Median voter theorem
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. 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. 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 is a total of seven assumptions that are made. 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. 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. 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. 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.
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 most preferred policy.
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. 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.
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. 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.
In his 1929 paper titled Stability in Competition, Harold Hotelling notes in passing that political candidates' platforms seem to converge during majoritarian elections. 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. 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.
Several important economic studies strongly support the median voter theorem. For example, Holcombe analyzes the Bowen equilibrium 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.
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. 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. 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, was relatively low compared to after the 1960s, this tightened the gap.
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 Pareto Dominance. This is the idea that if society 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 analog 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. 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.
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 1951, 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 had the insight to realize that numerous individuals of different demographics and geography cannot all have consistent and similar political, socioeconomic, or many other beliefs. Arrow found that the only way for the social choice problem to have a single solution was to either (1) restrict individual preferences to certain and specific bundles or (2) impose a dictatorship. In reality, it seems that the party system in the United States and in other countries restricts your political beliefs to one side or the other.
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.
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 others' 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. 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. 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.
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. 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. They will tend to vote for short-term policies that they hope will get them reelected.
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