We consider variable preference relations, called also reference dependent preference relations which are typical in the study of dynamic models in economic theories. We introduce a new concept of weak consistency, a generalization of acyclicity, as an immediate regret condition for variable preferences. The main result to establish is on an existence criterion for maximal elements of a space equipped with a weak consistent variable preference relation. It is expressed via preference completeness condition which is equivalent to existence of aspiration points. As applications, we show that a number of results known in the recent literature on maximum principles on a space with or without topological structure can be obtained from the unifying approach of this paper
Variable preference relations: existence of maximal elements Dinh The Luc∗and Soubeyran Antoine† June 21, 2012 Abstract We consider variable preference relations, called also reference dependent preference relations which are typical in the study of dynamic models in economic theories We introduce a new concept of weak consistency, a generalization of acyclicity, as an immediate regret condition for variable preferences The main result to establish is on an existence criterion for maximal elements of a space equipped with a weak consistent variable preference relation It is expressed via preference completeness condition which is equivalent to existence of aspiration points As applications, we show that a number of results known in the recent literature on maximum principles on a space with or without topological structure can be obtained from the unifying approach of this paper Keywords: Variable preference relation, improving path, maximal point, maximum principle JEL Classification: C60, C62 ∗ Avignon University, LMA EA2151, France The work of this author is partially supported by the VIASM, Hanoi, Vietnam † GREQAM, University of Aix-Marseille, France E-mail: antoine.soubeyran-at-gmail.fr; Tel (33) 4.91.14.07.27; Fax (33) 4.91.90.02.27 1 Introduction Preference is one of the fundamental concepts in economics and social sciences It refers to a way to put available alternatives in a certain order according to their degree of advantageousness or satisfaction In equilibrium theory a static point of view is dominant, that is, one is looking for situations in which all agents prefer to stay rather then to move; while in innovation theory (Schumpeter [24]) a dynamic point of view is more important, that is, one is interested in knowing when the agents prefer moving from their current situations than staying there These points of views are dual, once a preference is given In the classical models preference relations are a priori determined and constant They are transitive, and sometimes complete, often defined by utility functions, which greatly facilitate applications of mathematical tools However, in the real world preference relations vary with the context (experience, characters, emotions, mental state, social links, embeddedness ) and have evolutionary nature, generating dynamic processes of actions This is why models with changing preferences appeal much attention of researchers for several decades (see Basmann [4], Bleichbrodt [5], Koszegi and Rabin [17], Tversky and Kahneman [30] among many others) In the present study we consider preference relations, not necessarily being transitive or complete, or given by utility functions, but varying from state to state Our overall concern is the question under which circumstances maximal elements exist for a variable preference relation without transitivity Maximal elements play a central role in many economic models, including global maximum of a utility function and Nash equilibrium of a noncooperative game or equilibrium of an economy (Debreu [9]) Existence of maximal elements has been extensively studied for constant preference relations and literature on it is abundant However, to our knowledge there exists few papers dealing with existence of maximal elements in the case of variable preference relations Koszegi [16], Koszegi-Rabin [17] and some others study a personal equilibrium of consumption in which a particular variable preference is used via variable utility functions (see also Gul-Pesendorfer [14]); Soubeyran [26] considers a worthwhile to change function, which generates a variable preference relation to model a number of problems of social science such as habits, routines, behavioral traps etc in terms of maximal elements To establish existence criteria for maximal elements of a variable preference relation we exploit the concept of order-completeness borrowed from vector optimization which can be interpreted as a hypothesis on aspiration points recently introduced in the theory of change (see Soubeyran [25]) or Brezis-Browders inductivity hypothesis (Brezis and Browder [8]) when the preference relation is determined by a utility function Similar to the case of multi-criteria optimization given in Luc [19] and recently in Flores-Bazan et al [11] the method of order-complete sets is very useful in unifying results on existence of maximal elements in