Dichotomous preferences

In economics, dichotomous preferences (DP) are preference relations that divide the set of alternatives to two subsets, "Good" and "Bad".

From ordinal utility perspective, DP means that for every two alternatives ${\displaystyle X,Y}$:^[1]: 292

${\displaystyle X\preceq Y\iff X\in Bad{\text{ or }}Y\in Good}$

${\displaystyle X\prec Y\iff X\in Bad{\text{ and }}Y\in Good}$

From cardinal utility perspective, DP means that for each agent, there are two utility levels: low and high, and for every alternative ${\displaystyle X}$:

${\displaystyle u(X)=u_{low}\iff X\in Bad}$

${\displaystyle u(X)=u_{high}\iff X\in Good}$

A common way to let people express dichotomous preferences is using approval ballots, in which each voter can either "approve" or "reject" each alternative.

Exactly-dichotomous preferences are uncommon, but can be a useful approximation of voters' behaviors in two-party systems or when voters support candidates if and only if they share a party. Single-winner voting rules that satisfy independence of irrelevant alternatives are strategyproof with dichotomous preferences.