Purpose: This study aimed to develop a stroke effectiveness test in table tennis based on the temporal game structure to assess the ball speed and ball placement of the players, with a purpose to analyze its reproducibility and validity. Methods: Nineteen male table tennis players participated in this study. The test was performed twice during the first session and once during the second session to assess the intrasession and intersession reproducibility, respectively. Moreover, the test was examined on its (...) ability to discriminate between regional and local performance-level players and on the relationship between the test results and the table tennis performance to assess the discriminant and concurrent validity, respectively. In general, the test consisted of 11 simulated rallies of 2–5 balls with the effort and rest ratio of 0.5, and focused on attack with offensive strokes at defensive balls delivered by a robot randomly between the left and right positions on the table. Results: Ball speed, ball placement, and ball speed-ball placement index showed satisfactory reliability and agreement outcomes. Additionally, the Bland-Altman plots show the systematic error of the analyses closer to 0, and that most values were within the limits of agreements. Concerning validity analyses, regional players had higher scores of ball placement and ball speed-ball placement index as well as made fewer errors than local players. Moreover, ball placement, ball speed-ball placement index, and percentage error presented a strong and significant correlation with table tennis performance. However, ball speed was slightly different between the regional than local players and this variable was not related to table tennis performance. Conclusion: Our findings show evidences that the test is reproducible. Moreover, discriminant and concurrent validity are confirmed for ball placement and ball speed-ball placement index. (shrink)

This 2000 text applies modern advances in game theory to the analysis of competition policy and develops some of the theoretical and policy concerns associated with the pioneering work of Louis Phlips. Containing contributions by leading scholars from Europe and North America, this book observes a common theme in the relationship between the regulatory regime and market structure. Since the inception of the new industrial organization, economists have developed a better understanding of how real-world markets operate. These results (...) have particular relevance to the design and application of anti-trust policy. Analyses indicate that picking the most competitive framework in the short run may be detrimental to competition and welfare in the long run, concentrating the attention of policy makers on the impact on the long-run market structure. This book provides essential reading for graduate students of industrial and managerial economics as well as researchers and policy makers. (shrink)

Few phenomena have found such divergent descriptions in sociological lit­ erature as have social inconsistencies. They were studied by George Herbert Mead as eruptive "natural" events constituting a social temporality. Alfred SchUtz described them as "explosions" of the individual actor's anticipatory action patterns. Talcott Parsons attempted to grasp social inconsistencies into his frame of "pattern variables," while Erving Goffman dealt with them as disruptions of "fostered impressions of reality" maintained by one or the other dominant team. The present study traces (...) these divergent approaches back to various un­ checked assumptions concerning the structure and the constitution of social types. Thus, to further clarify the relationship between social types and the relevance structure of interactional situations has been my first objective. This initially rather limited intention widened when the role of social incon­ sistencies for analysing the differences between play, game, and social action proper in the immediate context of social interaction became apparent. The structure of social inconsistencies seems to hold a key to unifying the theo­ ries of play and social,action. (shrink)

Our studies of naturally occurring human dialogue have led to the recognition of a class of regularities which characterize impoltant aspects of communication. People appear to interact according to established patterns which span several turns in a dialogue and which recur frequently. These patterns appear to be organized around the goals which the dialogue serves for each participant. Many things which are said later in a dialogue can only be interpreted as pursuit of these goals, established by earlier dialogue.These patterns (...) have been represented b) a set of knowledge structures called Dialogue‐Games, capturing shared, conventional knowledge that people have about communication and how it can be used to achieve goals. A Dialogue‐Game has Parameters, which represent those elements that vary across instances of a particular pattern‐the particular dialogue participants and the content topic. The states of the world which must be in effect for a particular Dialogue‐Game to be employed successfully are represented by Specifications of these Parameters. Finally, the expected sequence of intermediate states that occur during instances of a particular conventional pattern are represented by the Components of the corresponding Dialogue‐Game.Representations for several Dialogue‐Games are presented here, based on our analyses of different kinds of naturally occurring dialogue. A process model is discussed, showing Dialogue‐Game identification, pursuit, and termination as part of the comprehension of dialogue utterances. This Dialogue‐Game model captures some of the important functional aspects of language, especially indirect uses to achieve implicit communication. (shrink)