various contexts, including (a) existence of maximal elements in social choice theory with nontransitive and incomplete preferences (Bergstrom [6] for acyclic relations, Tian and Zhou [28] for transfer upper continuous preferences, Zuanon [35] for weakly tc-upper semicontinuous acyclic relations, Alcantud 2002 [1] for upper continuous preference on ” ”upper compact sets, Andrikopoulos and Zacharias [3] for consistent upper tc-Ssemicontinuous preferences); (b) maximum ordering principles in partially ordered spaces without compactness (Altman [2], Turinici [31], Szaz [27], Zeidler [33], Zhu and Li [34] in the context of partial ordering); and (c) effciency conditions in the framework of generalized multi-criteria optimization (Flores-Bazan et al [11]) The paper is structured as follows In Section we discuss variable preferences and present some examples to motivate our study In Section basic concepts and elementary properties of variable preference relations are given We introduce the concept of weak consistency which generalizes well-known concepts of acyclicity, path consistency and consistency by Suzumura We establish equivalence between acyclicity of a variable preference relation and its transitive closure being a partial order Section deals with a central concept of our paper: ex-ante maximal points and their existence We prove equivalence between a mathematical concept of preference-completeness and an economic concept of aspiration point and establish a criterion for existence of maximal elements in a space equipped with a variable preference relation It is based on the idea of order-complete sets from vector optimization and the weak consistency introduced in Section Section is devoted to applications of the main existence result of Section to several models of recent literature Related notions such as ex-ante ideal elements, ex-post ideal elements and ex-post maximal elements are also briefly discussed Applications of our approach to behavioral traps and equilibrium theory are addressed in [12] Variable preferences and improving processes of habit formation Let X be a set of states of an economic system A preference relation on X that evolves from state to state is called a variable preference Reference dependent, variable, dynamic and evolutionary preferences have the same meaning in our context They all describe a system in which at a given state x ∈ X an agent has a set of criteria that forms a preference helping him in choosing a new state to move to At this new state, again, he has a set of criteria that may differ from the one he has had at x and generates a new preference for the next move, as we have already mentioned in the introduction section Formally, at every state x ∈ X the agent has a preference relation on X, denoted ” ≤x ” and determines whether a future state z is preferred to a future state y in dependance on whether he has y ≤x z or not The point x is seen as his reference point x and he moves from x to y if at least x ≤x y In the context of decision making the agent’ s decision to move from a state x to a state y depends not only on the criterion x ≤x y (y is preferred to x ), but also on whether the move from x to y is realizable In general the decision maker possesses a state or transition function r defined on the product space A × X and taking values in X, where A denotes a set of available actions to carry out the moves on X A state y is said to be reachable from x if there is some action a from a subset A(x) of A such that r(a, x) = y Here A(x) is the set of actions in the agent’ s disposal when being at x, and hence it depends on x and not necessarily coincides with A When both ” ≤x ” and r are applicable, one may incorporate actions into the preference relation by defining a new variable preference relation: x x y if and only if x ≤x y and r(a, x) = y for some a ∈ A(x) Such a state y is considered as an improving reachable state from x In the present approach we focus on a general variable preference relation without explicitly mentioning the state function Here are some examples of variable preferences that motivate our study Experience and anchoring effects Consider the problem of habit formation where an agent makes a succession of consumptions xn ∈ X, n = 0, 1, , n More precisely, at every stage n, we consider the consumption path Xn = {x0 , x1 , , xn } ⊂ X of a consumer and denote the context (external events/ environment) perceived by the consumer by ωn The experience of the consumer is the set En = {(x0 , ω0 ), , (xn , ωn )} in which (x0 , x1 , , xn ) expresses his internal experience and (ω0 , ω1 , , ωn ) his external experience If a utility function is available, the consumer uses it to determine his next consumption The utility function is then a function on X with values in R (real numbers), and depends on the consumer’ s experience too, that is, it takes form V (.