit to a different policy position, the deviating candidate receives less than half the vote.
Political commentator Mickey Kaus
of Slate Magazine wrote in 2004 that the United States
presidential election
s of 2000
and 2004
, in addition to local elections in those years, provide evidence that this phenomenon is taking place in the United States.
The theorem was first articulated in Duncan Black
's 1948 article, "On the Rationale of Group Decision-making" and popularized by Anthony Downs
's 1957 book, An Economic Theory of Democracy
. The theorem is central to moderation theory
that delineates the processes through which radical political groups are incorporated into the existing political system.
The non-verification of single-peaked preferences may lead to the Condorcet paradox, in which the agenda-maker who chooses in which order propositions are voted on may have the power to choose any outcome.
Generally, single-peaked preferences are a sufficient condition for the theorem to apply (Mas-Collel et al. 1995, pg. 802-3, Propositions 21.D.1 and 21.D.2). When preferences are not single-peaked or the policy space is multi-dimensional (e.g., individuals vote on both taxation and public expenditure), the median voter theorem yields no prediction. This has led some political economists to increasingly adopt the newer probabilistic voting theory
, which has a unique equilibrium in a multi-dimensional spaces as well.
Majority
A majority is a subset of a group consisting of more than half of its members. This can be compared to a plurality, which is a subset larger than any other subset; i.e. a plurality is not necessarily a majority as the largest subset may consist of less than half the group's population...
election
Election
An election is a formal decision-making process by which a population chooses an individual to hold public office. Elections have been the usual mechanism by which modern representative democracy operates since the 17th century. Elections may fill offices in the legislature, sometimes in the...
, if voter policy preferences can be represented as a point along a single dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
, if all voters vote deterministically for the politician who commits to a policy position closest to their own preference, and if there are only two politicians, then a politician maximizes their number of votes by committing to the policy position preferred by the median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
voter. This strategy is a Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
. It results in voters being indifferent between the candidates and casting their votes for either candidate with equal probability. Hence, in expectation each politician receives half of the votes. If either candidate deviates to commit to a different policy position, the deviating candidate receives less than half the vote.
Political commentator Mickey Kaus
Mickey Kaus
Robert Michael Kaus , better known as Mickey Kaus, is an American journalist, pundit, and author best known for writing Kausfiles, a "mostly political" blog which was featured on Slate until 2010. Kaus is the author of The End of Equality and had previously worked as a journalist for Newsweek, The...
of Slate Magazine wrote in 2004 that the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
presidential election
Presidential election
A presidential election is the election of any head of state whose official title is president.- United States :The United States has elections on the state and local levels...
s of 2000
United States presidential election, 2000
The United States presidential election of 2000 was a contest between Republican candidate George W. Bush, then-governor of Texas and son of former president George H. W. Bush , and Democratic candidate Al Gore, then-Vice President....
and 2004
United States presidential election, 2004
The United States presidential election of 2004 was the United States' 55th quadrennial presidential election. It was held on Tuesday, November 2, 2004. Republican Party candidate and incumbent President George W. Bush defeated Democratic Party candidate John Kerry, the then-junior U.S. Senator...
, in addition to local elections in those years, provide evidence that this phenomenon is taking place in the United States.
The theorem was first articulated in Duncan Black
Duncan Black
Duncan Black was a Scottish economist who laid the foundations of social choice theory. In particular he was responsible for unearthing the work of many early political scientists, including Charles Dodgson, and was responsible for the Black electoral system, a Condorcet method whereby, in the...
's 1948 article, "On the Rationale of Group Decision-making" and popularized by Anthony Downs
Anthony Downs
Anthony Downs is a scholar in public policy and public administration, and since 1977 is a Senior Fellow at the Brookings Institution in Washington D.C..-Education:...
's 1957 book, An Economic Theory of Democracy
An Economic Theory of Democracy
An Economic Theory of Democracy is a political science treatise written by Anthony Downs, published in 1957. The book set forth a model with precise conditions under which economic theory could be applied to non-market political decision-making. It also suggested areas of empirical research that...
. The theorem is central to moderation theory
Moderation theory
Moderation theory is a set of interrelated hypotheses that explain the process through which political groups eschew radical platforms in favor of more moderate policies and prefer electoral, compromising and non-confrontational strategies over non-electoral, exclusive, and confrontational strategies...
that delineates the processes through which radical political groups are incorporated into the existing political system.
Shortcomings of the Median Voter Model
There are many instances in which the Median Voter Theorem may not be applied. Among the most common is the case of preferences which are not single-peaked (multimodal preferences) or order-restricted. Consider the case where several voters are voting on the budget of a public school, for instance, and there are three options: low, medium and high. If one of the voters prefers both low and high to medium, then the preferences will not be single-peaked (they form a "valley", instead of a "mountain", if represented in a unidimensional diagram).The non-verification of single-peaked preferences may lead to the Condorcet paradox, in which the agenda-maker who chooses in which order propositions are voted on may have the power to choose any outcome.
Generally, single-peaked preferences are a sufficient condition for the theorem to apply (Mas-Collel et al. 1995, pg. 802-3, Propositions 21.D.1 and 21.D.2). When preferences are not single-peaked or the policy space is multi-dimensional (e.g., individuals vote on both taxation and public expenditure), the median voter theorem yields no prediction. This has led some political economists to increasingly adopt the newer probabilistic voting theory
Probabilistic voting theory
The probabilistic voting theory, also known as the probabilistic voting model, is a voting theory developed by professor Melvin Hinich, which has gradually replaced the median voter theory, thanks to its ability to find an equilibrium in a multi-dimensional space...
, which has a unique equilibrium in a multi-dimensional spaces as well.