Single peaked preferences

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

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.

With single-peaked preferences, there is a simple truthful mechanism for selecting an outcome: it is to select the median quantity. See the median voter theorem. It is truthful because the median function satisfies the strong monotonicity property.

Take an ordered set of outcomes: ${\displaystyle \{x_{1},\ldots ,x_{N}\}}$. An agent has a "single-peaked" preference relation over outcomes, ${\displaystyle \succsim }$, or "single-peaked preferences", if there exists a unique ${\displaystyle x^{*}\in \{x_{1},\ldots ,x_{N}\}}$ such that

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