A group has single-peaked preferences over a set of outcomes if the outcomes can be ordered along a line such that: Single-peaked preferences are typical of one-dimensional domains.
A typical example is when several consumers have to decide on the amount of public good to purchase.
The amount is a one-dimensional variable.
With single-peaked preferences, there is a simple truthful mechanism for selecting an outcome, which is to select the median quantity; this results in the median voter theorem.
[citation needed] It is truthful because the median function satisfies the strong monotonicity property.
The notion was first presented by Duncan Black[1] and later by Kenneth Arrow.
The preference-relation of agent i is denoted by
The group N is said to have single-peaked preferences over X, if there exists an ordering > of the outcomes such that, for every agent i in N:
When the agent compares between two outcomes that are both to the right or to the left of his ideal point, he strictly prefers whichever option is closest to
Ballester and Haeringer[3] proved the following necessary condition for single-peaked preferences.
If the group N has single-peaked preferences over X, then for every triplet of outcomes in X, there exists an outcome that is not ranked last by any agent in N. The following graph shows a set of three preferences that are single-peaked over outcomes {A,B,C,D,E}.
On the vertical axis, the number represents the preference ranking of the outcome, with 1 being most preferred.
Two outcomes that are equally preferred have the same ranking.
The ordering over the outcomes is A < B < C < D < E. The ideal outcome for the green agent is A, for the red it is B, for the blue it is C. For each agent, when we move away from his ideal outcome, the ranking decreases.
It can also be verified that, for each triplet of outcomes, one of them is never ranked last - the one in the middle.
This proves that the blue preferences are not single-peaked with respect to the ordering A To formally prove this, consider the set of three outcomes {A, D, E}. Therefore, no ordering on X can make the set of preferences single-peaked. Single-peaked preferences have a number of interpretations for different applications. A simple application of ideological preferences is to think of the outcome space 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. 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. A group of agents is said to have single-dipped preferences over a set of possible outcomes if the outcomes can be ordered along a line such that: Lackner and Peters[4] study a class of preferences that are single-peaked on a circle. The single-peaked recognition problem is the following decision problem: given a set of preferences on a set of outcomes, decide if there is a common order of the outcomes for which the preferences are single-peaked. Usually, it is required to also find this common order, if it exists. Trick[5] presents a polynomial-time algorithm for recognizing preferences that are single-peaked on a tree. Escoffier, Spanjaard and Tydrichova[6] study the problem of recognizing preferences that are single-peaked on a general graph.