Domain Closedness Conditions and Rational Choice

The rationalizability of a choice function on an arbitrary domain under various coherence properties has received a considerable amount of attention both in the long-established and in the recent literature. Because domain closedness conditions play an important role in much of rational choice theory, we examine the consequences of these requirements on the logical relationships among different versions of rationalizability. It turns out that closedness under intersection does not lead to any results differing from those obtained on arbitrary domains. In contrast, closedness under union allows us to prove an additional implication.

A Study of Closure and Fuzzy Closure Spaces with Reference to H-Closedness and Convexity

Department of Mathematics, Cochin University of Science and Technology.

On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property).

Out of the darkness and into the light : a study of inauthentic and authentic speech

Introduction Man can be described as the being who shows himself in speech, and from birth to death is continually speaking. Communication is so close to us, so woven into our very being, that we have little understanding of the way it is constituted; for it is as hard to obtain distance from communication as it is to obtain distance from ourselves. All communication is not alike. There are two basic modesl of communication, the inauthentic and the authentic, between which there occurs a constant tension. It is in the inauthentic mode, points out Heidegger, that we find ourselves "proximately and for the most part"; 1. Being and Time, pg. 68 Dasein decides as to the way it will comport itself in taking up its task of having being as an issue for it. " •.• it~, in its very being 'choose' itself and win itself; it can also lose itself and never win itself or only "seem" to do so. But only in so far as it is essentially something which can be authentic--that is, something of its own--can it have lost itself and not yet won itself." 2. therefore Heidegger also terms it "everydayness".2 Caught up in the world of everydayness, our speaking covers over and conceals3 our rootedness in being, leaving us in the darkness of untruth. The image of darkness may be inferred from Heidegger's use of the image of "clearing,,4 to depict being as 2. ibid. pg. 69 "Dasein's average everydayness, however, is not to be taken as a mere 'aspect'. Here too, and even in the mode of inauthenticity, the structure of existentiality lies ~ priori and here too Dasein's being is an issue for it in a definite way; and Dasein comports itself towards it the mode of average everydayness, even if this is only the mode of fleeing in the face of it and forgetfulness thereof." 3. ibid. pg. 59 "covering over" and "concealing" are 1;yays Dasein tries to flee its task of having being as an issue for itself. " ••• This being can be covered up so extensively that it becomes forgotten and no question arises about it or its meaning ••• n How everyday speaking accomplishes this will be taken up in detail in the second chapter which explores Dasein's everyday speech. 4. ibid, pg. 171 lI ••• we have in mind nothing other than the Existential - ontological structure of this entity (Dasein), that it is in such a way as to be its 'there'. To say that it is -' illuminated' [tlerleuchtet"] means that as Being-in-theworld it is cleared [gelichtetJ in itself7 not through any other entity, but in such a way that it is itself the clearing. Only for an entity which is eXistentially cleared in this way does what is present-at-hand become accessible in the light or hidden in the dark •••• " 3 dis-coveredness and truth. Our first task will be to explore the nature of communication in general and then to explore each of the modes manifested in turn. The structure of the inauthentic mode of communication can be explored by asking the following questions: What is this speaking about? Who is it that is speaking and who is spoken to? Does this speaking show man in his speech? The authentic mode is distinguished by the rarity with which we encounter it; as the inauthentic conceals, so the authentic reveals our rootedness in being. Yet this rarity makes it difficult to delineate its elusive structure clearly. Its constituent elements can be brought into focus by asking the same questions of this mode that we previously asked of the inauthentic mode. Our initial response to the disclosure of the authentic mode is to attempt to abandon the inauthentic mode and leave the darkness behind dwelling only in the "lighted place". All through the ages, some men pushing this to extreme, have, upon uncovering their relatedness to being, experienced a deep longing to dwell in such a "place" of pure truth and oft times denigrated or attempted to exclude the everyday world. Such 4. flight is twice mistaken: first it atbempts to fix truth as unchanging and static and secondly, it opposes this to untruth which it seeks to abolish. This is both the wrong view of truth and the wrong view of untruth as Heidegger points out in The Origin of The-Work of Art: The Way-to-be of truth, i.e., of discoveredness, is under the sway of refusal. But this refusal is no lack or privation, as if truth could be simply discoveredness rid of all covers. If it could be that, it would no longer be itself . ••• Truth in its way-to-be is untruth.5 Pure light is not the nature of Being nor is pure unconcealedness possible for man. Failure to remember this is the failure to realize that communication destroys itself in such flight because it no longer maintains the contingency of its task, i.e., the dis-closedness of being. We are reminded of the strong attraction this flight from darkness held for Plato. Light, truth and Being are all beyond the darkness and have nothing to do with it. In Book VII of the R~public, Socrates' explanation of the Allegory of the Cave to Glaucon points to a decided preference men have for the "lighted place". 5. The Origin Of The Work Of Art, pg. 42 5. Come then, I said, and join me in this further thought, and do not be surprised that those who attained to this height are not willing to occupy themselves with the affairs of men, but their souls ever feel the upward urge and yearning for that sojourn above. For this, I take it, is likely if in this point too the likeliness of our image holds. 6 Despite the attraction to pure truth, human communication is more complex than putting down one mode of communication and picking up another. Due to the fact that we are always on the way, the title of my thesis will have to be amended: OUT OF THE DARKNESS AND INTO THE LIGHT--AGAIN AND AGAIN. It must be this way because this is what it means to be human. This is the point made by Mephisto to Faust in pointing out that man, standing between God and the devil, needs both darkness and light: Er findet sich in einem ewigen Gl~t Uns hat er in die Finsternis gebracht, Und euch taugt einzig Tag und Nacht. 7 6. Republic z (517 c & d) It should be noted however, that while the philosopherking must be compelled to return to the cave for purely political reasons, once he has taken adequate view of the "brightest region of being" he has the full truth and his return to darkness adds nothing to the truth. 7. Faust, pg. 188 6. This thesis proposes to examine the grounds that give rise to communication, uncovering the structure of its inauthentic and authentic modes and paying close attention to tpeir interrelationship and to their relationship to language as "the house of Being": language that both covers and opens up man's rootedness in Being, transforming him as he moves along his way, taking up his "ownmost task" of becoming who he is. roots. He is the being who shows himself inn that reflects his forgetfulness or remembrance of his rootedness in being. Man comes into an already existent world and is addressedl through things in the world which are c

