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:

  1. Each agent has an ideal choice in the set; and
  2. 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.

Singlepeaked1.jpg

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

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2 Responses to Single-peaked preferences: words versus mathematical notation versus visualization

  1. jecgenovese says:

    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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