|En ) : X → R The consumer can take advantages of his experience over more or less past periods, forget some, remember others If he remembers only the last period (xn , ωn ) his utility function becomes a kind of a Markov reference dependent preference In some models the utility function depends only on his last action xn This is the case of the preference studied by Kahneman and Tversky [15] in which the anchoring effect or the impact of experience on utility is constant, so that the utility function depends only on the last consumption bundle xn This bundle is referred to as a reference point In a more general setting when a set of reference points R is given, one writes x ≤r y when the agent, based on the reference point r, prefers y to x, or in a dynamic setting, the agent accepts to pass from x to y A reference point represents the context of decisions and actions It can be a bundle of goods, a reference consumption, an initial endowment, a distribution of probability, a consumer choice, or a consumption to which an individual is used to or the one he aspires to in a social context (see Giraud [13]) In Kahneman and Tversky [15] the authors have shown how dependent preferences can explain a lot of anomalies, loss aversion, anchoring and status quo effects, endowment effects, framing and bracketing effects In all these cases gains and losses with respect to a reference point matter Reference consumption level In the simplest situation of the consumption model the set X is the real line R and a state y ≥ is interpreted as a quantity of a good to consume and r ≥ a reference level of consump4 tion In [17] Koszegi and Rabin use a particular form of the utility function V (y|r) := M (r) + µ(v(y) − v(r)) where M (r) is a traditional consumption utility, v(y) is a gain utility and µ(.) is a gain-loss utility (initially introduced in Kahneman and Tversky [15]) In our case r = x ∈ X Different types of utility functions describe different economic situations Here are two typical ones a) Linear function The gain utility function v is the identity function, that is v(x) = x for x ≥ 0, and µ(u) = α+ u for u positive, µ(u) = α− u for u negative with < α+ < α− Then V (y|x) = V (x|x) + α+ (y − x) − V (x|x) + α (x − y) if y x if y < x This utility function shows that at a reference point x, a gain u = y − x ≥ has a marginal utility α+ which is lower than the marginal desutility α− for loss u b) Power function In this case the utility function is determined by V (y|x) = V (x|x) + (y − x)α if y V (x|x) − λ(x − y)β if y ≤ x x where < α < β < ≤ λ Then, the agent improves his situation when he changes from x to y : x ≤x y if and only if V (y|x) ≥ V (x|x) This amounts to saying that moving from x to y is better than staying at x and consuming x again To answer the question: ’should I stay, should I go’, the consumer considers the advantage to change function A(x, y) := V (y|x) − V (x|x) Moving from x to y is better than staying at x and consuming x again, that is x ≤x y if and only if A(x, y) ≥ Thus, in this model the preference (≤x ) is not constant, it varies with the consumption quantity It is to notice that sometimes an additional hypothesis is imposed on the utility function, namely for x = y, if the agent prefers y to x at x, the same is true at y, that is V (y|x) ≥ V (x|x) ⇒ V (y|y) > V (x|y) This hypothesis assures acyclicity (Section 3) Thus, in this model A(x, y) ≥ and y = x implies A(y, x) < (see Kozsegi and Rabin [17], Proposition 1) Improving consumption processes and habits Consider a consumer who initially consumes a bundle of goods x0 ∈ X If he chooses to stay at x0 and consume again x0 his utility is V (x0 |x0 ) If he finds a new bundle of goods x1 ∈ X such that V (x1 |x0 ) ≥ V (x0 |x0 ), he will choose to move from x0 to x1 Having x1 , the consumer may stay at x1 and consumes again x1 or he moves to a new bundle x2 if he can find x2 ∈ X such that V (x2 |x1 ) ≥ V (x1 |x1 ) etc In this process the consumer does not optimize He only improves his consumption each time with respect to the previous situation, V (xn |xn ) ≤ V (xn |xn+1 ), n ∈ N Here we have a variable preference relation determined by the function V : x ≤x y if and only if V (x|x) ≤ V (y|x) There are a lot of questions arisen in connection with this model For instance, how long, each step, the consumer explores around the current consumption bundle and how much it costs to find a new improving consumption bundle; which part of his variable utility function the consumer is able to discover at each step, leading to local actions, how he can discover his tastes which change with his experience in consumption, learning on line or off line, how much time and effort it takes et cetera (see Soubeyran [26] for a general ’exploration exploitation’ model) Related to this consumption model two questions are of particular interest: (i) whether a habit exists, that is a bundle x∗ ∈ X that the consumer will prefer to repeat than to take a new one, or in other words V (x∗ |x∗ ) ≥ V (y|x∗ ) for all y ∈ X; (ii) whether an improving consumption process {x1 , x2 , } (with V (xn |xn ) ≤ V (xn |xn+1 ), n ∈ N ( converges to some habit x∗ ∈ X If it is so, the distance between two successive consumption bundles goes to zero As a result, a kind of habituation process is formed The consumer repeatedly consumes bundles of goods which become more and more the same This ends in a habit The