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: . An agent has a “single-peaked” preference relation over outcomes, , or “single-peaked preferences”, if there exists a unique such that

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.

So, which type of explanation did you like the best?

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You may be interested in Proofs without words ; exercises in visual thinking by Roger B. Nelsen. There is a pdf on line at https://is.muni.cz/el/1441/podzim2013/MA2MP_SMR2/um/Nelsen–Proofs_without_Words.pdf

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