Set-Rationalizable Choice and Self-Stability

A common assumption in modern microeconomic theory is that choice should be rationalizable via a binary preference relation, which \citeauthor{Sen71a} showed to be equivalent to two consistency conditions, namely $\alpha$ (contraction) and $\gamma$ (expansion). Within the context of \emph{social} choice, however, rationalizability and similar notions of consistency have proved to be highly problematic, as witnessed by a range of impossibility results, among which Arrow's is the most prominent. Since choice functions select \emph{sets} of alternatives rather than single alternatives, we propose to rationalize choice functions by preference relations over sets (set-rationalizability). We also introduce two consistency conditions, $\hat\alpha$ and $\hat\gamma$, which are defined in analogy to $\alpha$ and $\gamma$, and find that a choice function is set-rationalizable if and only if it satisfies $\hat\alpha$. Moreover, a choice function satisfies $\hat\alpha$ and $\hat\gamma$ if and only if it is \emph{self-stable}, a new concept based on earlier work by \citeauthor{Dutt88a}. The class of self-stable social choice functions contains a number of appealing Condorcet extensions such as the minimal covering set and the essential set.Comment: 20 pages, 2 figure, changed conten

Widwast Choice

A choice function is (weakly) width-maximizing if there exists a dissimilarity- i.e. an irreflexive symmetric binary relation- on the underlying object set such that the choice sets are (include, respectively) dissimilarity chains of locally maximum size. Width-maximizing and weakly width-maximizing choice functions on an arbitrary domain are characterized relying on the newly introduced notion of a revealed dissimilarity relation.

Set-Monotonicity Implies Kelly-Strategyproofness

This paper studies the strategic manipulation of set-valued social choice functions according to Kelly's preference extension, which prescribes that one set of alternatives is preferred to another if and only if all elements of the former are preferred to all elements of the latter. It is shown that set-monotonicity---a new variant of Maskin-monotonicity---implies Kelly-strategyproofness in comprehensive subdomains of the linear domain. Interestingly, there are a handful of appealing Condorcet extensions---such as the top cycle, the minimal covering set, and the bipartisan set---that satisfy set-monotonicity even in the unrestricted linear domain, thereby answering questions raised independently by Barber\`a (1977) and Kelly (1977).Comment: 14 page

Positively responsive collection choice rules and majority rule: a generalization of May's theorem to many alternatives

A collective choice rule selects a set of alternatives for each collective choice problem. Suppose that the alternative ’x’, is in the set selected by a collective choice rule for some collective choice problem. Now suppose that ‘x’ rises above another selected alternative ‘y’ in some individual’s preferences. If the collective choice rule is “positively responsive”, ‘x’ remains selected but ‘y’ is no longer selected. If the set of alternatives contains two members, an anonymous and neutral collective choice rule is positively responsive if and only if it is majority rule (May 1952). If the set of alternatives contains three or more members, a large set of collective choice rules satisfy these three conditions. We show, however, that in this case only the rule that assigns to every problem its strict Condorcet winner satisfies the three conditions plus Nash’s version of “independence of irrelevant alternatives” for the domain of problems that have strict Condorcet winners. Further, no rule satisfies the four conditions for the domain of all preference relations

Combining social choice functions

This paper considers the problem of combining two choice functions (CFs), or setwise optimisation functions, based on use of intersection and composition. Each choice function represents preference information for an agent, saying, for any subset of a set of alternatives, which are the preferred, and which are the sub-optimal alternatives. The aim is to find a combination operation that maintains good properties of the choice function. We consider a family of natural properties of CFs, and analyse which hold for different classes of CF. We determine relationships between intersection and composition operations, and find out which properties are maintained by these combination rules. We go on to show how the most important of the CF properties can be enforced or restored, and use this kind of procedure to define combination operations that then maintain the desirable properties

Impossibility theorems involving weakenings of expansion consistency and resoluteness in voting

A fundamental principle of individual rational choice is Sen's $\gamma$ axiom, also known as expansion consistency, stating that any alternative chosen from each of two menus must be chosen from the union of the menus. Expansion consistency can also be formulated in the setting of social choice. In voting theory, it states that any candidate chosen from two fields of candidates must be chosen from the combined field of candidates. An important special case of the axiom is binary expansion consistency, which states that any candidate chosen from an initial field of candidates and chosen in a head-to-head match with a new candidate must also be chosen when the new candidate is added to the field, thereby ruling out spoiler effects. In this paper, we study the tension between this weakening of expansion consistency and weakenings of resoluteness, an axiom demanding the choice of a single candidate in any election. As is well known, resoluteness is inconsistent with basic fairness conditions on social choice, namely anonymity and neutrality. Here we prove that even significant weakenings of resoluteness, which are consistent with anonymity and neutrality, are inconsistent with binary expansion consistency. The proofs make use of SAT solving, with the correctness of a SAT encoding formally verified in the Lean Theorem Prover, as well as a strategy for generalizing impossibility theorems obtained for special types of voting methods (namely majoritarian and pairwise voting methods) to impossibility theorems for arbitrary voting methods. This proof strategy may be of independent interest for its potential applicability to other impossibility theorems in social choice.Comment: Forthcoming in Mathematical Analyses of Decisions, Voting, and Games, eds. M. A. Jones, D. McCune, and J. Wilson, Contemporary Mathematics, American Mathematical Society, 202

Complexity of Manipulating and Controlling Approval-Based Multiwinner Voting

We investigate the complexity of several manipulation and control problems under numerous prevalent approval-based multiwinner voting rules. Particularly, the rules we study include approval voting (AV), satisfaction approval voting (SAV), net-satisfaction approval voting (NSAV), proportional approval voting (PAV), approval-based Chamberlin-Courant voting (ABCCV), minimax approval voting (MAV), etc. We show that these rules generally resist the strategic types scrutinized in the paper, with only a few exceptions. In addition, we also obtain many fixed-parameter tractability results for these problems with respect to several natural parameters, and derive polynomial-time algorithms for certain special cases.Comment: 45pages, 1figure, full version of a paper at IJCAI 201