Single-Plateaued Choice

Single-plateaued preferences generalize single-peaked preferences by allowing for multiple best elements. These preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of singleplateauedness in a choice-theoretic setting. Single-plateaued choice is characterized by means of a collinear interval continuity property in the presence of independence of irrelevant alternatives. Further results establish that our notion of single-plateauedness conforms to the motivation underlying the term and we analyze the consequences of alternative continuity properties. The importance of basic assumptions such as closedness and convexity is discussed. Journal of Economic Literature Classification Nos.: D11, D71.

Single-plateaued choice

Single-plateaued preferences generalize single-peaked preferences by allowing for multiple best elements. These preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of single-plateauedness in a choice-theoretic setting. Single-plateaued choice is characterized by means of a collinear interval continuity property in the presence of independence of irrelevant alternatives. Further results establish that our notion of single-plateauedness conforms to the motivation underlying the term and we analyze the consequences of alternative continuity properties. The importance of basic assumptions such as closedness and convexity is discussed. Journal of Economic Literature Classification Nos.: D11, D71.

Opting for an Open Society? Personality Traits and Attitudes toward the Openness of Switzerland

The tension between openness and closedness is one of the most important cleavages in Swiss political debates. In the present article, we study the psychological foundations of attitudes regarding this issue. More precisely, we examine the link between personality and attitudes toward the degree of openness of Switzerland as a general stance toward the cultural, economic and political alignment of the country. Personality is understood as a complex and multifaceted concept that forms the basis for consistent patterns of attitudes and behavior. We build on the Five-Factor Theory to explain the link between personality traits, contextual factors and political attitudes. Analyzing survey data from a random sample of Swiss citizens, we find clear evidence that personality traits affect political attitudes. Furthermore, we are able to demonstrate that the relationship between personality and attitudes toward the degree of openness of Switzerland is moderated by perceived ethnic diversity in the neighborhood.

From the Farkas Lemma to the Hahn–Banach Theorem

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn–Banach–Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn–Banach theorem and the Mazur–Orlicz theorem for extended sublinear functions.

Primal Attainment in Convex Infinite Optimization Duality

This article provides results guarateeing that the optimal value of a given convex infinite optimization problem and its corresponding surrogate Lagrangian dual coincide and the primal optimal value is attainable. The conditions ensuring converse strong Lagrangian (in short, minsup) duality involve the weakly-inf-(locally) compactness of suitable functions and the linearity or relative closedness of some sets depending on the data. Applications are given to different areas of convex optimization, including an extension of the Clark-Duffin Theorem for ordinary convex programs.

New glimpses on convex infinite optimization duality

Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that inf(P)=max(D), inf(P)=max(Q), and inf(P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing min(P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described.