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 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: { x 1 , … , x N } {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 x ∗ ∈ { x 1 , … , x N } {displaystyle x^{*}in {x_{1},ldots ,x_{N}}} such that x m < x n ≤ x ∗ ⇒ x n ≻ x m {displaystyle x_{m}<x_{n}leq x^{*}Rightarrow x_{n}succ x_{m}} x m > x n ≥ x ∗ ⇒ x n ≻ x m {displaystyle x_{m}>x_{n}geq x^{*}Rightarrow x_{n}succ x_{m}} In words, x ∗ {displaystyle x^{*}} is the ideal point. When the agent compares between two outcomes that are both to the right or to the left of the ideal point, they strictly prefer whichever option is closest to x ∗ {displaystyle x^{*}} .