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Single peaked preferences

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Title: Single peaked preferences  
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Subject: Condorcet method, Median voter theorem
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Single peaked preferences

Roughly speaking, a group of voters, consumers or agents have single-peaked preferences over a group of outcomes if: 1) they each have an ideal choice in the set; and 2) outcomes that are further from their ideal choice are preferred less.

These preferences are very important for formal models of political science and political economy, because they are necessary to show the median voter theorem.

Formal definition

Take an ordered set of outcomes: \{x_1,\ldots,x_N\}. An agent has a "single-peaked" preference relation over outcomes, \succsim, or "single-peaked preferences", if there exists a unique x^*\in\{x_1,\ldots,x_N\} such that


x_m>x_n \geq x^* \Rightarrow x_n \succ x_m

In words, x^* is the ideal point. When the agent compares between two outcomes that are both to the right or to the left of the ideal point, she strictly prefers whichever option is closest to x^*.

Some examples

The following graphs shows three preferences that are single-peaked over outcomes {A,B,C,D,E}. The number on the vertical axis represents the preference ranking, so 1 is the most preferred choice. Two outcomes that are equally preferred have the same ranking.

The following graph gives two examples of preferences that are not single-peaked. The blue preferences are clearly not single-peaked because the preference ranking spikes down for "D" and then spikes up for "E". The green preferences are not single-peaked because they have two outcomes that are the most preferred: "B" and "C". Such preferences are sometimes called single-plateaued.


Single-peaked preferences have a number of interpretations for different applications.

A simple application of ideological preferences is to think of the outcome space \{x_1,x_2,\ldots,x_n\} as locations on a street and each x_i as the address of an individual. Suppose a single bus stop has to be located on the street and every individual wishes to walk as little as possible to the stop. Individuals then have single-peaked preferences: individual i's ideal point is x_i and she dislikes other locations the farther they are to the west or the farther they are to the east.

The outcome space can also be thought as different policies in an ideological spectrum: policies from the Left vs policies from the Right; policies that are more liberal vs policies that are more conservative; policies that are pro free markets vs policies that are pro state intervention. Voters have single-peaked preferences if they have an ideal balance between the two directions of the ideological spectrum and if they dislike policies the farther away they are from their ideal point.


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