most famous example of such a situation is a muddling through process ( Lindblom [18]) where an agent (a policy-maker) makes small steps to reform A simple example below shows how such a process works Suppose that V (y|y) ≥ V (y|x) for all x, y ∈ X This means that consuming again the bundle y is better than consuming y for the first time, having consumed the bundle x just before In this case breaking habits and learning how to consume a new bundle of goods is costly Then habituation is beneficial, that is V (x1 |x1 ) ≥ V (x1 |x0 ) ≥ V (x0 |x0 ) More generally, at stage n, one has V (x0 |x0 ) V (x1 |x0 ) V (x1 |x1 ) V (xn |xn ) V (xn |xn+1 ) V (xn+1 |xn+1 ) Let G(y) = V (y|y) be the utility to consume again the bundle of goods y ∈ X Then this utility of consuming again improves along the process, G(x0 ) ≤ G(x1 ) ≤ ≤ G(xn ) ≤ It is a potential function In this context a habit is a permanent personal equilibrium (see [17]) It is a consumption bundle x∗ ∈ X such that A(x∗ , y) = V (y|x∗ ) − V (x∗ |x∗ ) ≤ for all y ∈ X, that is, once the bundle x∗ is chosen, the consumer does not increase his utility when changing this consumption If we assume that (i) G = sup{G(y), y ∈ X} < +∞ (bounded needs) (ii) V (y|y) ≥ V (y|x) for all x, y ∈ X (habituation generates utility), then, for every improving sequence {xn }n≥0 , V (xn |xn+1 ) ≤ V (xn+1 |xn+1 ), n ∈ N ,the sequence {G(xn )}n≥0 of one time repeated consumption is increasing and hence converges to a limit G∗ as n tends to ∞ However this does not prove the existence of a habit x∗ ∈ X and nor the convergence of the sequence {xn }n≥0 to x∗ This is our goal in this paper to provide conditions under which habits exist Changing weights and psychological preference In a certain situation the state space can be identified with the set of actions In such a model, the agent uses several criteria to choose his actions Let πi (y) and πi (z) be the levels of performance of actions y and z, and ix a preference relation at x over performance levels with respect to a criterion i from the set of all criteria I Inequality πi (z) ix πi (y) means that, having just done x, the agent prefers z to y with respect to criteria i ∈ I The variable preference at x ∈ X is given by z i x i x y ⇔ πi (z) πi (y), i ∈ I(x), where I(x) ⊂ I is a subset of criteria that are taken into account for making a choice at x ∈ X It is quite convenient to represent I(x) in form of weights when utility functions are available For instance if Vi (πi (z)) ∈ R is the utility of the level πi (z), then the utility of z can be represented by V (z|x) = i∈I αi (x)Vi (πi (z)) where αi (x) ≥ is a weight the agent assigns to the ith criterion at x When i ∈ / I(x), the weight αi (x) is zero The variable preference is then given by z x y if and only if V (z|x) ≥ V (y|x) The use of weights can also be experienced in the case of psychological preferences We fix a criterion i ∈ I and denote by π (E) ∈ R and π ¯i (E) ∈ R − the minimum and maximum feasible levels of the agent The positive quantities πi (y) − π (E) ≥ and π ¯i (E) − πi (y) ≥ represent the relative performance gaps − of the agent with respect to the reference levels π (E)) and π ¯i (E) At y ∈ X − the agent is happy to have filled a portion of the gap πi (y) − π (E) ≥ and − unhappy to have not filled the residual gap π ¯i (E) − πi (y) ≥ His contentment and deception feelings (emotions) at y are expressed by λi (E)[πi (y) − π (E)] − and µi (E)[¯ πi (E) − πi y] where λi (E) > and µi (E) > indicate the intensities of such feelings The variable preference corresponding to the criterion i and experience E is given by Vi (y|E) = λi (E)[πi (y) − π (E)] − µi (E)[¯ πi (E) − πi (y)] − In the case of short memory, E is just the current action x, and so Vi (y|x) is the weighted sum λi (x)[πi (y) − π (E)] − µi (x)[¯ πi (x) − πi (y)] These preferences − are a variant of the psychological preferences studied in [26] Moving ordering cones In the theory of vector optimization (see Luc [19]) the space of outcomes is often equipped with a partial order by a convex cone Namely, given a vector function f over a nonempty set X with values in a topological vector space Y in which a convex cone C is specified, one looks for a best solution x ∈ X satisfying f (x ) >C f (x) (that is f (x ) − f (x) ∈ C and f (x ) − f (x) ∈ / C) for no x ∈ X The cone C is generally fixed and does not change along optimizing processes However, in a modern theory C is no longer fixed, it varies from state to state and becomes a cone-valued mapping C(x) of the variable x For the moving ordering cone C(x) two interpretations of strict inequality f (x ) >C f (x) are to distinguish f (x ) >C(x ) f (x) and f (x ) >C(x) f (x) They lead to different notions of optimality, but both of them reflect the variability nature of the order Notice that inequalities defined by moving cones cannot be determined by utility functions and are neither transitive, nor complete Basic concepts and elementary properties In this section we develop fundamental concepts related to variable preference relations and some of their elementary properties Let us first recall basic definitions for constant preference