In this chapter, some mathematical models of language dynamics (Wichmann,2008a,b; Schulze et al., 2008; Vogt, 2009; Castellano et al., 2009;Solé et al., 2010) are discussed and illustrated with someapplications. Many language dynamics models have been developed to describe,for example, the evolution of languages, the competition processes betweendifferent linguistic features (considered as fixed entities), or thecognitive dimension of language. Due to limited space, it is not possible toprovide here an exhaustive discussion of language dynamics models and manyimportant pieces of the complete picture of the field are missing. In thefollowing sections, we will discuss a selection of examples with the goal ofproviding at least a general idea of the field and why it represents aframework for further possible developments.

The ultimate goal of mathematical language dynamics models is to provide aquantitative description of language change, that is, of the combineddynamics of evolution, competition, and spreading processes of languages inspace and time and of the consequent diversity and correlations of thelinguistic landscapes, discussed in the first part of the book. Forconvenience, models are classified into different types. Fully evolutionarymodels, in which languages themselves undergo changes, while possiblycompeting with each other, are discussed in Chapter 6. Competition andnatural selection models, which consider the processes taking place on ashorter time scale, and on which language features can be considered fixed,are discussed in Chapter 7; they can in turn be classified into modelsstudying the time scale of language shifts and those focusing on the shortertime scale of language use. We explore the important and interestingcognitive dimension of language only partially, by reviewing a few semioticdynamics models, which are discussed in Section 6.1.

Motivations behind Language Dynamics Modeling

There are different motivations that drive the study of language dynamics,ranging from scientific to social ones. These diverse motivations reveal theinterdisciplinary nature and wide applicability range of languagedynamics.

• • Language represents one of the most complex known phenomena.This fuels the interest of scientists who are keen to solvelife's every problem, be it complex or simple.

In the present monograph, we have illustrated a set of tools and conceptsrelated to complex systems that are employed in language dynamics. In orderto see how they work and how they relate to linguistics, such tools andconcepts have been applied to specific case studies as well as to moregeneral and abstract simplified models. This application follows the generalpoint of view of complex systems theory, that there are some universalframeworks that can be applied through different fields to describe verydifferent types of complex systems.

Language dynamics is a young field looking for an actual integration amongthe various tools and methods from complex systems, on the one hand, andlinguistics, on the other hand, resulting in a unified and consistentpicture of language change and use. Such a picture is multifold, in that itconcerns (a) the study of the reconstruction of a consistent history oflanguages, (b) the investigations interpreting the complexity of thecurrently observed linguistic landscape, and (c) the forecast of the futureevolution of language groups. The latter point has recently become ofinterest due to the fact that cultural and linguistic diversity are nowrecognized as an invaluable heritage. In fact, the tools of languagedynamics have already been used by various authors to suggest languagepolicies and evaluate the ability of a language to find or create theconditions and a suitable niche, in which to be successful.

The structure and contents of this book reflect the current situation: anintegration of the main methods of investigation has began, but it is stillunderway. The path toward the goals of language dynamics is long andnecessarily passes through a series of applications to real life situationsand historical tests, which are represented by the many linguistic databasesnow available, containing data about the history of languages and thecurrent sociolinguistic structures. We have not mentioned, for reasons ofspace, all the databases coming from social networks or from onlinecontents, which offer ‘big data’ with an unprecedented amountof information and a unique level of fine-grained detail, which are by now arelevant element in linguistic analysis—see for example,Gonçalves and Sánchez (2014); Donoso and Sánchez(2017); Paradowski (2010); Paradowski et al. (2012).

Introduction to Language Competition

The time scale of competition dynamics on which languages can be described asan analogy of competing biological species is usually shorter than that ofthe fully evolutionary dynamics. However, it is a time scale on whichlanguages can either spread or disappear and therefore is relevant for thedestiny of most of the currently existing languages (Solé et al.,2010). Competition models are technically more simple than evolutionary orcognitive models, but in a linguistic system, one first needs to identifythe competitors and the main competition processes in order to describe themwithin a competition and natural selection paradigm; this may not be astraightforward task. An example of language competition is provided by twolinguistic features competing with each other without undergoing majorchanges, for example, two different ways of pronouncing the same word or twosynonyms referring to the same object/idea. The models considered in thischapter can be categorized either as two-state models, when there aremonolinguals of language X and language Y, or as three-state models withbilinguals, where in addition to X and Y monolinguals, there is also a Zgroup of bilingual speakers. These models provide a simplified descriptionof the adoption of a new language or of the loss of a known language asabrupt changes of the state of a speaker: X→Z or Y→Z(monolingual→bilingual) transitions and the inverse Z→X orZ→Y (bilingual→monolingual) transitions. The overall processX→Z→Y (Y→Z→X) represents a language shift, theprocess whereby a community speaking a certain language X (Y) shifts tospeaking another language Y (X) because of its interaction with anotherlinguistic community.

The paradigm of language shift is extensively studied and still represents ahuge challenge for mathematical modelers. Usually, languages that areconsidered to be more prestigious expand at the expense of other languages.However, there are many possible causes behind a language shift. From ahistorical perspective, some questions have remained unanswered, puzzlinglinguists. These questions can receive at least partial answers withmathematical modeling.

