# Median voter theorem

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One possible model; here, if parties A and B want to catch the median voters, they should move towards the center. The red and blue areas represent the voters that A and B expect they have already captured.

The median voter theorem states that "a majority rule voting system will select the outcome most preferred by the median voter".[1]

The median voter theorem rests on two main assumptions, with several others detailed below. First, the theorem assumes that voters can place all election 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 choose the alternative closest to their own view. This assumption predicts that the further away the outcome is from the voter's most preferred outcome, the less likely the voter is to select that alternative.[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.

## Assumptions

Example 1 of Single-Peaked Preferences
Example 2 of Single-Peaked Preferences
Example of Multi-Peaked Preferences

For the median voter theorem to be successful, there is a total of seven assumptions that are 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.

## Explanations

To appreciate the logic of the median voter model, consider a setting where three individuals, Al, Bob, and Charlie, are to choose a restaurant for lunch. Al prefers a restaurant where lunch costs \$5, Bob favors somewhat better fare at a restaurant serving \$10 lunches, and Charlie wants a gourmet restaurant where lunch will cost around \$20. Bob can be said to be the median voter, because there are exactly the same number of people who prefer a more expensive restaurant than Bob as there are who prefer a less expensive restaurant than Bob: here one each. For convenience assume that, given any two options, each member of the lunch group prefers restaurants with prices closer to their preferred restaurant to those that are farther from it. Now consider some majority decisions over alternative restaurants:

Options Pattern of votes Result
\$20 vs. \$5 A: 5 B: 5 C: 20 5
\$10 vs. \$20 A: 10 B: 10 C: 20 10
\$10 vs. \$5 A: 5 B: 10 C: 10 10

The weak form of the median voter theorem says the median voter always casts his or her vote for the policy that is adopted. Note that Bob always votes in favor of the outcome that wins the election. Note also that Bob's preferred \$10 restaurant will defeat any other. 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.

### Representative Democracy

Example of Two Politicians in a Representative Democracy.

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]

## History

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]

## Accuracy

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.

## Limitation

### Abstract Social Choice Problem

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 prefers 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]

#### Cycling[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. 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.

#### Arrow's Impossibility Theorem

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).

#### Two Common Solutions

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.

### Political

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, think about the 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, 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]

### Social Efficiency

While the median voter outcome simplifies the political process, the question of efficiency still remains. The social efficiency problem is best explained using the following example. Society’s marginal benefit of a public good is calculated by summing all individual marginal benefits:

A town of 1,001 voters wishes to pay for a \$40,040 public good by taxing every individual \$40. 500 voters decide that they value the new public good at \$100 each, while the remaining 501 voters will not contribute any money towards the good. Based on these revealed preferences, the total social benefit (500 * \$100 = \$50,000) is greater than the previously established cost. While the efficient outcome is to fund the public good, in this scenario the median voter (number 501 in the opposition) is not willing to pay, so the vote fails.

Inefficiency in the median voter outcome results from the intensity of preferences. While a large number of people were willing to pay a considerable sum (\$100 each) to fund the good, this fact is ultimately irrelevant when the median voter favors the opposite result. In any scenario, intensity of preferences can cause the median voter outcome to be inefficient.[4]

## References

1. ^ a b c Holcombe, Randall G. (2006). Public Sector Economics, Upper Saddle River: Pearson Prentice Hall, p. 155.
2. ^ a b c d Hotelling, Harold (1929). "Stability in Competition". The Economic Journal. 39: 41–57. doi:10.2307/2224214.
3. ^ Downs, Anthony (1957). "An Economic Theory of Political Action in a Democracy". Journal of Political Economy. 65: 135–150. doi:10.1086/257897.
4. Gruber, Jonathan (2012). Public Finance and Public Policy. New York, NY: Worth Publishers. ISBN 978-1-4292-7845-4.
5. ^ Congleton, Roger (2002). The Median Voter Model. In * C. K. *Rowley (Ed.); F. Schneider (Ed.) (2003). The Encyclopedia of Public Choice. Kluwer Academic Press. ISBN 978-0-7923-8607-0.
6. ^ Black, Duncan (1948). "On the Rationale of Group Decision-making". Journal of Political Economy. 56: 23–34. doi:10.1086/256633.
7. ^ Downs, Anthony (1957). An Economic Theory of Democracy. Harper Collins.
8. ^ Bowen, Howard R. (1943). "The Interpretation of Voting in the Allocation of Resources". Quarterly Journal of Economics. 58 (1): 27–48. doi:10.2307/1885754.
9. ^ Holcombe, Randall G. (1980). "An Empirical Test of the Median Voter Model". Economic Inquiry. 18: 260–275. doi:10.1111/j.1465-7295.1980.tb00574.x.
10. ^ Fujiwara, T. (2015). "Voting Technology, Political Responsiveness, and Infant Health: Evidence from Brazil". Econometrica. 83 (2): 423–464.
11. ^ Husted, Thomas A. & Lawrence W. Kenny (1997). "The Effect of the Expansion of the Voting Franchise on the Size of Government Model". Journal of Political Economy. 105: 54–82. doi:10.1086/262065.
12. ^ Rice, Tom W. (1985). "An Examination of the Median Voter Hypothesis". The Western Political Quarterly. 38: 211–223. doi:10.1177/106591298503800204.
13. ^ Krehbiel, Keith (2004). "Legislative Organization". Journal of Economic Perspectives. 18: 113–128. doi:10.1257/089533004773563467.
14. ^ Buchanan, James M.; Tollison, Robert D. (1984). The Theory of Public Choice.
15. ^ Lee, D. S., Moretti, E., & Butler, M. J (2004). "Do voters affect or elect policies? Evidence from the US House". Quarterly Journal of Economics. 119 (3): 807–859.
16. ^ a b Downs, Anthony (1965). "A Theory of Bureaucracy". American Economic Review. 55: 439–446.