Sen's Theorem Research Articles

Vertex operator algebras of rank $2$: The Mathur–Mukhi–Sen theorem revisited

Vertex operator algebras of rank $2$: The Mathur–Mukhi–Sen theorem revisited

Treading a fine line: (Im)possibilities for Nash implementation with partially-honest individuals

This paper investigates the robustness of Dutta and Sen's (2012) Theorem 1 to weaker notions of truth-telling. It models individual i's honesty standard as a profile of (possibly non-empty) collections of ordered pairs of outcomes, one for each member of society, over which individual i feels truth-telling concerns. Individual i is honest provided that she states her true preferences as well as rankings (not necessarily complete) of outcomes that are consistent with the true preferences of individuals in her honesty standard. Under this notion of honesty, we offer a condition, called S(N)-partial-honesty monotonicity, which is necessary for Nash implementation when there are partially-honest agents. In an independent domain of preferences, we show that this condition is equivalent to Maskin monotonicity provided that honesty means stating the orderings of individuals (in a honesty standard) truthfully and individuals' honesty standards are non-connected.

Proving classical theorems of social choice theory in modal logic

A number of seminal results in the field of social choice theory demonstrate the difficulties of aggregating the preferences of several individual agents for the purpose of making a decision together. We show how to formalise three of the most important impossibility results of this kind--Arrow's Theorem, Sen's Theorem, and the Muller---Satterthwaite Theorem--by using a modal logic of social choice functions. We also provide syntactic proofs of these theorems in the same logic. While prior work has been successful in applying tools from logic and automated reasoning to social choice theory, this is the first human-readable formalisation of the Arrovian framework allowing for a direct derivation of the main impossibility theorems of social choice theory. This is useful for gaining a deeper understanding of the foundations of collective decision making, both in human society and in groups of autonomous software agents.

On Arrow–Sen style equivalences between rationality conditions for fuzzy choice functions

The Arrow---Sen Theorem establishes the equivalence between different rationality conditions for a choice function. In this contribution we deal with fuzzy versions of these rationality conditions and we study the connection between them. We recall results found in the literature and we prove that they are valid under weaker conditions. We also present new implications and counterexamples that show that our results cannot be obtained under weaker conditions.

A further study on rationality conditions of fuzzy choice functions

The investigation of rationality plays a central role in the study of fuzzy choice functions. In this paper, we deal with some important rationality conditions of fuzzy choice functions, including Weak (Strong) Fuzzy Congruence Axiom, Weak (Strong) Axiom of Fuzzy Revealed Preference, fuzzy versions of famous crisp conditions α , β , γ and δ etc. in the framework of the Banerjee choice function. We systematically investigate the relationships between these conditions under the assumption that every involved choice set is normal and verify some equivalence results for arbitrary t-norms. As a result, a fuzzy version of the Arrow–Sen theorem is presented. The research makes a significant difference compared with the current investigation in the framework of the Georgescu choice function.

Aggregating Partially Ordered Preferences

Preferences are not always expressible via complete inear orders: sometimes it is more natural to allow for the presence of incomparable outcomes. This may hold both in the agents' preference ordering and in the social order. In this article, we consider this scenario and study what properties it may have. In particular, we show that, despite the added expressivity and ability to resolve conflicts provided by incomparability, classical impossibility results (such as Arrow's theorem, Muller–Satterthwaite's theorem and Gibbard–Satterthwaite's theorem) still hold. We also prove some possibility results, generalizing Sen's theorem for majority voting. To prove these results, we define new notions of unanimity, monotonicity, dictator, triple-wise value-restriction and strategy-proofness, which are suitable and natural generalizations of the classical ones for complete orders.

Congruence indicators for fuzzy choice functions

We introduce the congruence indicators WFCA(·) and SFCA(·) corresponding to fuzzy congruence axioms WFCA and SFCA. These indicators measure the degree to which a fuzzy choice function verifies the axioms WFCA and SFCA, respectively. The main result of the paper establishes for a given choice function the relationship between its congruence indicators and some rationality conditions. One obtains a fuzzy counterpart of the well-known Arrow–Sen theorem in crisp choice functions theory.