Prior research suggests brokers do not always act in the best interests of clients, although morally obligated to do so. We empirically investigated this issue focusing on trades executed at best execution price, before and after the introduction of electronic limit-order trading, on the London Stock Exchange. As a result of limit-order trading, the proportion of trades executed at the best execution price for the customer significantly increased. We attribute this to a sustained increase in the liquidity of stocks as (...) a result of limit-order trading, regardless of market capitalisation. We discuss the ethical implications of our findings and conclude that market structures that enhance market competitiveness may help reconcile broker and client interests. (shrink)

We address simple voting games as mathematical objects in their own right, and study structures made up of these objects, rather than focusing on SVGs primarily as co-operative games. To this end it is convenient to employ the conceptual framework and language of category theory. This enables us to uncover the underlying unity of the basic operations involving SVGs.

We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented (...) by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power.We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core.We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera. (shrink)

We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented (...) by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power.We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core.We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera. (shrink)

In this paper, we propose a game semantics for the (associative) Lambek calculus . Compared to the implicational fragment of intuitionistic propositional calculus, the semantics deals with two features of the logic: absence of structural rules, as well as directionality of implication. We investigate the impact of these variations of the logic on its game semantics.

The objective of this paper is to study stable coalition structures in symmetric majority games. We assume that players deviate from a coalition structure to another to maximize their power given by the Owen power index. We introduce three myopic core concepts and one farsighted stability concept, the farsighted vNM stable set. Our main result is that the pessimistic core, the largest myopic core, coincides with some farsighted vNM stable set for any number of players. Moreover, we show that (...) a coalition structure belongs to the pessimistic core and the farsighted vNM stable set if and only if it contains an exact majority coalition. (shrink)

In this paper, we study the structural optimization of community governance in smart cities and optimize the structure based on the game analysis of dynamic planning evolution. Theoretically, the connotation and extension of the concept of “self-organization” are cleared, and the environment and conditions of community self-organization formation, the formation process of community self-organization, and the types of community self-organization are discussed. The so-called “self-organization” is a process of spontaneously moving from disorder to order, and the environment and (...) conditions of community self-organization are mainly the openness, nonequilibrium state, and nonlinearity of the system. The key parameters of the contract with the change of the values of each factor make up for the inability to collect data for the empirical study in this paper; then, it presents four linear stages along with the cause such as self-creation, self-expansion, self-maintenance, and self-degeneration, respectively. The types of community self-organization mainly include community self-organization with random aggregation, socialized community self-organization, and stable community self-organization with autonomy orientation. By systematically sorting out the relationship between community self-organization and government, while confirming the positive significance of community self-organization to community cultural governance, various complex conflicts and game relationships will deeply affect the actual effectiveness of cultural governance and provide useful knowledge reference for future community cultural governance. (shrink)

Consideration of reference systems for semivalues leads us to a global representation of the action of a semivalue on a game. Using this representation, we show how to modify semivalues to take account of coalition structures on the player set. The players' allocations according to a semivalue, including a modified semivalue, can be computed by means of products of matrices obtained from the multilinear extension of the game.

The article discusses the issue of how to categorize video games—not the medium of video games, but individual video games. As a lead in to this discussion, the article discusses video game specificity and genericity and moves on to genre theory. On the basis of this discussion, a cognitive experientialist genre framework is sketched, which incorporates both general points from genre theory and theories more specific to the video game domain. The framework is illustrated through a brief example. (...) One virtue of the framework is that it offers a way to bridge the gap between game ontology and player experience; a brief conclusion discusses how the breadth of this gap may depend upon player biographies and also outlines how further work may proceed. (shrink)

It is known that peer group games are a special class of games with a permission structure. However, peer group games are also a special class of digraph games. To be specific, they are digraph games in which the digraph is the transitive closure of a rooted tree. In this paper we first argue that some known results on solutions for peer group games hold more general for digraph games. Second, we generalize both digraph games as well as games (...) with a permission structure into a model called games with a local permission structure, where every player needs permission from its predecessors only to generate worth, but does not need its predecessors to give permission to its own successors. We introduce and axiomatize a Shapley value-type solution for these games, generalizing the conjunctive permission value for games with a permission structure and the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-measure for weighted digraphs. (shrink)

