The **median voter theorem** states that "a majority rule voting system will select the outcome most preferred by the median voter".^{ [1] } It is associated with public choice economics and statistical political science.

- Assumptions
- Representative democracy
- History
- Accuracy
- Limitation
- Abstract social choice problem
- Political
- References
- Further reading
- External links

The median voter theorem rests on two main assumptions, with several others detailed below. The theorem is assuming that voters can place all alternatives along a one-dimensional political spectrum.^{ [2] } It seems plausible that voters could do this if they can clearly place political candidates on a left-to-right continuum, but this is often not the case as each party will have its own policy on each of many different issues. Similarly, in the case of a referendum, the alternatives on offer may cover more than one issue. Second, the theorem assumes that voters' preferences are single-peaked, which means that voters have one alternative that they favor more than any other.^{ [3] } It also assumes that voters always vote, regardless of how far the alternatives are from their own views. The median voter theorem implies that voters have an incentive to vote for their true preferences. Finally, the median voter theorem applies best to a majoritarian election system.

For the median voter theorem to be successful, there are seven assumptions made,^{ [4] } and each includes exceptions for when politicians decide to move away from the median voter:

As mentioned above, (1) the first assumption is that there is single-dimensional voting. Put simply, this means that there is only one issue that is being voted on at a time. Additionally, it is assumed that (2) voters' preferences are single-peaked, which is just the notion that people's preferences are a spectrum of utility, with the strongest preference at the maximum (see figures to the right). This assumption is critical because it prevents a phenomenon called "cycling" which is detailed below. The third assumption (3) is that voters are only choosing between two options. This is important because when there are more than two choices for voters, the median voter may not have voted for the most popular option. For example, in a population of 100 people voting between A, B, and C imagine 33 people vote for A, 33 people vote for B, and 34 people vote for C. Assuming A, B, and C lie on a spectrum (i.e. a scale from liberal to neutral to conservative) the median voter would have voted for B even though choice C was the most popular. If each choice is a candidate, and all three start at the median, any slight move captures the entire voting group on that end of the spectrum. No equilibrium exists in this situation, because every candidate has the incentive to move across the ideological spectrum based on the positions of their competitors. The fourth assumption (4) is that there is no ideology or influence with regards to the voting options. Essentially, this means that politicians only care about maximizing votes, not necessarily sticking true to their beliefs. In reality, this ignores politician’s ability to change voters’ ideologies to mirror their own. Additionally, career politicians may purposely take positions away from the ideological center in order to gain favor with specific voting bases. The fifth assumption (5) is that there is no selective voting and all eligible voters for an election will turn out to vote. The sixth assumption (6) says that money and lobbying have no effect on elections because introducing these incentives can dramatically change voting patterns. In reality, money is one of many variables that contribute to the outcome of elections. The theorem ignores the fact that politicians sometimes take positions with the primary goal of raising money for their campaigns. Political contributions can be used for advertising and campaign trips, and gaining monetary support may require moving away from the median. The final assumption (7) is the notion that all parties of elections have full information. This means that voters have knowledge on the issues, candidates have knowledge on the issues, and candidates have knowledge on voter preferences.

The weak form of the median voter theorem says the median voter always casts his or her vote for the policy that is adopted. If there is a median voter, his or her preferred policy will beat any other alternative in a pairwise vote. (The median voter's ideal point is always a Condorcet winner.) Consequently, once the median voter's preferred outcome is reached, it cannot be defeated by another in a pairwise majoritarian election. The strong form of the median voter theorem says the median voter always gets his or hers most preferred policy.^{ [5] }

The median voter theorem seems to explain some of the things that happen in majoritarian voting systems. First, it may explain why politicians tend to adopt similar platforms and campaign rhetoric. In order to win the majority vote, politicians must tailor their platforms to the median voter.^{ [2] } For example, in the United States, the Democratic and Republican candidates typically move their campaign platforms towards the middle during congressional election campaigns. Just as sellers in a free market try to win over their competitors' customers by making slight changes to improve their products, so too do politicians deviate only slightly from their opponent's platform so as to gain votes.^{ [2] }