A simple proof of Sen's possibility theorem on majority decisions

Condorcet’s voting paradox shows that pairwise majority voting may lead to cyclical majority preferences. In a famous paper, Sen (1966) identified a general condition on a profile of individual preference orderings, called triplewise value-restriction, which is sufficient for the avoidance of such cycles. This note aims to make Sen’s result easily accessible. We provide an elementary proof of Sen's possibility theorem and a simple reformulation of Sen’s condition. We discuss how Sen’s condition is logically related to a number of precursors. Finally, we state a necessary and sufficient condition for the avoidance of cycles, and suggest that, although there is still some logical space between that condition and Sen’s sufficient condition, Sen’s condition cannot be further generalized in an appealing way.

Ramification des séries formelles

AbstractLet p be a prime number. Let k be a finite field of characteristic p. The subset X + X2k[[X]] of the ring k[[X]] is a group under the substitution law ○ sometimes called the Nottingham group of k, it is denoted by Rk. The ramification of one series γ ∈ Rk is caracterized by its lower ramification numbers: , as well as its upper ramification numbers:By Sen's theorem, the um(γ) are integers. In this paper, we determine the sequences of integers (um) for which there exists γ ∈ Rk such that um(γ) = um for all integer m & 0.

(100–200/ m)% veto power

One interpretation of Sen's theorem is that, by making an individual decisive over both a pair and its symmetric pair, the individual can veto the existence of another decisive individual. This paper considers a weaker form of power (almost decisiveness) and determines the veto power (expressed in terms of the ability to prevent the existence of a disjoint almost decisive group) that is attributed to a group G of individuals by a unanimous preference aggregation function when G is successively allowed to be almost decisive over one pair, two pairs and three pairs of alternatives. Some extensions and variations on these results are also suggested.

Minimal Liberties, Maximal State and the Market: The Case for a Coasian Liberal

The paper compares the Sen's paradox on the Paretian liberal (1970) and the standard version of the Coase theorem (1960). The comparison of the two theorems is somehow puzzling: If we accept Sen then we have to forget Coase and vice-versa. Following the analysis of Hillinger and Lapham (1971), Nozick (1974) and Gibbard (1974), among others, the paper first shows a possibility result for a Paretian Liberal by applying the Coase Theorem in its hard version and then shows an impossibility result for a Coasian liberal by applying the Sen Theorem. What generates a possibility result for a Paretian Liberal shows also the conditions for an impossibility result for a Coasian Liberal unless any hierarchy of social preferences is introduced by any allocation of rights in order to induce social preferences to follow rights. This conclusion leads us to show the relevance of ex-ante (alienable) rights definition for a Coasian market to work and the necessity of a maximal state (which properly defines an alienable right for every relevant state of the world) - instead of the Nozick's minimal state - to enforce minimal liberty when agents fail to properly define a complete (alienable) rights allocation over the social choice domain.

How Liberalism Kills Democracy or Sen's Theorem Revisited

First, a general qualification rule based on individualpreferences is proposed which allows any given coalition todistinguish among its members some individuals who are said tobe qualified by it since sensitive in preferences withall individuals of this coalition. A particular qualificationrule, the liberal one, is then introduced as a ruleconferring the power to qualify or disqualify any individualon the individual himself. Now, since each preference propertycorresponds to a qualification property, the liberalqualification rule is precisely characterized by individualpreferences. Second, a delegator, that is a mappingdescribing a subset of delegates within the society, isdefined to justify the standard notion of decisiveness.The idea of delegation allows us to generalize Arrow-Senframework. This is done by means of a collective rationalitypostulate such that a coalition must be competent, i.e.includes all delegates, to be decisive. Then, we prove that(1) there exists a delegate who is qualified by any coalitionhe belongs to, (2) such a delegate is a dictator iff thequalification rule is liberal, (3) the qualification rule isliberal iff the preferences are selfish.

On the existence of moments of generalized order statistics

Sufficient conditions for the existence of moments of generalized order statistics are established including an extension of Sen's theorem, which is valid in the very particular case of ordinary order statistics.

