Abstract
We investigate the social choice problem in which the range of a rule consists of only two alternatives. While Barberà et al. (Int J Game Theory 41:791–808, 2012a) capture the feature of “strategy-proof” rules based on the monotonicity condition of winning coalitions, we newly consider the “monotonicity condition with respect to the direction of preference changes.” We show that a rule is strategy-proof if and only if it satisfies the monotonicity condition of preference changes. In addition, we define the class of “upper set rules,” and show that these rules are characterized by strategy-proofness.
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Notes
Our results are also applicable in the pure exchange economy (Barberà and Jackson 1995) in which the range of a rule has some disconnected jumps, the social choice problem in which there are only two alternatives (Harless 2015; Lahiri and Pramanik 2017; Larsson and Svensson 2006; Manjunath 2012; Marchant and Mishra 2015), and the problem of locating a public facility on the set of preference profiles such that each agent in a subset M of the set N of agents has single-peaked preferences and each agent in \( N\backslash M\) has single-dipped preferences (Alcalde-Unzu and Vorsatz 2018).
For the definition of minmax rules, see, for example, Massó and Moreno de Barreda (2011).
For more detailed discussions concerning the range of a rule on the set of all single-dipped preferences, see Barberà et al. (2012b).
An “essential” condition means that the condition holds when we only compare profiles where agents indifferent between x and y keep their preferences unchanged.
In the problem of choosing a subset of a finite set of indivisible objects with strict and separable preferences, Barberà et al. (1991) define “voting by committees” based on a similar monotonicity condition to essentialxy-monotonicity. In the social choice problem in which there are only two alternatives when agents may be indifferent between them, a similar monotonicity condition is also considered in “voting by extended committees” by Larsson and Svensson (2006) and Manjunath (2012).
For the discussion concerning open and “closed” characterizations, see Massó and Moreno de Barreda (2011). They provide a closed characterization of strategy-proof rules in which they explicitly describe how to select alternatives in the problem involving choosing one alternative on the set of “symmetric” single-peaked preferences in which the range of a rule has some disconnected jumps. Applying our results, Hagiwara et al. (2019) propose a closed characterization of strategy-proof rules in this problem not only on the set of symmetric single-peaked preferences, but also “asymmetric” single-peaked preferences. In the same problem as ours, Barberà et al. (2012a) provide a closed characterization of “strongly group” strategy-proof rules with binary ranges. For the definition of strong group strategy-proofness, see Barberà et al. (2012a).
In the same model as ours, Vannucci (2013) provides another open characterization of strategy-proof rules with binary ranges. In the problem involving choosing one alternative on the set of all single-peaked preferences in which the range of a rule has some disconnected jumps, Barberà and Jackson (1994) also proposes an open characterization of strategy-proof rules with binary ranges.
In the social choice problem with binary ranges of rules and strict preferences, strategy-proof rules are characterized (Rao et al. 2018). There are two differences between Rao et al. (2018) and this paper. First, in their model, indifferences are not allowed, but we allow them. Second, in their characterization, a nonempty collection of agent coalitions is considered. In our characterization, however, we consider an upper set of a set of preferences.
This is applicable in the pure exchange economy (Barberà and Jackson 1995) in which the range of a rule has some disconnected jumps.
This is applicable in the social choice problem in which there are only two alternatives and the problem of locating a public facility on the set of preference profiles such that each agent in a subset M of N has single-peaked preferences and each agent in \( N\backslash M\) has single-dipped preferences.
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We are grateful to two anonymous reviewers of this journal, Patrick Harless, Ryo Kawasaki, Vikram Manjunath, Goro Ochiai, Satoru Takahashi, William Thomson, Jingyi Xue, and Takehiko Yamato, as well as the participants at 2019 Conference on Economic Design (Corvinus University of Budapest, Hungary) for their invaluable comments and suggestions. Hagiwara was partially supported by JSPS KAKENHI Grant Number JP17J01520 and JSPS Overseas Challenge Program for Young Researchers Grant Number 201780041. Yamamura was partially supported by JSPS KAKENHI Grant Numbers JP26285045 and 18K12744.
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Hagiwara, M., Yamamura, H. Upper set rules with binary ranges. Soc Choice Welf 54, 657–666 (2020). https://doi.org/10.1007/s00355-019-01225-2
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DOI: https://doi.org/10.1007/s00355-019-01225-2