Second, the median voter theorem reflects that radical candidates or parties rarely get elected. For example, a politician or party which is at an extreme end of the political spectrum will usually not get nearly as many votes as a more moderate party. Finally, the theorem may explain why two major political parties tend to emerge in majoritarian voting systems (Duverger's law). In the United States there are countless political parties, but only two established major parties play a part in almost every major election: the Democratic and Republican parties. According to the median voter theorem third parties will rarely, if ever, win elections for the same reason why extreme candidates do not tend to win. The major parties tend to co-opt the platforms of the minor parties in order to secure more votes.^{ [1] } In many other long-established democratic countries there are several parties who each get a substantial share of the vote, although most of these have some form of proportional representation.

While the median voter theorem traditionally applies to direct democracy, in reality most nations function as some sort of representative democracy. An adjusted version of the theorem suggests that, in representative democracy, politicians legislate and execute the laws based on the preferences of the median voter.

Consider the example in the Figure^{ [4] } to the right of two politicians, Hillary and Donny, who initially hold different views about what percentage of federal government spending should be dedicated to entitlement programs. While Hillary wants to increase the current amount by 25%, Donny represents supporters who want to see a decrease of 25%. Meanwhile, the median voter prefers that entitlement spending remain the same. To capture some of Donny's voters, Hillary decides that she will now campaign on a spending increase of 10%. To prevent her from gaining an electoral advantage, Donny counters by advocating for a 5% decrease in spending. This cycle continues until both candidates arrive at the outcome preferred by the median voter. Politicians have the incentive to reach this position because, if they don’t, they risk allowing their opponents to capture additional voters.^{ [4] }

In his 1929 paper titled *Stability in Competition*, Harold Hotelling notes in passing that political candidates' platforms seem to converge during majoritarian elections.^{ [2] } Hotelling compared political elections to businesses in the private sector. He postulated that just as there is often not much difference between the products of different competing companies, so, too, there is not a stark contrast between electoral platforms of different parties. This is because politicians, just like salesmen with consumers, seek to capture the majority of voters. Duncan Black, in his 1948 paper titled *On the Rationale of Group Decision-making*, provided a formal analysis of majority voting that made the theorem and its assumptions explicit.^{ [6] } Black wrote that he saw a large gap in economic theory concerning how voting determines the outcome of decisions, including political decisions. Black's paper thus triggered research on how economics can explain voting systems. In 1957 with his paper titled *An Economic Theory of Political Action in Democracy*, Anthony Downs expounded upon the median voter theorem.^{ [7] }

Several important economic studies strongly support the median voter theorem. For example, Holcombe analyzes the Bowen equilibrium^{ [8] } level of education expenditures for 257 Michigan school districts and finds that the actual expenditures are only about 3% away from the estimated district average.^{ [9] } Fujiwara also supported the theorem through his study of the 1998 Brazil general elections. He analysed the effect of an exogenous increase in the voter base on the policies implemented by the subsequent government chosen through the introduction of EVMs (Electronic Voting Machines), which enabled a large section of the less educated communities to cast their vote. The outcome of this election was an increase in policies targeted at issues affecting these communities, specifically healthcare. Thus, Fujiwara’s conclusions show that an increase in voter base shifted the median voter, and hence the middle ground for politicians, to a stance more favourable to the new total voter base, indicating that the voters do have a say in the policies implemented by candidates.^{ [10] }

The theorem also explains the rise in government redistribution programs over the past few decades. Thomas Husted and Lawrence W. Kenny examined growth of redistribution programs especially between the years of 1950 and 1988.^{ [11] } Tom Rice also writes that voters with the median income will take advantage of their status as deciders by electing politicians who will tax those who are earning more than the median voter, and then redistribute the money, including to those who are at the median.^{ [12] } More specifically, Rice demonstrates that if a systematic closing of the gap between the median and mean income levels in the United States could be shown, more credibility could be given to the median voter theorem. Until the mid-1960s, Rice says that the gap between median and mean income levels tightened. Three main forces served to tighten this gap. First, the strength of the Democratic Party in the United States Congress in the decades leading up to the 1960s, as Democrats are more disposed to redistribution of wealth. Second, increased turnout at the polls, just as Husted and Kenny postulated, tightened the gap because an increase in voters means more individuals of lower income are voting. Finally, since unemployment, which causes median income families to fall below the median income^{[ citation needed ]}, was relatively low compared to after the 1960s, this tightened the gap.

