Median voter theorem

The median voter theorem is a proposition relating to direct ranked preference voting put forward by Duncan Black in 1948.[1] It states that if voters and policies are distributed along a one-dimensional spectrum, then any voting method which satisfies the Condorcet criterion will produce a winner close to the median voter. Since several methods are known to satisfy the Condorcet criterion, it follows that it is possible to conduct an election in such a way that it is won by a candidate close to the opinions of the average voter. In particular, a majority vote between two options satisfies the conditions.

A loosely related assertion had been made earlier (in 1929) by Harold Hotelling.[2] It is not a true theorem and is more properly known as the median voter theory or median voter model. It says that in a representative democracy, politicians will converge to the viewpoint of the median voter.[3]

The theorem is associated with public choice economics and statistical political science. Partha Dasgupta and Eric Maskin have argued that it provides a powerful justification for voting methods based on the Condorcet criterion.[4]

Statement and proof of the theorem

Assume that there is an odd number of voters and at least two candidates, and assume that opinions are distributed along a spectrum. Assume that each voter ranks the candidates in an order of preference such that the candidate closest to the voter receives his or her first preference, the next closest receives his or her second preference, and so forth. Then there is a median voter and we will show that the election will be won by the candidate who is closest to him or her.

Let the median voter be Marlene. The candidate who is closest to her will receive her first preference vote. Suppose that this candidate is Charles and that he lies to her left. Then Marlene and all voters to her left (comprising a majority of the electorate) will prefer Charles to all candidates to his right, and Marlene and all voters to her right will prefer Charles to all candidates to his left.

The Condorcet criterion is defined as being satisfied by any voting method which ensures that a candidate who is preferred to every other candidate by a majority of the electorate will be the winner, and this is precisely the case with Charles here; so it follows that Charles will win any election conducted using a method satisfying the Condorcet criterion.

Hence if any voting methods exist which satisfy the Condorcet criterion, then it must be possible to conduct an election in such a way that the winner corresponds to the median voter. For binary decisions the majority vote satisfies the criterion. Several methods are known to satisfy it for multiway votes. The Condorcet criterion can be considered as a method in its own right (the Condorcet method), and has a natural extension due to Ramon Llull (1299), sometimes known as Copeland's method.

Assumptions

The theorem also applies when the number of voters is even, but the details depend on how ties are resolved.

The assumption that preferences are cast in order of proximity can be relaxed to say merely that they are single peaked.[5]

The assumption that opinions lie along a real line can be relaxed to allow more general topologies.[6]

History

The theorem was first set out by Duncan Black in 1948. He wrote that he saw a large gap in economic theory concerning how voting determines the outcome of decisions, including political decisions. Black's paper 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]

Hotelling’s law

The more informal assertion is related to Harold Hotelling’s ‘principle of minimum differentiation’, also known as ‘Hotelling's law’. It states that politicians gravitate toward the position occupied by the median voter, or more generally toward the position favored by the electoral system. It was first put forward (as an observation, without any claim to rigor) by Hotelling in 1929.[2]

Hotelling saw the behavior of politicians through the eyes of an economist. He was struck by the fact that shops selling a particular good often congregate in the same part of a town, and saw this as analogous the convergence of political parties. In both cases it may be a rational policy for maximizing market share.

As with any characterization of human behavior it depends on psychological factors which are not easily predictable, and is subject to many exceptions. It is also contingent on the voting system: politicians will not converge to the median voter unless the electoral process does so. If an electoral process gives more weight to rural than to urban voters, then parties are likely to converge to policies which favor rural areas rather than to the true median.

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, 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 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.[13]

Cycling

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.[13] 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
Main article: 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.[13] 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

The median voter theorem has several limitations. Keith Krehbiel postulates that there are many factors which prevent the political process from reaching maximum efficiency.[14] 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.[15] 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.[16]

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.[17] 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.[17] They will tend to vote for short-term policies that they hope will get them reelected.[3]

References
  1. Duncan Black, ‘On the Rationale of Group Decision-making’ (1948).
  2. Hotelling, Harold (1929). "Stability in Competition". The Economic Journal. 39 (153): 41–57. doi:10.2307/2224214. JSTOR 2224214.
  3. Holcombe, Randall G. (2006). Public Sector Economics: The Role of Government in the American Economy. p. 155. ISBN 9780131450424.
  4. P. Dasgupta and E. Maskin, “The fairest vote of all” (2004); “On the Robustness of Majority Rule” (2008).
  5. See Black’s paper.
  6. Berno Buechel, ‘Condorcet winners on median spaces’ (2014).
  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. JSTOR 1885754. S2CID 154860911.
  9. Holcombe, Randall G. (1980). "An Empirical Test of the Median Voter Model". Economic Inquiry. 18 (2): 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. doi:10.3982/ecta11520.
  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 (2): 211–223. doi:10.1177/106591298503800204. S2CID 153499829.
  13. Gruber, Jonathan (2012). Public Finance and Public Policy. New York, NY: Worth Publishers. ISBN 978-1-4292-7845-4.
  14. Krehbiel, Keith (2004). "Legislative Organization". Journal of Economic Perspectives. 18: 113–128. doi:10.1257/089533004773563467.
  15. Buchanan, James M.; Tollison, Robert D. (1984). The Theory of Public Choice.
  16. Lee, D. S., Moretti, E., & Butler, M. J (2004). "Do voters affect or elect policies? Evidence from the US House" (PDF). Quarterly Journal of Economics. 119 (3): 807–859. doi:10.1162/0033553041502153. S2CID 11274969.CS1 maint: multiple names: authors list (link)
  17. Downs, Anthony (1965). "A Theory of Bureaucracy". American Economic Review. 55: 439–446.

Further reading
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.