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Evolutionarily stable states of the signaling game with a uniform probability distribution over meanings
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In this article we discuss the notion of a linguistic universal, and possible sources of such invariant properties of natural
languages. In the first part, we explore the conceptual issues that arise. In the second part of the paper, we focus on the
explanatory potential of horizontal evolution. We particularly focus on two case studies, concerning...
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Context 1
... similarity between input to coding and output from decoding, is positively correlated to the degree of mutual reinforcement between S and R via priming. The degree of reinforcement in turn determines the probability with which a certain coding/decoding strategy is used in subsequent utterances. The utility function is thus directly related to the “fitness” of a strategy—strategies with a high fitness increase their probability to be used in communication. Given this, we are in a position to apply the concepts of Evolutionary Game Theory (EGT) here. Under this interpretation of game theory, games are played iteratively, and the utility of a strategy at a given point of time is nothing but its probability to be used at the next point in time. The original motivation for EGT comes from biology. In the biological context, strategies are genetically determined, and the probability of a strategy in a population corresponds to the abundance of individuals with the respective genetic endowment in the population. A state of a dynamic system is evolutionarily stable if it does not change under the evolutionary dynamics, and if it will return into that state if minor perturbations (“mutations” in the biological context) occur. This is not the right occasion to go into the mathematical details of EGT. However, it is easy to see that an evolutionarily stable state 23 must be a Nash equilibrium. Otherwise, there would be a better response to an incumbent strategy than the incumbent counter-strategy. A mutant “playing” this better strategy would then obtain a higher utility than the incumbent and thus spread. As already mentioned, every Nash equilibrium in our signaling game amounts to a Voronoi tesselation of the meaning space. This means that every evolutionarily stable state induces a partition of the meaning space into convex categories. 24 For the purpose of illustration, we did a few computer simulations of the dynamics described above. The meaning space was a set of squares inside a circle. The similarity between two squares is inversely related to its Euclidean distance. All meanings were assumed to be equally likely. The experiments confirmed the evolutionary stability of Voronoi tesselations. The graphics in Figure 5 show stable states for different numbers of forms. The shadings of a square indicates the form that it is mapped to by the dominant sender strategy. Black squares indicate the interpretation of a form under the dominant receiver strategy. To sum up so far, we assumed that a production strategy and an interpretation strategy reinforce each other the more similar the input for production and the interpretation of the corresponding form are on average. This induces ...
Context 2
... such a switch from the first to the second equilibrium may by more likely than in the reverse direction. This would have the long term effect that in the long run, the system spends more time in the second than in the first equilibrium. Such an asymmetry grows larger as the mutation rate gets smaller. In the limit, the long term probability of the first equilibrium converges to 0 then, and the probability of the second equilibrium to one. Equilibria which have a non-zero probability for any mutation rate in this sense are called stochastically stable . 28 Computer simulations indicate that for the game in question, the only stochastically stable states are those that are based on the partition { Red } / { Yellow } / { Green, Blue } . In the simulation, the system underwent 20,000 up- date cycles, starting from a random state. Of these 20,000 “generations”, the system spent 18,847 in a { Red } / { Yellow } / { Green, Blue } state, against 1,054 in a { Green } / { Blue } / { Red, Yellow } state. In fact, the system first stabilized in the second equilibrium, mutated into the bipartition { Red, Yellow } / { Green,Blue } after 1,096 cycles, moved on into a state using the first partition after another 16 cycles, and remained there for the rest of the simulation. A switch from the first into the second kind of equilibrium did not occur. Figure 7 visualizes the stable states for the game with two, three and four different forms. As in Figure 5, the shade of a point indicates the form to which the sender maps this point, while the black squares indicate the preferred interpretation of the forms according to the dominant receiver strategy. The circles indicate the location of the four focal meanings Red, Yellow, Green and Blue. Let us summarize the findings of this section. We assumed a signaling game which models the communication about a continuous meaning space by means of finitely many signals. The utility of a strategy pair is inversely related to the distance between the input meaning (Nature’s choice) and the output meaning (receiver’s interpretation). We furthermore assumed that this kind of utility function corresponds to an evolutionary dynamics: A high utility amounts to a strong self-reinforcement of a pair of strategies. Under these assumptions, it follows directly that all evolutionarily stable states correspond to Voronoi tessellations of the meaning space. If the distance metric is Euclidean, this entails that semantic categories correspond to convex regions of the meaning space. The precise nature of the evolutionarily stable states and of the stochastically stable states depends on the details of the similarity function, Nature’s probability distribution over the meaning space, and the number of forms ...
Citations
... An example of how such a correlation can be realized is presented by Jäger and van Rooij (2007), who use computer simulations to show how semantic fixed points (in the form of Nash equilibria) can represent a meeting of minds. They refer to the domain they choose-a circular disk-as "the colour space," but there is nothing in the process that depends on relations to colours. ...
