Condorcet Domains, Median Graphs and the Single Crossing Property

Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters belong to this set, the majority relation has no cycles. We observe that, without loss of generality, such domain can be assumed to be closed in the sense that it contains the majority relation of every profile with an odd number of individuals whose preferences belong to this domain. We show that every closed Condorcet domain is naturally endowed with the structure of a median graph and that, conversely, every median graph is associated with a closed Condorcet domain (which may not be a unique one). The subclass of those Condorcet domains that correspond to linear graphs (chains) are exactly the preference domains with the classical single crossing property. As a corollary, we obtain that the domains with the so-called `representative voter property' (with the exception of a 4-cycle) are the single crossing domains. Maximality of a Condorcet domain imposes additional restrictions on the underlying median graph. We prove that among all trees only the chains can induce maximal Condorcet domains, and we characterize the single crossing domains that in fact do correspond to maximal Condorcet domains. Finally, using Nehring's and Puppe's (2007) characterization of monotone Arrowian aggregation, our analysis yields a rich class of strategy-proof social choice functions on any closed Condorcet domain

Axiomatic Districting

In a framework with two parties, deterministic voter preferences and a type of geographical constraints, we propose a set of simple axioms and show that they jointly characterize the districting rule that maximizes the number of districts one party can win, given the distribution of individual votes (the \optimal gerrymandering rule"). As a corollary, we obtain that no districting rule can satisfy our axioms and treat parties symmetrically

Non-bossy social classification

We consider the problem of how societies should be partitioned into classes if individuals express their views about who should be put with whom in the same class. A non-bossy social aggregator depends only on those cells of the individual partitions the society members classify themselves in. This fact allows us to concentrate on a corresponding 'opinion graph' for each profile of views. By means of natural sovereignty, liberalism, and equal treatment requirements, we characterize the non-bossy aggregators generating partitions in which the social classes are refinements of the weakly connected components of the opinion graph. --social aggregation,group identity,liberalism, non-bossiness

Non-bossy Social Classification

We consider the problem of how societies should be partitioned into classes if individuals express their views about who should be put with whom in the same class. A non-bossiness condition makes the social aggregator dependent only on those cells of the individual partitions the society members classify themselves in. This information is used to construct for each profile of views an opinion graph. By means of natural sovereignty and liberalism requirements, we characterize the non-bossy aggregators generating partitions in which the social classes are refinements of the connected components in the opinion graph.Social Aggregation, Group Identity, Liberalism, Non-bossiness

Nash implementable domains for the Borda count

We characterize the preference domains on which the Borda count satisfies Maskin monotonicity. The basic concept is the notion of a "cyclic permutation domain" which arises by fixing one particular ordering of alternatives and including all its cyclic permutations. The cyclic permutation domains are exactly the maximal domains on which the Borda count is strategy-proof (when combined with every tie breaking rule). It turns out that the Borda count is monotonic on a larger class of domains. We show that the maximal domains on which the Borda count satisfies Maskin monotonicity are the "cyclically nested permutation domains." These are the preference domains which can be obtained from the cyclic permutation domains in an appropriate recursive way.Maskin monotonicity; Borda count; restricted preference domains

The single-peaked domain revisited: A simple global characterization

It is proved that, among all restricted preference domains that guarantee consistency (i.e. transitivity) of pairwise majority voting, the single-peaked domain is the only minimally rich and connected domain that contains two completely reversed strict preference orders. It is argued that this result explains the predominant role of single-peakedness as a domain restriction in models of political economy and elsewhere. The main result has a number of corollaries, among them a dual characterization of the single-dipped do-main; it also implies that a single-crossing (`order-restricted\u27) domain can be minimally rich only if it is a subdomain of a single-peaked domain. The conclusions are robust as the results apply both to domains of strict and of weak preference orders, respectively

Envy Freeness in Experimental Fair Division Problems

In the recent experimental literature several social preference models have been suggested that address observed behavior not reducible to the pursuit of self-interest. Inequality aversion is one such model where preferences are distributional. Frequently, envy is suggested as the underlying rationale for inequality aversion. Envy is a central criterion in the theoretical literature on fair division, whose definition (Foley 1967) differs from the more casual use of the word in the experimental literature. We present and discuss results from free-form bargaining experiments on fair division problems where the role of envy in Foley’s sense can be analyzed and compared to social preferences. We find that envy freeness does matter as a secondary criterion.Fairness, Envy Freeness, Social Preferences, Bargaining

Optimal redistricting under geographical constraints: Why "pack and crack" does not work

We show that optimal partisan redistricting with geographical constraints is a computationally intractable (NP-complete) problem. In particular, even when voter's preferences are deterministic, a solution is generally not obtained by concentrating opponent's supporters in \unwinnable" districts ("packing") and spreading one's own supporters evenly among the other districts in order to produce many slight marginal wins ("cracking")

Condorcet admissibility: Indeterminacy and path-dependence under majority voting on interconnected decisions

Judgement aggregation is a model of social choice where the space of social alternatives is the set of consistent evaluations (`views') on a family of logically interconnected propositions, or yes/no-issues. Unfortunately, simply complying with the majority opinion in each issue often yields a logically inconsistent collection of judgements. Thus, we consider the Condorcet set: the set of logically consistent views which agree with the majority in as many issues as possible. Any element of this set can be obtained through a process of diachronic judgement aggregation, where the evaluations of the individual issues are decided through a sequence of majority votes unfolding over time, with earlier decisions possibly imposing logical constraints on later decisions. Thus, for a fixed profile of votes, the ultimate social choice can depend on the order in which the issues are decided; this is called path dependence. We investigate the size and structure of the Condorcet set ---and hence the scope and severity of path-dependence ---for several important classes of judgement aggregation problems.judgement aggregation; diachronic; path-dependence; indeterminacy; Condorcet; median rule; majoritarian