How do we choose the best outcome from an election for society? This question is the root of the median voter theorem and provides the basis for how and why this theorem was created. It starts with the idea of a "social decision rule." Essentially, this is a tool that is used to aggregate preferences of all members of society that, ultimately, provides a clear-cut and consistent answer for what outcome is most preferred. This choice rests on three main principles that allow the most preferred social choice to be salient. The first (1) is weak Pareto efficiency or unanimity. This is the idea that if all voters prefer one choice to all other choices, the social decision should reflect this and this option will be the outcome. The second principle (2) is a concept called transitivity, which is analogous to the mathematical property. This phenomenon simply means that if option A is preferred to option B, and option B is preferred to option C, then option A is preferred to option C. The final principle (3) is the idea of independence of irrelevant alternatives (IIA). This suggests that if something is not relevant to the election or the issues involved, then it should not affect the outcome or results. For example, imagine there is a vote for the Most Valuable Player in a baseball league and player A has the most votes, player B has the second most and player C has the third most. Now, say, player C is disqualified for cheating – this should not change the outcome of the vote. If the voting system was set up in a way in which aggregate votes are shifted and player B ends up with more votes, this is not a consistent aggregation method.^{ [4] }

If any of the above-mentioned principles is violated, it could result in cycling. Cycling happens when there is no clear winner from a majority vote that results in a constant cycle of trying to determine which outcome is most preferred.^{ [4] } This is a crucial concept because it exposes how majority voting in general and the median voter theorem can fail when assumptions are not met. There are several more failures that come about from this model that stem from this phenomenon.

With the difficulties associated with aggregating society's preferences, what are some alternatives that can be considered? Potentially, members of society could simply vote for their first choice rather than rank their preferences. Alternatively, there could be weights distributed based on the intensity and passion that members feel for specific issues. Both of these are problematic for several reasons, including the frequent occurrences of ties.

In 1972, Kenneth Arrow received the Nobel Prize in economics for a theorem based on these challenges with aggregating ranked preferences consistently. Arrow's Impossibility Theorem states that there is no general solution to the abstract social choice problem which is based on ranked preferences (although his theorem does not apply to rated scores). Arrow found that the only way for the social choice problem to have any consistent solution is to (1) assume individual preferences fit some particular pattern or (2) impose a dictatorship or (3) accept a rule that violates IIA.^{ [4] } The Median voter theorem is an example of option (1).

Restrict preferences to single peaks, meaning that individuals vote on a spectrum and allow the median voter theorem to be implemented naturally. This is essentially the function of the party system mentioned briefly above. Another common solution is to allow people's intensities on issues play a factor in their vote. This is difficult to achieve since both social welfare functions and the Samuelson rule are necessary to calculate.

In reality, many of the assumptions of this model do not hold. One assumption the theorem makes is that there is only single dimensional voting. This is never true of government representatives – politicians do not only take stances on only one issue but rather several. To test the median voter theorem further, consider the U.S. Senate. If the median voter theorem holds, it would mean that the two senators from a state should vote the same way every time because the median voter in the state would be the voter that chooses the outcome. However, when there is one Democratic senator and one Republican senator, they typically vote opposite to each other, effectively canceling each other's votes.

The median voter theorem has several limitations. Keith Krehbiel postulates that there are many factors which prevent the political process from reaching maximum efficiency.^{ [13] } Just as transaction costs prevent efficiency in market exchanges, the limitations of the majoritarian voting process stop it from reaching optimality. With the median voter theorem in particular, Krehbiel argues that voters' inability to directly amend legislation acts against the theorem. Sometimes, as Krehbiel writes, the policies being voted on are too complex to be placed within a one-dimensional continuum. Buchanan and Tollison also note that this is a problem for the median voter theorem, which assumes that decisions can be made on a one-dimensional field.^{ [14] } If voters are considering more than one issue simultaneously, the median voter theorem is inapplicable. This may happen if, for example, voters may vote on a referendum regarding education spending and police spending simultaneously.