... Following the standard definition in game theory, a Nash equilibrium of the game is a pair (S, R), where S is the sender's partitioning (into n subsets) of C, and R is the responder's n-tuple of prototype points of C, such that both are a best response to each other. The central result of Jäger and van Rooij (2007) is that if the colour space is convex and compact and the similarity function is continuous, then there exists a Nash equilibrium, and it corresponds to a Voronoi tessellation of the colour space that is common to s and r. 6 Their solution is guaranteed to satisfy both Convexity and Representation and it also satisfies Parsimony (though this is built in), Informativeness, and Contrast. In addition, their implementation can be seen as satisfying Well-formedness as well, because the fact that prototypes minimize the average distance to the other points in their cells makes them evenly distributed in the space and, consequently, items falling under one prototype are maximally dissimilar to items falling under another. ...
... Neither the results of the simulations by Jäger and van Rooij (2007) nor the theoretical analysis by Warglien and Gärdenfors (2013) guarantee that there is a unique best partitioning of any given similarity space. This means that there exist many "languages" that contain the same information about a conceptual domain. ...
This article takes a cognitive approach to natural concepts. The aim is to introduce criteria that are evaluated with respect to how they support the cognitive economy of humans when using concepts in reasoning and communicating with them. I first present the theory of conceptual spaces as a tool for expressing the criteria. Then I introduce the central idea that natural concepts correspond to convex regions of a conceptual space. I argue that this criterion has far-reaching consequences as regards natural concepts. Partly following earlier work, I present some other criteria that further delimit the class of natural concepts. One of these is coherence, which does not seem to have been discussed previously. Finally, I show that convexity and other criteria make it possible to ensure that people mean the same thing when they communicate using concepts. Apart from its philosophical interest, the analysis presented in the article will be relevant for tasks of conceptual engineering in artificial systems that work with concepts.
... Further evidence in favour of a similarity-based structure bias in form-meaning mappings comes from interaction studies ( Jäger & van Rooij, 2007;Silvey et al., 2019;Voiklis & Corter, 2012). Similarity-based structure is hypothesised to aid the language user not only during learning but also during communicative interaction, by facilitating the establishment of linguistic conventions and in turn alignment with the interlocutor (Freyd, 1983;Warglien & Gärdenfors, 2013). ...
... Similarity-based structure is hypothesised to aid the language user not only during learning but also during communicative interaction, by facilitating the establishment of linguistic conventions and in turn alignment with the interlocutor (Freyd, 1983;Warglien & Gärdenfors, 2013). These studies suggest that communicative contexts promote the focus on commonalities between the entities being categorised ( Jäger & van Rooij, 2007;Voiklis & Corter, 2012), and when combined with learning, they can accelerate the emergence of connected and efficient linguistic category systems (Silvey et al., 2019). ...
Morphological systems often reuse the same forms in different functions, creating what is known as syncretism. While syncretism varies greatly, certain cross-linguistic tendencies are apparent. Patterns where all syncretic forms share a morphological feature value (e.g., first person, or plural number) are most common cross-linguistically, and this preference is mirrored in results from learning experiments. While this suggests a general bias towards natural (featurally homogeneous) over unnatural (featurally heterogeneous) patterns, little is yet known about gradients in learnability and distributions of different kinds of unnatural patterns. In this paper we assess apparent cross-linguistic asymmetries between different types of unnatural patterns in person-number verbal agreement paradigms and test their learnability in an artificial language learning experiment. We find that the cross-linguistic recurrence of unnatural patterns of syncretism in person-number paradigms is proportional to the amount of shared feature values (i.e., similarity) amongst the syncretic forms. Our experimental results further suggest that the learnability of syncretic patterns also mirrors the paradigm’s degree of feature-value similarity. We propose that this gradient in learnability reflects a general bias towards similarity-based structure in morphological learning, which previous literature has shown to play a crucial role in word learning as well as in category and concept learning more generally. Rather than a dichotomous natural/unnatural distinction, our results thus support a more nuanced view of (un)naturalness in morphological paradigms and suggest that a preference for similarity-based structure during language learning might shape the worldwide transmission and typological distribution of patterns of syncretism.
... A powerful and parsimonious way to justify that human cognition favours a Voronoi tessellation, even without the existence of primary prototypes, comes from evolutionary game theory, which is used by Jäger (2007) and Jäger and van Rooij (2006). The linguists apply signaling game models to show that convexity is an expected conse- quence of language evolution. ...
