[Paper Review] Strategy-proofness and single-peackedness in bounded distributive lattices

Ernesto Savaglio, Stefano Vannucci|arXiv (Cornell University)|Jun 15, 2014

Game Theory and Voting Systems33 references3 citations

TL;DR

This paper establishes that strategy-proof voting rules on bounded distributive lattices under unimodal or locally strictly unimodal preferences are precisely those representable as iterated medians of projections and constants, or equivalently as the behavior of a median tree-automaton. It further shows that individual and coalitional strategy-proofness are not equivalent in this setting, and that no anonymous voting rule can be coalitionally strategy-proof under minimal neutrality in non-linear bounded distributive lattices.

ABSTRACT

Two distinct specifications of single peakedness as currently met in the relevant literature are singled out and discussed. Then, it is shown that, under both of those specifications, a voting rule as defined on a bounded distributive lattice is strategy-proof on the set of all profiles of single peaked total preorders if and only if it can be represented as an iterated median of projections and constants, or equivalently as the behaviour of a certain median tree-automaton. The equivalence of individual and coalitional strategy-proofness that is known to hold for single peaked domains in bounded linear orders fails in such a general setting. A related impossibility result on anonymous coalitionally strategy-proof voting rules is also obtained.

Motivation & Objective

• To characterize strategy-proof voting rules on bounded distributive lattices under single-peaked preferences.
• To investigate the relationship between individual and coalitional strategy-proofness in non-linear lattice settings.
• To determine whether anonymous voting rules can be coalitionally strategy-proof in bounded distributive lattices.
• To formalize voting rules as median operations and tree-automata behaviors in lattice-structured outcome spaces.
• To clarify the distinction between two specifications of single-peakedness: the 'compromise' view and the 'top proximity' view.

Proposed method

• The paper defines two specifications of single-peakedness: unimodal and locally strictly unimodal total preorders on bounded distributive lattices.
• It models voting rules as iterated medians of projections and constants, leveraging the lattice structure to ensure strategy-proofness.
• It introduces median tree-automata as computational models, where each finite labeled tree represents a voting profile and the automaton computes the outcome via backward induction.
• The behavior of a median tree-automaton is defined as a homomorphism from the free algebra of finite labeled trees into a state space, with output determined by a final function.
• The paper uses algebraic structures—specifically, the free Σ-algebra generated by a set of variables—to formalize the recursive computation of voting outcomes.
• It proves equivalence between voting rules defined as iterated medians and those computed by median tree-automata, establishing a computational and algebraic characterization.

Experimental results

Research questions

• RQ1 What voting rules are strategy-proof on bounded distributive lattices under unimodal preferences?
• RQ2 What voting rules are strategy-proof under locally strictly unimodal preferences in the same setting?
• RQ3 Is individual strategy-proofness equivalent to coalitional strategy-proofness in bounded distributive lattices that are not linear orders?
• RQ4 Can anonymous voting rules be coalitionally strategy-proof in bounded distributive lattices under minimal neutrality?
• RQ5 How do the 'compromise' and 'top proximity' views of betweenness consistency affect the characterization of strategy-proof rules?

Key findings

• All strategy-proof voting rules on the full unimodal and locally strictly unimodal domains in bounded distributive lattices are representable as iterated medians of projections and constants.
• The same class of voting rules is equivalent to the behavior of a median tree-automaton, providing a computational model for strategy-proof rules.
• The equivalence between individual and coalitional strategy-proofness, valid in linear orders, fails in general bounded distributive lattices.
• No anonymous voting rule is coalitionally strategy-proof on unimodal or locally strictly unimodal domains in bounded distributive lattices that are not linear orders, under minimal neutrality.
• The simple majority rule (extended median) remains strategy-proof and coalitionally strategy-proof in this generalized lattice setting.
• The paper formalizes two distinct notions of single-peakedness: the 'compromise' view (where intermediates are not worse than extremes) and the 'top proximity' view (where intermediates are strictly better than farther outcomes), and shows their implications for strategy-proofness.