CONDORCET CHOICE FUNCTIONS AND MAXIMAL ELEMENTS

Choice functions on tournaments always select the maximal element (Condorcet winner), provided they exist, but this property does not hold in the more general case of weak tournaments. In this paper we analyze the relationship between the usual choice functions and the set of maximal elements in weak tournaments. We introduce choice functions selecting maximal elements, whenever they exist. Moreover, we compare these choice functions with those that already exist in the literature.choice functions, tournaments, maximal elements.

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

Consistent Probabilistic Social Choice

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (Rev. Econ. Stud., 51(4), 683--692, 1984). Fishburn's function returns so-called maximal lotteries, i.e., lotteries that correspond to optimal mixed strategies of the underlying plurality game. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming

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

Condorcet domains of tiling type

A Condorcet domain (CD) is a collection of linear orders on a set of candidates satisfying the following property: for any choice of preferences of voters from this collection, a simple majority rule does not yield cycles. We propose a method of constructing "large" CDs by use of rhombus tiling diagrams and explain that this method unifies several constructions of CDs known earlier. Finally, we show that three conjectures on the maximal sizes of those CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic

Minimal Stable Sets in Tournaments

We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncovered set, the Banks set, the minimal covering set, the tournament equilibrium set, the Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set.Comment: 29 pages, 4 figures, changed conten

- CHOICE FUNCTIONS: RATIONALITY RE-EXAMINED.

On analyzing the problem that arises whenever the set of maximal elements is large, and aselection is then required (see Peris and Subiza, 1998), we realize that logical ways of selectingamong maximals violate the classical notion and axioms of rationality. We arrive at the sameconclusion if we analyze solutions to the problem of choosing from a tournament (where maximalelements do not necessarily exist). So, in our opinion the notion of rationality must be discussed,not only in the traditional sense of external conditions (Sen, 1993) but in terms of the internalinformation provided by the binary relation.Rationality; Choice Functions; Maximal Elements.

Computing Tournament Solutions using Relation Algebra and REL VIEW

We describe a simple computing technique for the tournament choice problem. It rests upon a relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView's labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.Tournament, relational algebra, RelView, Copeland set, Condorcet non-losers, Schwartz set, top cycle, uncovered set, minimal covering set, Banks set, tournament equilibrium set.