Conceptual spaces are a frequently applied framework for representing concepts. One of its central aims is to find criteria for what makes a concept natural. A prominent demand is that natural concepts cover convex regions in conceptual spaces. The first aim of this paper is to analyse the convexity thesis and the arguments that have been advanced in its favour or against it. Based on this, I argue that most supporting arguments focus on single-domain concepts (e.g., colours, smells, shapes). Unfortunately, these concepts are not the primary examples of natural concepts. Building on this observation, the second aim of the paper is to develop criteria for natural multi-domain concepts. The representation of such concepts has two main aspects: features that are associated with the concept and the probabilistic correlation pattern which the concept captures. Conceptual spaces, together with probabilistic considerations, provide a helpful framework to approach these aspects. With respect to feature representation, the existence of characteristic features (i.e., that apples have a specific taste) is essential. Moreover, natural concepts capture peaks of a probabilistic distribution over complex spaces. They carve up nature at its joints, that is, at areas with no or low probabilistic density. This last aspect is shown to be closely related to the convexity demand.
... In particular, the evolution of the universal of monotonicity might be a case where communication and learning push toward similarly structured categories, namely those satisfying the universal of monotonicity. In the case of categories expressed by nouns, the universal property of convexity has been argued to be a consequence of a pressure both from learning (Gärdenfors, 2004) and from communication (Jäger & van Rooij, 2007). As pointed out in and Carcassi (2020), the property of convexity is structurally similar to the property of monotonicity as defined above (see cited papers for a more discussion of why this is the case), suggesting that they might be explained by similar pressures. ...
Natural languages exhibit many semantic universals, that is, properties of meaning shared across all languages. In this paper, we develop an explanation of one very prominent semantic universal, the monotonicity universal. While the existing work has shown that quantifiers satisfying the monotonicity universal are easier to learn, we provide a more complete explanation by considering the emergence of quantifiers from the perspective of cultural evolution. In particular, we show that quantifiers satisfy the monotonicity universal evolve reliably in an iterated learning paradigm with neural networks as agents.
... In this paper we seek to motivate why meanings tend to be convex and why extreme exemplars of these meanings, or categories, are considered to be representative by making use of such signaling games. Jäger (2007) and Jäger and van Rooij (2007) introduced so-called sim-max games, signaling games using an Euclidean meaning space with a similarity-based utility function. They show that by using a simple learning dynamic the evolved equilibria of these games give rise to descriptive meanings which are convex sets. ...
... Just as in standard sim-max games, descriptive meanings are now convex sets. But whereas imperative meanings in Jäger (2007) and Jäger and van Rooij (2007) were central points, i.e., prototypes, now they are extreme points, i.e., stereotypes. ...
In this paper I defend the epistemic value of the representational-computational view of cognition by arguing that it has explanatory merits that cannot be ignored. To this end, I focus on the virtue of a computational explanation of optic ataxia, a disorder characterized by difficulties in executing visually-guided reaching tasks, although ataxic patients do not exhibit any specific disease of the muscular apparatus. I argue that addressing cases of patients who are suffering from optic ataxia by invoking a causal role for internal representations is more effective than merely relying on correlations between bodily and environmental variables. This argument has consequences for the epistemic assessment of radical enactivism, whichRE invokes the Dynamical System Theory as the best tool for explaining cognitive phenomena.
... In this paper we seek to motivate why meanings tend to be convex and why extreme exemplars of these meanings, or categories, are considered to be representative by making use of such signaling games. Jäger (2007) and Jäger and van Rooij (2007) introduced so-called sim-max games, signaling games using an Euclidean meaning space with a similarity-based utility function. They show that by using a simple learning dynamic the evolved equilibria of these games give rise to descriptive meanings which are convex sets. ...
... Just as in standard sim-max games, descriptive meanings are now convex sets. But whereas imperative meanings in Jäger (2007) and Jäger and van Rooij (2007) were central points, i.e., prototypes, now they are extreme points, i.e., stereotypes. ...
It is often assumed that a requirement for counting objects is that they do not overlap. However, this condition can be violated. The paper deals, specifically, with counting objects that consist of parts, that is, with configurations. One example is outfit as a configuration of articles of clothing; notice that one article of clothing may be part of different outfits. The article develops an analysis of such configurational entities as individual concepts. It investigates the interaction of noun phrases based on such nouns with modal operators and in collective and cumulative interpretations.
... In this paper we seek to motivate why meanings tend to be convex and why extreme exemplars of these meanings, or categories, are considered to be representative by making use of such signaling games. Jäger (2007) and Jäger and van Rooij (2007) introduced so-called sim-max games, signaling games using an Euclidean meaning space with a similarity-based utility function. They show that by using a simple learning dynamic the evolved equilibria of these games give rise to descriptive meanings which are convex sets. ...
... Just as in standard sim-max games, descriptive meanings are now convex sets. But whereas imperative meanings in Jäger (2007) and Jäger and van Rooij (2007) were central points, i.e., prototypes, now they are extreme points, i.e., stereotypes. ...
