Local‐global equivalence in voting models: A characterization and applications

Abstract

The paper considers a voting model where each voter's type is her preference. The type graph for a voter is a graph whose vertices are the possible types of the voter. Two vertices are connected by an edge in the graph if the associated types are “neighbors.” A social choice function is locally strategy‐proof if no type of a voter can gain by misrepresentation to a type that is a neighbor of her true type. A social choice function is strategy‐proof if no type of a voter can gain by misrepresentation to an arbitrary type. Local‐global equivalence (LGE) is satisfied if local strategy‐proofness implies strategy‐proofness. The paper identifies a condition on the graph that characterizes LGE. Our notion of “localness” is perfectly general. We use this feature of our model to identify notions of localness according to which various models of multidimensional voting satisfy LGE. Finally, we show that LGE for deterministic social choice functions does not imply LGE for random social choice functions.

The structure of (local) ordinal Bayesian incentive compatible random rules

“…For random rules, to the best of our knowledge, it is still not known whether sep-LDSIC implies DSIC or not. However, the same is shown for DSCFs on domains having unrestricted marginals (see Kumar et al (2020) for details). Thus, it follows from Corollary 9.3 that for almost all sep-monotonic DSCFs, OBIC and DSIC are equivalent on such domains.…”

“…Since lex-LDSIC implies DSIC for DSCFs on lexicographic domains in which component orderings are swap-connected and marginal domains are regular and swap-DLGE (see Kumar et al (2020) for details), it follows that for almost all lex-monotonic DSCFs, OBIC and DSIC are equivalent.…”

“…Proof. Kumar et al (2020) show that a domain D is graph-DLGE if and only if it satisfies the following property: for all distinct P, P ′ ∈ D and all a ∈ A, there exists a path π from P to P ′ with no (a, b)-restoration for all b ∈ L(a, P). Here, a path is said to have no (a, b)-restoration if the relative ranking of a and b is reversed at most once along π.…”

“…As the name suggests, they apply to deviations/misreports to only "local" preferences (the notion of which is fixed a priori). The importance of these local notions is well-established in the literature: on one hand, they are useful in modeling behavioral agents (see Carroll (2012)), on the other hand, on many domains they turn out to be equivalent to their corresponding global versions and thereby used as a simpler way to check whether a given RSCF is DSIC (see Carroll (2012), Kumar et al (2020), Sato (2013, Cho (2016), etc. ).…”

Probabilistic Fixed Ballot Rules and Hybrid Domains

“…The third condition, no-restoration requires the existence of a path from one preference to another in the domain, by a sequence of specific preference switches. No-restoration has been shown to be a key property for the equivalence between local and global strategy-proofness -see Sato (2013), Kumar et al (2020Kumar et al ( , 2021 and Hong and Kim (2020). Related conditions on domains such as connectedness which has been used extensively in the literature on Condorcet domains (e.g.…”