View source for Dichotomous preferences

{{Short description|Two-leveled set of preferences in economics}}
In [[economics]], '''dichotomous preferences (DP)''' are [[preference relation]]s that divide the set of alternatives to two subsets, "Good" and "Bad".

From [[ordinal utility]] perspective, DP means that for every two alternatives <math>X,Y</math>:<ref name=comsoc>{{Cite ComSoc Handbook 2016}}</ref>{{rp|292}}
:<math>
X \preceq Y 
\iff
X \in Bad
\text{ or }
Y \in Good
</math>
:<math>
X \prec Y 
\iff
X \in Bad
\text{ and }
Y \in Good
</math>

From [[cardinal utility]] perspective, DP means that for each agent, there are two utility levels: low and high, and for every alternative <math>X</math>:
:<math>
u(X) = u_{low} \iff X\in Bad
</math>
:<math>
u(X) = u_{high} \iff X\in Good
</math>
A common way to let people express dichotomous preferences is using [[approval ballot]]s, 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:<ref name=comsoc/>{{rp|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<sub>i</sub>'' of classes; the school has utility 1 if it gets all ''d<sub>i</sub>'' classes in the same place and 0 otherwise.<ref>{{Cite journal|last1=Bogomolnaia|first1=Anna|last2=Moulin|first2=Herve|date=2004|title=Random Matching Under Dichotomous Preferences|url=https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1468-0262.2004.00483.x|journal=Econometrica|language=en|volume=72|issue=1|pages=257–279|doi=10.1111/j.1468-0262.2004.00483.x|issn=1468-0262}}</ref><ref name=":0">{{Cite book|last1=Kurokawa|first1=David|last2=Procaccia|first2=Ariel D.|last3=Shah|first3=Nisarg|title=Proceedings of the Sixteenth ACM Conference on Economics and Computation |chapter=Leximin Allocations in the Real World |date=2015-06-15|publisher=ACM|pages=345–362|doi=10.1145/2764468.2764490|isbn=9781450334105|s2cid=1060279 }}</ref><ref>{{Cite journal|last=Ortega|first=Josué|date=2020-01-01|title=Multi-unit assignment under dichotomous preferences|journal=Mathematical Social Sciences|language=en|volume=103|pages=15–24|doi=10.1016/j.mathsocsci.2019.11.003|issn=0165-4896|doi-access=free|arxiv=1703.10897}}</ref>

== Collective choice under DP ==
=== Without transfers ===
Suppose a mechanism selects a lottery over outcomes. The utility of each agent, under this mechanism, is the probability that one of his Good outcomes is selected. The [[Utilitarian rule|utilitarian]] mechanism averages over outcomes with the highest approval ratings. It is [[Pareto efficient]], [[strategyproof]], [[Anonymity (social choice)|fair to voters]], and [[Neutrality (social choice)|fair to candidates]].

However, it is impossible to achieve all of these properties in addition to [[proportional division|proportionality]], and as a result [[proportional representation]] systems cannot be strategyproof with dichotomous preferences.<ref name="bms05">{{cite journal|doi=10.1016/j.jet.2004.05.005|title=Collective choice under dichotomous preferences|journal=Journal of Economic Theory|volume=122|issue=2|pages=165|year=2005|last1=Bogomolnaia|first1=Anna|author1-link= Anna Bogomolnaia |last2=Moulin|first2=Hervé|author2-link=Hervé Moulin|last3=Stong|first3=Richard|citeseerx=10.1.1.134.211}}</ref>

=== With transfers ===
Suppose all agents have DP [[cardinal utility]], where each agent is characterized by a single number <math>u_{high}</math> so that <math>u_{low}=0</math>. Then a condition called ''generation monotonicity''{{Jargon inline|date=August 2024}} is necessary and sufficient for implementation by a [[truthful mechanism]]s in any dichotomous domain (see [[Monotonicity (mechanism design)]]).<ref>{{cite journal|doi=10.3982/TE1239|title=Implementation in multidimensional dichotomous domains|journal=Theoretical Economics|volume=8|issue=2|pages=431|year=2013|last1=Mishra|first1=Debasis|last2=Roy|first2=Souvik|doi-access=free|hdl=10419/150197|hdl-access=free}}</ref> If such a domain satisfies a richness condition,{{Jargon inline|date=August 2024}} then a weaker version of generation monotonicity, ''2-generation monotonicity'' (equivalent to ''3-cycle monotonicity''), is necessary and sufficient for implementation.{{Cn|date=August 2024}} This result can be used to derive the optimal mechanism in a one-sided matching problem with agents who have dichotomous types.{{Cn|date=August 2024}}{{Explain|date=August 2024}} 

== References ==
{{reflist}}
[[Category:Utility function types]]