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L∞ωω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L∞ωω to obtain the logics L∞ωω, where Q = {Qi: iε I} is a family (...) of generalized quantifiers on finite structures. Using the logics L∞ωω, we can express polynomial-time properties that are not definable in L∞ωω, such as “there is an even number of x” and “there exists at least n/2 x” , without going to second-order logic. We show that equivalence of finite structures relative to L∞ωω can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht—Fraïssé games. We combine this game-theoretic characterization with sophisticated combinatorial tools from Ramsey theory, such as van der Waerden's Theorem and Folkman's Theorem, in order to investigate the scope and limits of generalized quantifiers in finite model theory. We obtain sharp lower bounds for expressibility in the logics L∞ωω and discover an intrinsic difference between adding finitely many simple unary generalized quantifiers to L∞ωω adding infinitely many. In particular, we show that if Qis a finite sequence of simple unary generalized quantifiers, then the equicardinality, or Härtig, quantifier is not definable in L∞ωω. We also show that the query “does the equivalence relation E have an even number of equivalence classes” is not definable in the extension L∞ωω of L∞ωω by the Härtig quantifier I and any finite sequence Q of simple unary generalized quantifiers. (shrink)

This paper is presented as a sociological account of social action and as part of the “cognitive and cultural turn” in sociology. It retains Weber’s definition of social action as meaningful behavior directed toward another, but employs concepts developed by Noam Chomsky, Pierre Bourdieu and Ludwig Wittgenstein to refine and amplify Weber’s understanding of meaning and subjectivity. It attempts to ground symbolic interaction in innate properties of mind suggested by Chomsky and others. It attempts to enrich Bourdieu’s concept of the (...) habitus with concepts taken from cognitive psychology. Finally it attempts to further integrate Wittgenstein’s language games in to the ethnomethodolical analysis of social situations while grounding ethnomethodological analysis in Bourdieu’s concept of social fields. (shrink)

When I began to think about normative questions of rationality and morality, I soon became a consequentialist. This is the natural result of questioning socially established institutional and moral rules and their metaphysical underpinnings. A critical mind cannot accept a duty to do this or that just because it is told that this duty is implied by some general principle. The immediate question is: Where does this principle come from and how is it justified? And if this question cannot be (...) answered satisfactorily, some critical minds become normative sceptics and others search for rational grounds. Those who search for rational grounds easily end up with one form or other of consequentialism. A rational person does not act by established convention or acquired habit but chooses her action as a means to cause the preferred effects. Thus it seems that the rational person tries to optimize, and this seems to imply that moral action as a variant of rational action should optimize results from a universal point of view, e. g. by maximizing the sum of individual utilities. (shrink)

A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In (...) this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties. (shrink)

Within the framework of Viewpoint Spaces (Dancygier, Barbara. 2012. The language of stories: A cognitive approach. Cambridge: Cambridge University Press), this paper investigates viewpoint interactions in a mobile game’s plot to show how the game’s structural framing leads to meaning construction, specifically the construal of irony. The notion of (meta-)Ground Viewpoint Space is proposed not only to provide a generalized account of a global mental space where local spaces and viewpoints relate to one another, but also to elucidate (...) how the invoked genre-specific expectation makes a significant meaning contribution to the player’s overall construal of the game. This study presents a case study of the mobile game Liyla and the Shadows of War (Abueideh, Rasheed. 2016. Liyla and the Shadows of War) as a viewpoint phenomenon whose construal involves the interaction of multiple viewpoints. The game depicts the consequences of the 2014 Gaza conflict from the perspective of an unnamed Palestinian father struggling to escape the war zone with his daughter Liyla. It deviates from prototypical games in that it is impossible to complete the expected narrative, regardless of the player’s skill, although its ostensible goal is to let the protagonists survive and escape to a safer place. The mismatch between the player’s goal to clear all the stages, the character’s goal to survive, and the developer’s goal to convey a message is a source of ironic meaning, and this study focuses on how the network of multiple viewpoints – of the game developer, the character, and the player – constructs the intended meaning for players. (shrink)

I examine the value and limitations of Kuhn’s Structure of Scientific Revolutions. In the interests of developing a social epistemology of science, I argue that we should draw on Kuhn’s later work, published in The Road since Structure. There, Kuhn draws attention to the important role that specialty formation plays in resolving crises in science, a topic he did not discuss in Structure. I argue that we need to develop a better understanding of specialty research communities. Kuhn’s (...) later work provides a research agenda for those who want to study scientific specialization. I briefly discuss some recent studies that promise to provide valuable insight into the social structure of science. (shrink)

