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

In fair item assignment

In the context of fair item assignment, DP can be represented by a mathematical logic formula:^[1]: 292 for every agent, there is a formula that describes his desired bundles. An agent is satisfied if-and-only-if he receives a bundle that satisfies the formula.

A special case of DP is single-mindedness. A single-minded agent wants a very specific bundle; he is happy if-and-only-if he receives this bundle, or any bundle that contains it. Such preferences appear in real-life, for example, in the problem of allocating classrooms to schools: each school i needs a number d_i of classes; the school has utility 1 if it gets all d_i classes in the same place and 0 otherwise. ^[2][3][4]

Collective choice under DP

References