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Social Choice and Welfare

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17-04-2023 | Original Paper

Unanimity and local incentive compatibility in sparsely connected domains

Authors: Miho Hong, Semin Kim

Published in: Social Choice and Welfare | Issue 2/2023

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Abstract

This paper studies the implications of imposing unanimity and local incentive compatibility on a deterministic social choice function. In an environment with strict ordinal preferences over a finite set of alternatives, we find that tops-onlyness and full incentive compatibility necessarily follow from unanimity and local incentive compatibility in sparsely connected domains. Furthermore, we identify a property of preference domains that completely characterizes dictatorial domains within sparsely connected domains.

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Holmström and Myerson (1983) classify Pareto efficiency concepts depending on the stage of information regarding the preferences of agents, among which ex-post Pareto efficiency is the weakest. Clearly, ex-post Pareto efficiency implies unanimity.

Local dominant strategy proofness corresponds to AM-proofness in Sato (2013).

By abuse of notation, we refer to SCD and CD to indicate both the class of preference domains and an element of that class.

Roy and Storcken (2019) consider domains that satisfy top-connectedness, pervasiveness, and richness.

Both of the papers discuss their relationship with our paper by giving examples which would be helpful for interested readers.

We follow Mishra (2016)’s notations.

The term “restoration” comes from the restoration of relative rankings between alternatives a and b along the sequence.

Sato (2013) considers a similar condition, requiring that for each pair of alternatives a and b, there exists a path connecting each pair of preference orderings without \(\{a,b\}\)-restoration. SCD includes domains that satisfy this condition.

Example 3.2 in Sato (2013) shows an SCF defined on SCD that is LDSIC but not DSIC.

This result plays a crucial role in the proofs of Mishra (2016) as well as in ours.

For any proper subset \(\bar{N} \subset N\), \(P_{-\bar{N}}\) denotes the preference profile of agents \(j\in N \setminus \bar{N}\). When \(\bar{N}\) is a singleton, we omit the parentheses. In addition, from here onward, we write \((\hat{P},P_{-i})\) in place of \((P_i = \hat{P},P_{-i})\) and \((\hat{P},\hat{P}',P_{-\{i,i'\}})\) in place of \((P_i = \hat{P},P_{i'}=\hat{P}',P_{-\{i,i'\}})\).

Such a path exists since \(\{a,b\} \cap \bar{A} \ne \varnothing .\)

This is possible if the sequence is with \(\{a^*,a\}\)-restoration for some \(a \in A\), \(a \ne f(\bar{P},\bar{P}_{-i})\).

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Title: Unanimity and local incentive compatibility in sparsely connected domains
Authors: Miho Hong
Semin Kim
Publication date: 17-04-2023
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2023
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-023-01457-3

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