It is certainly the case that morality governs the interactions that take place between individuals. But what if morality exists because of these interactions? This book, first published in 2007, argues for the claim that much of the behaviour we view as 'moral' exists because acting in that way benefits each of us to the greatest extent possible, given the socially structured nature of society. Drawing upon aspects of evolutionary game theory, the theory of bounded rationality, and computational models (...) of social networks, it shows both how moral behaviour can emerge in socially structured environments, and how it can persist even when it is not typically viewed as 'rational' from a traditional economic perspective. This book also provides a theory of how moral principles and the moral sentiments play an indispensable role in effective choice, acting as 'fast and frugal heuristics' in social decision contexts. (shrink)

Haneke’s film Funny Games is a reflection on the nature of pain and representation. I argue that the film closely follows Elaine Scarry’s arguments about the structure of torture. Further, by refusing to appeal to categories of generalization such as ‘sadism’ and ‘psychopathy’, Haneke undermines the process of finding meaning in violence. Haneke positions his audiences as more than just witnesses to torture, but active participants in cruelty.

This article reports a comparative test of the central-union theory vis-à-vis several other game-theoretic solution concepts in 3-person sidepayment games. Based on a laboratory experiment, this comparison utilizes nine games in characteristic function form. The solution concepts under test include the equal excess model, the Myerson–Shapley solution, the kernel, and two variants of the central-union theory. With regard to the player's payoffs, results show that the CU-1, CU-2, kernel, and equal excess theories have essentially equal predictive accuracy and that (...) all of these are more accurate than Myerson–Shapley. When the solution concepts are extended and coalition structure probability predictions are incorporated in the test, one version of the central-union theory is overall more accurate than the other solutions. (shrink)

Video games are ideal platforms for the study of skill acquisition for a variety of reasons. However, our understanding of the development of skill and the cognitive representations that support skilled performance can be limited by a focus on game scores. We present an alternative approach to the study of skill acquisition in video games based on the tools of the Expert Performance Approach. Our investigation was motivated by a detailed analysis of the behaviors responsible for the superior performance (...) of one of the highest scoring players of the video game Space Fortress. This analysis revealed how certain behaviors contributed to his exceptional performance. In this study, we recruited a participant for a similar training regimen, but we collected concurrent and retrospective verbal protocol data throughout training. Protocol analysis revealed insights into strategies, errors, mental representations, and shifting game priorities. We argue that these insights into the developing representations that guided skilled performance could only easily have been derived from the tools of the Expert Performance Approach. We propose that the described approach could be applied to understand performance and skill acquisition in many different video games and help reveal mechanisms of transfer from gameplay to other measures of laboratory and real-world performance. (shrink)

A framework for pragmatic analysis is proposed which treats discourse as a game, with context as a scoreboard organized around the questions under discussion by the interlocutors. The framework is intended to be coordinated with a dynamic compositional semantics. Accordingly, the context of utterance is modeled as a tuple of different types of information, and the questions therein — modeled, as is usual in formal semantics, as alternative sets of propositions — constrain the felicitous flow of discourse. A requirement (...) of Relevance is satisfied by an utterance (whether an assertion, a question or a suggestion) iff it addresses the question under discussion. Finally, it is argued that the prosodic focus of an utterance canonically serves to reflect the question under discussion (at least in English), placing additional constraints on felicity in context. (shrink)

The paper attempts to rationalize cooperation in the one-shot prisoners' dilemma (PD). It starts by introducing (and preliminarily investigating) a new kind of equilibrium (differing from Aumann's correlated equilibria) according to which the players' actions may be correlated (sect. 2). In PD the Pareto-optimal among these equilibria is joint cooperation. Since these equilibria seem to contradict causal preconceptions, the paper continues with a standard analysis of the causal structure of decision situations (sect. 3). The analysis then raises to a (...) reflexive point of view according to which the agent integrates his own present and future decision situations into the causal picture of his situation (sect. 4). This reflexive structure is first applied to the toxin puzzle and then to Newcomb's problem, showing a way to rationalize drinking the toxin and taking only one box with-out assuming causal mystery (sect. 5). The latter result is finally extended to a rationalization of cooperation in PD (sect. 6). (shrink)