Lee, Moretti & Butler also show that the theorem does not hold in certain cases. They studied the US Congress to see whether voters were only voting for policies pre-decided by candidates or if they had an actual influence on where candidates stood on various political issues, i.e., made candidates converge. Their empirical evidence showed that voters had little effect on the policy stances taken by candidates, meaning that despite a large exogenous change in the probability a candidate would win an election, their policies remained unchanged. Hence, the median voter theorem, which supports the claim that voters make political candidates converge towards a middle ground, is outweighed by candidates refusing to compromise on their political standpoints.^{ [15] }

A larger problem for the median voter theorem, however, is the incentives structure for government representatives. Downs, in *A Theory of Bureaucracy*, writes that people's decisions are motivated by self-interest, an idea deeply rooted in the writings of Adam Smith.^{ [16] } This holds for the government system as well, because it is composed of individuals who are self-interested. One cannot guarantee the degree to which a government representative will be committed to the public good, but it is certain that, to some degree, they will be committed to their own set of goals. These goals can include a desire to serve the public interest, but most often they include the desire for power, income, and prestige. To continue obtaining these things, then, officials must secure re-election. When representatives are constantly focused on becoming re-elected, this distorts the mandate they receive from their constituents: representatives will translate the wishes of their constituents into benefits for themselves.^{ [16] } They will tend to vote for short-term policies that they hope will get them reelected.^{ [1] }

**Approval voting** is a single-winner electoral system where each voter may select ("approve") any number of candidates. The winner is the most-approved candidate. It is related to score voting in which voters give each option a score on a scale, and the option with the highest total of scores is selected. It is distinct from plurality voting in which a voter may choose only one option among several, whereby the option with the most votes is chosen—even absent a majority.

**Logrolling** is the trading of favors, or *quid pro quo*, such as vote trading by legislative members to obtain passage of actions of interest to each legislative member. In organizational analysis, it refers to a practice in which different organizations promote each other's agendas, each in the expectation that the other will reciprocate. In an academic context, the *Nuttall Encyclopedia* describes logrolling as "mutual praise by authors of each other's work".

**Proportional representation** (**PR**) characterizes electoral systems in which divisions in an electorate are reflected proportionately in the elected body. If *n*% of the electorate support a particular political party or set of candidates as their favorite, then roughly *n*% of seats will be won by that party or those candidates. The essence of such systems is that all votes contribute to the result—not just a plurality, or a bare majority. The most prevalent forms of proportional representation all require the use of multiple-member voting districts, as it is not possible to fill a single seat in a proportional manner. In fact, PR systems that achieve the highest levels of proportionality tend to include districts with large numbers of seats.

**Rational choice theory**, also known as **theory of rational choice**, **choice theory** or **rational action theory**, is a framework for understanding and often formally modeling social and economic behavior. The basic premise of rational choice theory is that aggregate social behavior results from the behavior of individual actors, each of whom is making their individual decisions. The theory also focuses on the determinants of the individual choices. Rational choice theory then assumes that an individual has preferences among the available choice alternatives that allow them to state which option they prefer. These preferences are assumed to be complete and transitive. The rational agent is assumed to take account of available information, probabilities of events, and potential costs and benefits in determining preferences, and to act consistently in choosing the self-determined best choice of action. In simpler terms, this theory dictates that every person, even when carrying out the most mundane of tasks, perform their own personal cost and benefit analysis in order to determine whether the action is worth pursuing for the best possible outcome. And following this, a person will choose the optimum venture in every case. This could culminate in a student deciding on whether to attend a lecture or stay in bed, a shopper deciding to provide their own bag to avoid the five pence charge or even a voter deciding which candidate or party based on who will fulfill their needs the best on issues that have an impact on themselves especially.

