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Dichotomous preferences

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Two-leveled set of preferences in economics

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 X , Y {\displaystyle X,Y} :

X ⪯ Y ⟺ X ∈ B a d  or  Y ∈ G o o d {\displaystyle X\preceq Y\iff X\in Bad{\text{ or }}Y\in Good}
X ≺ Y ⟺ X ∈ B a d  and  Y ∈ G o o d {\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 X {\displaystyle X} :

u ( X ) = u l o w ⟺ X ∈ B a d {\displaystyle u(X)=u_{low}\iff X\in Bad}
u ( X ) = u h i g h ⟺ X ∈ G o o d {\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.

In fair item assignment

In the context of fair item assignment, DP can be represented by a mathematical logic formula: 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 di of classes; the school has utility 1 if it gets all di classes in the same place and 0 otherwise.

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 mechanism averages over outcomes with the highest approval ratings. It is Pareto efficient, strategyproof, fair to voters, and fair to candidates.

However, it is impossible to achieve all of these properties in addition to proportionality, and as a result proportional representation systems cannot be strategyproof with dichotomous preferences.

With transfers

Suppose all agents have DP cardinal utility, where each agent is characterized by a single number u h i g h {\displaystyle u_{high}} so that u l o w = 0 {\displaystyle u_{low}=0} . Then a condition called generation monotonicity is necessary and sufficient for implementation by a truthful mechanisms in any dichotomous domain (see Monotonicity (mechanism design)). If such a domain satisfies a richness condition, then a weaker version of generation monotonicity, 2-generation monotonicity (equivalent to 3-cycle monotonicity), is necessary and sufficient for implementation. This result can be used to derive the optimal mechanism in a one-sided matching problem with agents who have dichotomous types.

References

  1. ^ Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432. (free online version)
  2. Bogomolnaia, Anna; Moulin, Herve (2004). "Random Matching Under Dichotomous Preferences". Econometrica. 72 (1): 257–279. doi:10.1111/j.1468-0262.2004.00483.x. ISSN 1468-0262.
  3. Kurokawa, David; Procaccia, Ariel D.; Shah, Nisarg (2015-06-15). "Leximin Allocations in the Real World". Proceedings of the Sixteenth ACM Conference on Economics and Computation. ACM. pp. 345–362. doi:10.1145/2764468.2764490. ISBN 9781450334105. S2CID 1060279.
  4. Ortega, Josué (2020-01-01). "Multi-unit assignment under dichotomous preferences". Mathematical Social Sciences. 103: 15–24. arXiv:1703.10897. doi:10.1016/j.mathsocsci.2019.11.003. ISSN 0165-4896.
  5. Bogomolnaia, Anna; Moulin, Hervé; Stong, Richard (2005). "Collective choice under dichotomous preferences". Journal of Economic Theory. 122 (2): 165. CiteSeerX 10.1.1.134.211. doi:10.1016/j.jet.2004.05.005.
  6. Mishra, Debasis; Roy, Souvik (2013). "Implementation in multidimensional dichotomous domains". Theoretical Economics. 8 (2): 431. doi:10.3982/TE1239. hdl:10419/150197.
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