This paper is a beginning—an initial attempt to think of the function and character of epistemic norms as a kind of social norm. We draw on social scientific thinking about social norms and the social games to which they respond. Assume that people individually follow epistemic norms for the sake of acquiring a stock of true beliefs. When they live in groups and share information with each other, they will in turn produce a shared store of true beliefs, an epistemic (...) public good. True beliefs, produced individually or in groups, constitute an epistemic good—one commonly stockpiled and distributed within a community. Epistemic norms can then be understood as a kind of socially developed and transmitted normative sensibility having to do with the production of this individual and public good. Epistemic norms should serve to regulate this practice—coordinating the practice of individuals so as to afford the benefits of life in an interdependent epistemic community—and to manage the risks of being epistemically dependent on others within such a community. Here, we provide some attempts to characterize the central aspects of "the epistemic game"—the epistemic choice situation confronted by communities of epistemic agents. (shrink)

The desirability relation was introduced by Isbell to qualitatively compare the a priori influence of voters in a simple game. In this paper, we extend this desirability relation to simple games with coalition structure. In these games, players organize themselves into a priori disjoint coalitions. It appears that the desirability relation defined in this paper is a complete preorder in the class of swap-robust games. We also compare our desirability relation with the preorders induced by the generalizations to (...) games with coalition structure of the Shapley–Shubik and Banzhaf–Coleman power indices Values of games with a priori unions, 1977; Owen, Power, voting and voting power. In: Holler MJ Modification of the Banzhaf–Coleman index for games with a priori unions, 1981). It happens that in general they are different even if one considers the subclass of weighed voting games. However, if structural coalitions have equal size then both Owen–Banzhaf and the desirability preordering coincide. (shrink)

We introduce a solution concept for cooperative games with transferable utility and a coalition structure that is proportional for two-player games. Our value is obtained from generalizing a proportional value for cooperative games with transferable utility in a way that parallels the extension of the Shapley value to the Owen value. We provide two characterizations of our solution concept, one that employs a property that can be seen as the proportional analog to Myerson’s balanced contribution property; and a second (...) one that relies on a consistency property. (shrink)

Research in anthropology has shown that kin terminologies have a complex combinatorial structure and vary systematically across cultures. This article argues that universals and variation in kin terminology result from the interaction of (1) an innate conceptual structure of kinship, homologous with conceptual structure in other domains, and (2) principles of optimal, “grammatical” communication active in language in general. Kin terms from two languages, English and Seneca, show how terminologies that look very different on the surface may (...) result from variation in the rankings of a universal set of constraints. Constraints on kin terms form a system: some are concerned with absolute features of kin (sex), others with the position (distance and direction) of kin in “kinship space,” others with groups and group boundaries (matrilines, patrilines, generations, etc.). Also, kin terms sometimes extend indefinitely via recursion, and recursion in kin terminology has parallels with recursion in other areas of language. Thus the study of kinship sheds light on two areas of cognition, and their phylogeny. The conceptual structure of kinship seems to borrow its organization from the conceptual structure of space, while being specialized for representing genealogy. And the grammar of kinship looks like the product of an evolved grammar faculty, opportunistically active across traditional domains of semantics, syntax, and phonology. Grammar is best understood as an offshoot of a uniquely human capacity for playing coordination games. (shrink)

For countable structures U and B, let $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ abbreviate the statement that every Σ0 α (Lω1,ω) sentence true in U also holds in B. One can define a back and forth game between the structures U and B that determines whether $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω Σ0 n sentence, then there are countably infinite models U and B such that $\mathfrak{U} \vDash \theta, \mathfrak{B} \vDash \neg \theta$ (...) , and $\mathfrak{U}\overset{n}{\rightarrow}\mathfrak{B}$ . For countable languages L there is a natural way to view L structures with universe ω as a topological space, XL. Let [U] = {B ∈ XL∣B ≅ U} denote the isomorphism class of U. Let U and B be countably infinite nonisomorphic L structures, and let $C \subseteq \omega^\omega$ be any Π0 α subset. Our main result states that if $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ , then there is a continuous function f: ωω → XL with the property that $x \in C \Rightarrow f(x) \in \lbrack\mathfrak{U}\rbrack$ and $x \notin C \Rightarrow f(x) \in \lbrack\mathfrak{B}\rbrack$ . In fact, for α ≤ 3, the continuous function f can be defined from the $\overset{\alpha}{\rightarrow}$ relation. (shrink)