**Score voting** or **range voting** is an electoral system for single-seat elections, in which voters give each candidate a score, the scores are added, and the candidate with the highest total is elected. It has been described by various other names including **evaluative voting**, **utilitarian voting**, **interval measure voting**, **the point system**, **ratings summation**, **0-99 voting**, **average voting** and **utility****voting**. It is a type of cardinal voting electoral system.

In voting methods, **tactical voting** occurs in elections with more than two candidates, when a voter supports another candidate more strongly than their *sincere preference* in order to prevent an undesirable outcome.

The **Condorcet paradox** in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic, even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority wishes can be in conflict with each other: Majorities prefer, for example, candidate A over B, B over C, and yet C over A. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.

**Public choice**, or **public choice theory**, is "the use of economic tools to deal with traditional problems of political science". Its content includes the study of political behavior. In political science, it is the subset of positive political theory that studies self-interested agents and their interactions, which can be represented in a number of ways – using standard constrained utility maximization, game theory, or decision theory.

In social choice theory, **Arrow's impossibility theorem**, the **general possibility theorem** or **Arrow's paradox** is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the **ranked preferences** of individuals into a community-wide ranking while also meeting a specified set of criteria: *unrestricted domain*, *non-dictatorship*, *Pareto efficiency*, and *independence of irrelevant alternatives*. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book *Social Choice and Individual Values*. The original paper was titled "A Difficulty in the Concept of Social Welfare".

The **independence of irrelevant alternatives** (**IIA**), also known as **binary independence** or the **independence axiom**, is an axiom of decision theory and various social sciences. The term is used with different meanings in different contexts; although they all attempt to provide an account of rational individual behavior or aggregation of individual preferences, the exact formulations differ from context to context.

In social choice theory, the **Gibbard–Satterthwaite theorem** is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner. It states that for every voting rule, one of the following three things must hold:

- The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
- The rule limits the possible outcomes to two alternatives only; or
- The rule is susceptible to tactical voting: in certain conditions some voter's sincere ballot may not defend their opinion best.

**Rational ignorance** is refraining from acquiring knowledge when the supposed cost of educating oneself on an issue exceeds the expected potential benefit that the knowledge would provide.

A **protest vote** is a vote cast in an election to demonstrate dissatisfaction with the choice of candidates or the current political system. Protest voting takes a variety of forms and reflects numerous voter motivations, including political alienation.

The **majority criterion** is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is ranked first by a majority of voters, then that candidate must win".

The **paradox of voting**, also called **Downs paradox**, is that for a rational, self-interested voter, the costs of voting will normally exceed the expected benefits. Because the chance of exercising the pivotal vote is minuscule compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the costs.

The **Borda count** is a family of single-winner election methods in which voters rank options or candidates in order of preference. The Borda count determines the outcome of a debate or the winner of an election by giving each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. Once all votes have been counted the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.

The **probabilistic voting theory**, also known as the **probabilistic voting model**, is a voting theory developed by professors Assar Lindbeck and Jörgen Weibull in the article "Balanced-budget redistribution as the outcome of political competition", published in 1987 in the journal *Public Choice*, which has gradually replaced the median voter theory, thanks to its ability to find equilibrium within multi-dimensional spaces.

**Cardinal voting** refers to any electoral system which allows the voter to give each candidate an independent rating or grade. These are also referred to as "rated", "evaluative", "graded", or "absolute" voting systems. *Cardinal* methods and *ordinal methods* are two main categories of modern voting systems, along with plurality voting.

The **altruism theory of voting** is a model of **voter behavior** which states that if citizens in a democracy have “social” preferences for the welfare of others, the extremely low probability of a single vote determining an election will be outweighed by the large cumulative benefits society will receive from the voter’s preferred policy being enacted, such that it is rational for an “altruistic” citizen, who receives **utility** from helping others, to vote. Altruistic voting has been compared to purchasing a lottery ticket, in which the probability of winning is extremely low but the payoff is large enough that the *expected* benefit outweighs the cost.

Electoral systems are the rules for conducting elections. Comparisons between different systems can focus on different aspects: on suffrage or rules for voter eligibility; on candidate eligibility and the rules governing political parties; on the way elections are scheduled, sequenced, and combined; or on the rules for determining the winner within a given election.

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