Abstract
Sen’s Liberal paradox describes a conflict between weak Pareto, minimal liberalism, and either transitivity or a best element over a domain of individual preferences. This paper examines variants of that paradox with varying amounts of unconditional preferences. We define a notion of unconditional preferences under which, in the absence of Pareto, there can be no cycles. We then define a stronger condition, that makes an individual’s preferences for her own private attributes independent of all other attributes. Under this assumption, there can be no cycles with or without Pareto. We also show there exists a social decision function satisfying those conditions. We then determine the probability of a cycle assuming a much weaker independence condition that does not restrict the domain. This probability converges to one as the number of non-private attributes within the social states increases. Finally, we use simulations to determine the probability that liberalism and Pareto conflict with best elements, maximal elements, and transitivity separately.
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Notes
A social state is “a complete description of society including every individual’s position in it” (Sen, 1970, 152). Sen (1970) uses the terms “social states” and “alternatives” interchangeably, but settles on alternatives for his theorem. We use social states to reduce confusion about a personal realm and a social realm (Barry, 1986).
Sen proved his theorem for weak individual orderings but the theorem also applies to purely strict individual orderings.
Sen (1979) defines a social ordering as reflexive (\(\forall s_j \in \mathcal{S}: s_j \succeq s_j\)), weakly complete (\(\forall s_j,s_k \in \mathcal{S}: s_j \succeq s_k\) or \(s_k \succeq s_j\)), and weakly transitive (\(\forall s_j, s_k, s_l, \in \mathcal{S}: (s_j \succeq s_k\) & \(s_k \succeq s_l) \rightarrow s_j \succeq s_l\)).
Social state \(s_j \in \mathcal{S}\) is a best element of \(\mathcal{S}\) if and only if \(\forall s_k: s_k \in \mathcal{S} \rightarrow s_j \succeq s_k\) (Sen 1979, 10).
Consider Lady Chatterley’s Lover for example. If Prude and Lewd had IID preferences, Prude would prefer (neither read) to (only Prude reads), and (only Lewd reads) to (only Prude reads); Lewd would prefer (only Lewd reads) to (neither reads), and (neither reads) to (only Prude reads). In this case, \(\mathbf{P} \) and \(\mathbf{L} _j\), \(j=1,2\) would create a transitive order with a best element, regardless of their remaining binary preferences.
\(s_k \sim s_j\) indicates social indifference between \(s_k\) and \(s_j\).
For \(N=d_0=d=2\), IC preferences are a generated by \(p=0.5\).
As a first step in creating the PCP distribution, we randomly choose an order of preferences for each individual using IC. As a second step, for each level \(\ell \) and each individual i, we randomly choose i’s type (\(\alpha \) or \(\gamma \)) with probabilities Pr(i is type \(\alpha \) on level \(\ell ) = p\) and Pr(i is type \(\gamma \) on level \(\ell ) = 1-p\). If \(\alpha \) preferences are drawn for individual i on level \(\ell \), we flip a coin to determine whether \(\{a_i = 0\} \succ _i\, \{a_i = 1\}\) or \(\{a_i = 1\} \succ _i \,\{a_i = 0\}\) on level \(\ell \) and rearrange i’s preferences accordingly. If \(\gamma \) preferences are drawn for individual i on level \(\ell \), we shuffle an individual’s preferences for the set of social states they are decisive over on level \(\ell \) until it rules out \(\alpha \) preferences for that individual, thus creating \(\gamma \) preferences for the individual on level \(\ell \). The effect of this procedure is it maintains the preference type drawn for an individual on a level while creating IC distributed preferences for all other binary pairs.
With a half-million trials, we are \(95\%\) confident that the true probability is within 0.0015 of the relative frequencies reported. This statement is based on the standard deviation of a univariate proportion, \(\sqrt{{\eta (1-\eta )}\over {T}}\), where \(\eta = 0.5\), the worst case value, and T is the number of trials.
With incompleteness, a social ranking can be transitive for two trivial reasons: (1) the antecedent for transitivity is false (i.e., \(\not \exists \) \(x,y,z: x \succ y\) and \(y \succ z\)) or, (2) the consequent is unknown (i.e. x?z).
In contrast, Igersheim (2013) prohibits invasive preferences if they interfere with a libertarian right.
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We thank Jac Heckelman for comments on an early version of this paper.
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Dougherty, K.L., Edward, J. The effect of unconditional preferences on Sen’s paradox. Theory Decis 93, 427–447 (2022). https://doi.org/10.1007/s11238-021-09863-8
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DOI: https://doi.org/10.1007/s11238-021-09863-8