Dichotomous Preferences
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In
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, dichotomous preferences (DP) are preference relations that divide the set of alternatives to two subsets: "Good" versus "Bad". From
ordinal utility In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to a ...
perspective, DP means that for every two alternatives X,Y: : X \preceq Y \iff X \in Bad \text Y \in Good : X \prec Y \iff X \in Bad \text Y \in Good From
cardinal utility In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u(x_i) of one i ...
perspective, DP means that for each agent, there are two utility levels: low and high, and for every alternative X: : u(X) = u_ \iff X\in Bad : u(X) = u_ \iff X\in Good A common way to let people express dichotomous preferences is using
approval ballot An approval ballot, also called an unordered ballot, is a ballot in which a voter may vote for any number of candidates simultaneously, rather than for just one candidate. Candidates that are selected in a voter's ballot are said to be ''approved'' ...
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 Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
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 money

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 In ethical philosophy, utilitarianism is a family of normative ethical theories that prescribe actions that maximize happiness and well-being for all affected individuals. Although different varieties of utilitarianism admit different charac ...
mechanism averages over outcomes with largest “approval”. It is
Pareto efficient Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
,
strategyproof In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about ...
, anonymous and neutral. It is impossible to attain these properties in addition to proportionality - giving each agent a utility of at least 1/''n''; or at least the fraction of good to feasible outcomes. conjecture that no ex ante efficient and strategyproof mechanism guarantees a strictly positive utility to all agents, and prove a weaker statement.


With money

Suppose all agents have DP
cardinal utility In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u(x_i) of one i ...
, where each agent is characterized by a single number - u_ (so that u_=0). identify a new condition, ''generation monotonicity'', that is necessary and sufficient for implementation by a
truthful mechanism In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about ...
s in any dichotomous domain (see
Monotonicity (mechanism design) In mechanism design, monotonicity is a property of a social choice function. It is a necessary condition for being able to implement the function using a strategyproof mechanism. Its verbal description is: In other words: Notation There is a ...
). 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

{{reflist Utility function types