A string metric is any metric distance between entitieswhich can be associated with a string. String metric-based methods have beendeveloped and used for tackling various problems, from plagiarism detectionand DNA/RNA analysis, image analysis and recognition, to data mining andintegration, and incremental search, to name a few. In this chapter, weconsider some simple examples of metric distances and apply them to somereal examples related to language.

Levenshtein Distance

The most widely known string metric for measuring the difference between twosequences is the Levenshtein distance, also known asedit distance, named after Vladimir Levenshtein, whoconsidered this distance in 1965 (Levenshtein, 1966).

The Levenshtein distance L(a,b) between two given strings a andb, each composed of a set of characters, is defined asthe minimum number of edit operations, including characteraddition, removal, and replacement, needed to turn a intob or vice versa (see e.g. Apostolico and Galil [1997]).Here is an example of how the Levenshtein distance can be used.

Example: Levenshtein distances between three given words. Let usconsider three different locations in the Basque countries, labeled herewith k = 1, 2, 3, where three correspondingly differentdialects of Basque are spoken, and compare the three variants of the sameword, the Basque word for ‘I am’, in these locations. Thewords are a1 = naiz,a2 = nais, anda3 = nas. Comparing thesethree words with each other, we notice the following relations:

• • a1 = naiz vs.a2 = nais:naiz → nais by onereplacement operation z →s; thus, L12 =L(naiz, nais)= 1.

• • a1 = naiz vs.a3 = nas:naiz → nais →nas by two edit operations: replacementz → s and deletion ofi; so L13 =L(naiz, nas)= 2.

This book builds on a long tradition of research on the role of extrapolation in various fields. To make it clear, extrapolation means that the future course of a time series is seen as depending only on past observations of this series. It seems reasonable to begin the discussion of earlier contributions on this topic with research from the field of economics. Here we can connect with the material already discussed in Chapter 1. Clearly, Irving Fisher (1930) stands out for initiating the explicit mathematical modeling of extrapolative expectations in his work on inflation expectations. Variations of this theme were proposed in the form of adaptive and regressive expectations. While the hypothesis of adaptive expectations is a special variant of extrapolative expectations, the notion of regressive expectations brings in the element of a long-term anchor for predicted values.1 Expectations are said to be regressive when they show a tendency to revert to a fixed value. In a more general use of this term, we will speak of a regressive tendency if expectations increase in an under-proportional way in response to an increase in the underlying variable (e.g., inflation). Clearly, expectations can combine the elements of extrapolation and regressiveness, and we will pursue this topic in the chapters to come.

The study of the effect of inflation on the nominal interest rate in African economies has already brought forth a variety of results in the research literature. For one thing, the list of countries to be discussed is rather short. Berument and Jelassi (2002) and Kasman et al. (2006) report statistical evidence for a weak-form Fisher effect for Egypt. Yet, with the interest rate data studied here, the hypothesis under consideration does not gain any support in the case of Egypt. Hence, this country is not treated any further. For Nigeria, Balparda et al. (2017) suggest that the effect under investigation only shows up for very short-term interest rates. For Malawi, Matchaya (2011) indicates that the econometric evidence is consistent with the presence of a Fisher effect. The study of short-term interest rates for Kenya by Caporale and Gil-Alana (2016) does not offer an explicit test of the Fisher hypothesis. Hence, for this country, there exists no previous finding which would permit us to make comparisons. For South Africa, the findings concerning the link between inflation expectations and interest rates are more telling: Phiri and Lusanga (2011) and Kim et al. (2018) document a significant Fisher effect for various nominal interest rates.1

This book is a contribution to a new way of doing economics. It will show that when taking fundamental insights from psychology into account, we can develop a bottom-up approach to the understanding of economic behavior. Instead of making a priori assumptions on human rationality, we draw on insights from cognitive psychology to elicit such behavior in the laboratory. The concrete behavior thus studied and modeled is the formation of expectations. The book will explain in detail how a model of expectations is built, starting from measurement of expectations under experimental conditions and then proceeding to apply the data thus elicited to real-world situations. With its focus on expectations regarding inflation, this text is a contribution to behavioral macroeconomics. It will appeal to readers who demand more from behavioral economics than just a critique of mainstream economics, thanks to its emphasis on bottom-up quantification and modeling.

Here we investigate the validity of the relationship between expected inflation and nominal interest rates proposed by Fisher (1930). Irving Fisher suggested that the nominal interest rate moves one-to-one with expected inflation. This means that an increase in expected inflation by one percentage point should increase the nominal interest rate by one percentage point. The basic intuition is that lenders (e.g., buyers of bonds) demand compensation from borrowers for the loss of purchasing power resulting from inflation.

In the following, estimates of expected inflation for 10 large economies will be presented. Due to space constraints, we have to limit the coverage to this relatively small set of countries. The economies selected are the largest according to their real GDP as documented by the World Bank in 2017: the USA, China, Japan, Germany, the United Kingdom, India, France, Brazil, Italy, and Canada. The main purpose of this project is to offer a service to fellow researchers and analysts. The estimates provide a starting point for those who do not want to compute pattern-based expectations by themselves. For those who need estimates for other countries or for higher frequency data, the spreadsheet files for the computation of pattern-based expectations are available for download from the website of this book.