A social choice function implementable via backward induction with values in the ultimate uncovered set

We prove the existence of a social choice function implementable via backward induction which always selects within the ultimate uncovered set. Whereas the uncovered set is the set of maximal elements of the covering relation, the ultimate uncovered set is the set obtained by iterative application of this covering operation. Dutta and Sen (1993) showed that any social choice function which is the solution of a generalized binary voting procedure is implementable via backward induction. Our result follows from Dutta and Sen's theorem, in that we construct a binary voting procedure always selecting within the ultimate uncovered set. We use the classical multistage elimination procedure, which always selects an alternative within the uncovered set. When this procedure is also used to select among all of the possible agendas or orderings of alternatives within the procedure, the alternative selected (from the agenda selected) will be within the uncovered set of the uncovered set. Our result follows from repeated application of this construction. Intuitively, the procedure constructed consists of requiring agents to vote on how they should vote and so on.

Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988).

P-adic Periodic Points and Sen's Theorem

To count the number of periodic points ofu(x) brings in very delicate questions about automorphisms of local fields. Using the ideas developed here, we shall give another short proof of Sen's Theorem. We also give an upper bound for the number of periodic points in anyp-adic number field.

Sen's Theorem on Iteration of Power Series

Sen's Theorem on Iteration of Power Series

Rights and Social Choice: Is There a Paretian Libertarian Paradox?

In 1970 Amartya Sen exposed an apparent antinomy that has come to be known as the Paradox of the Paretian Libertarian (Sen, 1970b, pp. 152–57). Sen introduced his paradox by establishing a simple but startling theorem. Roughly put, what he proved was that if a mechanism for selecting social choice functions satisfies two standard adequacy conditions, there are possible situations in which it will violate either the very weak libertarian precept that every individual has at least some rights or the seemingly innocuous Paretian principle that an option should be judged unacceptable if there is an available alternative that everyone prefers to it. Many economists and philosophers have proposed solutions to Sen's problem, but there is no general consensus on what solution (if any) is correct. In the present paper I argue that Sen's original theorem fails to establish the existence of any conflict between libertarianism and Paretianism. Furthermore, I contend that Sen has misinterpreted certain other theorems which he has used to defend the existence of a paradoxical conflict between these two doctrines. In general, I try to show that whenever Sen posits a Paretian-libertarian conflict to explain an apparently troubling result in social choice theory, the difficulty can be better dealt with either by claiming that the theorem in question imposes overly strong background constraints on the form of social choice functions or by claiming that it relies on an unacceptable construal of individual rights.

A general constitutional possibility theorem

Arrow's Impossibility Theorem shows that there exist no transitive social preferences in nonoligarchic societies for all possible profiles of individual preference orderings. Similarly, a generalization of Sen's Theorem of the Impossibility of a Paretian Liberal implies under the same conditions that non-Pareto-optimal outcomes may be present in the resulting cyclical preference relations. This essay changes the customary perspective. It demonstrates that, for any profile of individual preferences, we can always find a nonoligarchic assignment of rights to different subsets of society, to decide between pairs of outcomes, together with adequate decision rules, which escapes these problems. This assignment of rights can be a purely liberal one even for each profile, but not one in which everybody participates in all decisions and which uses simple or qualified-majority voting (Total Direct Democracy). The adequate purely liberal constitution, on the other hand, may imply an oligarchy if too few outcomes are present.

Liberty, Preference, and Choice

Ever since its first publication in 1970, Amartya Sen's paper “The Impossibility of a Paretian Liberal” (reprinted in Sen, 1982) has served as the starting point for almost all discussions of liberty in social choice theory. However, a number of people, myself included, have argued that Sen's theorem rests on a misleading characterization of liberty (Nozick, 1974, pp. 164–166; Bernholz, 1974; Sugden, 1978, 1981, pp. 193–198; Garden-fors, 1981; Levi, 1982). In a recent paper, addressed to a philosophical audience, Sen (1983) has provided a careful defence of his theorem against this charge. I shall argue that this defence does not work.