relations A preference relation P on X is a relation linking pairs of elements of X, that is, P is a binary relation on X and defined by a subset R of the product space X × X as follows: for x, y ∈ X, one has xP y (we say y is preferred to x) if and only if (x, y) ∈ R A preference relation P is said to be reflexive if xP x for all x ∈ X; it is irreflexive if xP x is true for no x ∈ X; it is antisymmetric if xP y and yP x implies x = y; and it is transitive if xP y and yP z imply xP z A reflexive, antisymmetric and transitive preference relation is called a partial order Sometimes irreflexive partial orders are also considered Reflexivity of a preference relation occurs in a system when an agent is happy with his position, that is, according to his criteria, the current position is acceptable for his next move Irreflexivity occurs when an agent is unhappy with his position and definitely wishes to move to another position at the next step If a preference relation P is not reflexive, mathematically one may generate an associated reflexive preference relation by adding the diagonal of the space X × X to the set R Similarly, if a preference relation is not irreflexive, by deleting the diagonal from the set R one obtains an irreflexive preference relation In general, assuming a preference relation reflexive is widely accepted by researchers and practitioners Moreover, if P is reflexive, a strict preference relation associated to P is defined to be an irreflexive preference relation P smaller than P in the sense that xP y implies xP y (or equivalently the set R defining P is a subset of R) When a reflexive preference relation P is transitive, one defines equivalent classes on X as follows: x ∈ [x] if and only if xP x and x P x The preference relation induced by P on equivalent classes is denoted P¯ and defined by [x]P¯ [y] if and only if x P y for some x ∈ [x], y ∈ [y] Then it is clear that P¯ is reflexive, antisymmetric and transitive on the set [X] of all equivalent classes of X For a reflexive preference relation P on X, we distinguish two kinds of strict preference relations associated with P : xP y if and only if xP y and not yP x; (1) xP y if and only if xP y and x = y (2) Here is a link between these strict preference relations Lemma If P is a reflexive preference relation on X, then xP y implies xP y Conversely, if P is transitive, then for the induced preference relation P¯ on the equivalent classes on X one has [x]P¯ [y] implies [x]P¯ [y] Consequently the two strict preference relations associated with P coincide Proof The first part of the lemma is clear For the second assertion, assume that [x]P¯ [y] If (1) failed for the induced preference relation, we would have xP y and yP x implying [x] = [y], which shows that (2) does not hold for P Throughout this section we will assume that the space X is equipped with a variable preference relation {” ≤x ” : x ∈ X} and that ” ≤x ” are reflexive Given an element x ∈ X, the following sets are of particular interest at the individual or collective levels, ex-ante (before moving) and ex-post ( after moving) levels : i) The ”ex-ante” dominant and dominated sets at x, S+ (x) = {y ∈ X : x ≤x y} S− (x) = {y ∈ X : y ≤x x} Thus, S+ (x) can be viewed as the set of states that dominates x by the criteria at x, that is, states to which from his own point of view, being at the state x, an agent or a group of agents wish to move; while S− (x) contains all states that are dominated by x, that is, states to which, being at x and according to their own judgements, an agent or a group of agents not want to move ii) The ”ex-post” dominant and dominated sets at x, F+ (x) = {y ∈ X : x ≤y y} F− (x) = {y ∈ X : y ≤y x} The terminology ”ex-post” refers to the fact that, starting from x and moving to y, an agent or a group of agents prefer staying at y ∈ F+ (x) to coming back to x, this being judged by criteria at y after having moved from x From now on we shall fix the strict preference relation ” (y)) In general, when the variable preference relation is not constant, the ex-ante and ex-post dominant/ dominated sets are distinct Let us now define paths of acceptable changes or improving paths Given two states x and y of X, an upward path from x to y is a finite sequence of elements x1 , , xn ∈ X such that x = x1 ≤x1 x2 ≤x2 x3 ≤xn−2 xn−1 ≤xn−1 xn = y (4) An upward path from x to y means that an agent, being at the state xk , prefers to move to xk+1 along this path A downward path from x to y is a sequence of elements x1 , , xn ∈ X such that x = x1 ≤x2 x2 ≤x3 x3 ≤xn−1 xn−1 ≤xn xn = y This means that once arrived at the state xk an agent prefers staying at this state to coming back to the previous state xk−1 When n = the above paths are called direct; otherwise they are called indirect (because y is preferred to x not directly, but through intermediate states xi for < i < n) Since variable preference relations are generally not transitive, upward paths and downward paths provide a way to obtain transitive preference relations on X Let us define the upper transitive closure and the lower transitive closure of the variable preference relation ” ≤x ” as follows: • x ≤u y if and only if there is an upward path from x to y; • x ≤ y if and only if there is a downward path from x to y Upper and lower transitive closures being transitive preference relations, we may define equivalent classes on X The induced preference relations are denoted respectively ” u ” and ” ” In view of Lemma there is no distinction between strict preference relations by (1) and (2), so we denote them by ” ≺u ” and ” ≺ ” Below is a relationship between a variable preference relation and its transitive closure Proposition Assume that x ≤u y is given by (4) Then y ≤u x implies xk+1 ≤xk+1 xk for some k ∈ {1, , n − 1} y=x implies xk+1 = xk for some k ∈ {1, , n − 1} Moreover, if [xk+1 ] u [xk ] for some k ∈ {1, , n − 1}, then [y] 10 u [x] In view of weak consistency we deduce y ≤y x∗ because the alternative x∗ [...]... Characterization of the existence of maximal elements of acyclic relation, Economic Theory, Vol.19, pp 407-416 [2] Altman M ( 1982), A generalization of the Brezis-Browder principle on ordered sets, Nonlinear Analysis, Vol.6, pp 157-165 [3] Andrikopoulos A and Zacharias E (2009), Characterization of the existence of R -maximal elements of consistent binary relations, Preprint, Department of Economics, University of. .. contradicts the P-completeness of S(a) Now we apply the first part of the proof to the induced preference to obtain a u -maximal element [x∗ ] of [X] Since the variable preference is weakly consistent, in view of Proposition 6, x∗ is an ex-ante maximal element of X as well The proof is complete Proof of Corollary 11 By working on equivalent classes if necessary one may assume without loss of generality that X... study a lot of anomalies raised by Kahneman and Tversky in [15] and their followers (we’ll address this issue and applications of variable preferences in another study) Existence of maximal elements with topology Throughout this part we assume that X is a topological space The concepts of upper closedness and upper compactness to introduce below are essential for the study of variable preferences in... ∈ J because the variable preference ” ≤x ” is upper closed Consequently, S(a) is not covered by the family {S(a) \ S(yj ) : j ∈ J}, a contradiction The proof is complete 5 Applications The equivalence between existence of maximal elements, existence of aspiration poits and preference- completeness we established in Theorem 9 under weak consistency is quite general, unifying a number of results on models... consequently it has ex-ante maximal points Proof Denote by [X] the space of equivalent classes of elements of X with respect to the preference ” ≤x ”; and equip [X] with the quotient topology and the quotient preference It is known that when the preference is consistent, the quotient preference is a partial order Moreover, with respect to the quotient topology, the quotient preference is continuous and... the preference relation generated by that binary relation is upper closed It is not transfer continuous under the usual topology of the real numbers because 2 > 1 and in every neighborhood of 1 there are elements of A which are not comparable with 2 6 Related notions Other concepts of maximality with respect to a variable preference relation is related to the best action of an agent or a group of agents... that when the variable preference relation ” ≤x ” is constant in the sense that for every x, y ∈ X the preference relations ” ≤x ” and ” ≤y ” coincide, there is no distinction between ex-ante and ex-post maximalities It is important to note that existence conditions for ex-post maximal points follow the same pattern of ex-ante maximal points Therefore we do not further go into details of these conditions... converse of Proposition 3 is not true The utility of acyclicity and consistency is seen from the next result, see also Proposition 6 Proposition 4 Given a variable preference relation ” ≤x ” on X, the induced preference relation ” u ” is a partial order on equivalent classes of X Moreover, the transitive closure ” ≤u ” is a partial order on X if and only if the variable preference relation is acyclic Proof... better place to go This is the concept of maximal points that we are going to develop now Throughout this section the space X is equipped with a reflexive variable preference relation ” ≤x ” Below we define maximal points with respect to the variable preference relation, and give some characterizations via dominant and dominated sets Then we introduce the concept of preference- complete sets borrowed from... ... a number of problems of social science such as habits, routines, behavioral traps etc in terms of maximal elements To establish existence criteria for maximal elements of a variable preference. .. existence of maximal elements in the case of variable preference relations Koszegi [16], Koszegi-Rabin [17] and some others study a personal equilibrium of consumption in which a particular variable. .. unifying results on existence of maximal elements in various contexts, including (a) existence of maximal elements in social choice theory with nontransitive and incomplete preferences (Bergstrom