In this paper we argue that a typical member of a class, or category, is an extreme, rather than a central, member of this category. Making use of a formal notion of representativeness, we can say that a typical member of a category is a stereotype of this category. In the second part of the paper we show that this account of typicality can be given a rational motivation by providing a game-theoretical derivation.
... In this paper we seek to motivate why meanings tend to be convex and why extreme exemplars of these meanings, or categories, are considered to be representative by making use of such signaling games. Jäger (2007) and Jäger and van Rooij (2007) introduced so-called sim-max games, signaling games using an Euclidean meaning space with a similarity-based utility function. They show that by using a simple learning dynamic the evolved equilibria of these games give rise to descriptive meanings which are convex sets. ...
... Just as in standard sim-max games, descriptive meanings are now convex sets. But whereas imperative meanings in Jäger (2007) and Jäger and van Rooij (2007) were central points, i.e., prototypes, now they are extreme points, i.e., stereotypes. ...
The term ‘cognitive structures’ is used to describe the fact that mental models underlie thinking, reasoning and representing. Cognitive structures generally improve the efficiency of information processing by providing a situational framework within which there are parameters governing the nature and timing of information and appropriate responses can be anticipated. Unanticipated events that violate the parameters of the cognitive structure require the cognitive model to be updated, but this comes at an efficiency cost. In reversal learning a response that had been reinforced is no longer reinforced, while an alternative is now reinforced, having previously not been (A+/B− becomes A−/B+). Unanticipated changes of contingencies require that cognitive structures are updated. In this study, we examined the effect of lesions of the orbital frontal cortex (OFC) and the effects of the selective serotonin reuptake inhibitor (SSRI), escitalopram, on discrimination and reversal learning. Escitalopram was without effect in intact rats. Rats with OFC lesions had selective impairment of reversal learning, which was ameliorated by escitalopram. We conclude that reversal learning in OFC-lesioned rats is an easily administered and sensitive test that can detect effects of serotonergic modulation on cognitive structures that are involved in behavioural flexibility.
... While such long-term, diachronic sensitivity to context has been explained by abstract principles of optimality, such as the equilibria concepts of evolutionary game theory (Jäger, 2007;Jäger & Van Rooij, 2007), it has not yet been grounded in a cognitive and mechanistic account of the immediate, synchronic processes unfolding in the minds of individual agents while they interact. In other words, while there is abundant empirical evidence for context-sensitivity in the outcomes of convention formation processes, our third puzzle concerns which cognitive mechanisms may be necessary or sufficient to give rise to such conventions. ...
Languages are powerful solutions to coordination problems: they provide stable, shared expectations about how the words we say correspond to the beliefs and intentions in our heads. Yet language use in a variable and non-stationary social environment requires linguistic representations to be flexible: old words acquire new ad hoc or partner-specific meanings on the fly. In this paper, we introduce a hierarchical Bayesian theory of convention formation that aims to reconcile the long-standing tension between these two basic observations. More specifically, we argue that the central computational problem of communication is not simply transmission, as in classical formulations, but learning and adaptation over multiple timescales. Under our account, rapid learning within dyadic interactions allows for coordination on partner-specific common ground, while social conventions are stable priors that have been abstracted away from interactions with multiple partners. We present new empirical data alongside simulations showing how our model provides a cognitive foundation for explaining several phenomena that have posed a challenge for previous accounts: (1) the convergence to more efficient referring expressions across repeated interaction with the same partner, (2) the gradual transfer of partner-specific common ground to novel partners, and (3) the influence of communicative context on which conventions eventually form.
... In this paper we seek to motivate why meanings tend to be convex and why extreme exemplars of these meanings, or categories, are considered to be representative by making use of such signaling games. Jäger (2007) and Jäger and van Rooij (2007) introduced so-called sim-max games, signaling games using an Euclidean meaning space with a similarity-based utility function. They show that by using a simple learning dynamic the evolved equilibria of these games give rise to descriptive meanings which are convex sets. ...
... Just as in standard sim-max games, descriptive meanings are now convex sets. But whereas imperative meanings in Jäger (2007) and Jäger and van Rooij (2007) were central points, i.e., prototypes, now they are extreme points, i.e., stereotypes. ...
This open access book presents novel theoretical, empirical and experimental work exploring the nature of mental representations that support natural language production and understanding, and other manifestations of cognition. One fundamental question raised in the text is whether requisite knowledge structures can be adequately modeled by means of a uniform representational format, and if so, what exactly is its nature.
Frames are a key topic covered which have had a strong impact on the exploration of knowledge representations in artificial intelligence, psychology and linguistics; cascades are a novel development in frame theory. Other key subject areas explored are: concepts and categorization, the experimental investigation of mental representation, as well as cognitive analysis in semantics. This book is of interest to students, researchers, and professionals working on cognition in the fields of linguistics, philosophy, and psychology.
