Single-peaked preferences: words versus mathematical notation versus visualization

The entry in Wikipedia for “single-peaked preferences” explains this concept in three different ways: in words, then using formal mathematical notation, and then with a simple image. First, the concept is defined in words:

Single-peaked preferences are a kind of preference relations. A group of agents is said to have single-peaked-preferences if:

Each agent has an ideal choice in the set; and

For each agent, outcomes that are further from his ideal choice are preferred less.

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. Usually, each consumer decides on a certain quantity which is best for him, and if the actual quantity is more/less than that ideal quantity, the agent is then less satisfied.

Next, the concept is stated in formal mathematical notation:

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

$x_{m}<x_{n}\leq x^{*}\Rightarrow x_{n}\succ x_{m}$

$x_{m}>x_{n}\geq x^{*}\Rightarrow x_{n}\succ x_{m}$

Lastly, the concept is expressed in visual form:

The following graph show 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.