The Distribution of Foreign Language Skills as a Game Equilibrium

Reinhard Selten

Jonathan Robert Pool

Originally published in, and reprinted with corrections with kind permission of Springer Science and Business Media from:

Reinhard Selten (ed.), Game Equilibrium Models IV: Social and Political Interaction (Berlin: Springer-Verlag, 1991), pp. 64-87.

© Springer-Verlag Berlin · Heidelberg 1991

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Individuals make choices of which of the world's thousands of languages to learn, responding to opportunities and incentives associated with contact, prestige, schooling, earnings, productivity, and communication. In the long run, such choices not only respond to the distribution and characteristics of native and non-native speakers of languages, but also affect them.

Like some prior theoretical models (Grenier, Lang), we investigate here a model of language-learning choices with a fixed distribution of native languages. Unlike them, however, our model (A) generalizes learning incentives beyond earnings, (B) permits not only two but an unlimited number of languages to exist, and (C) permits languages with native speakers and languages without native speakers to exist.

In our model, every person has exactly one native language, so the world population can be partitioned into native-speaker communities, which we model as continua. The world is a continuum of players of a noncooperative normal-form game, each player's strategy being the set of additional languages learned. We consider only pure strategies.

The players' payoffs are the net benefits they derive from learning the languages that they learn (a communicative benefit minus a learning cost). A player's communicative benefit is proportional to the importance-weighted fraction of the world population sharing at least one language with the player, where the player's community assigns some importance to each member of the world population.

A player's learning cost is the product of a linguistic factor (dependent only on the player's native and learned languages) and a personal factor (differentiating all members of a community).

We show that a player's payoff is independent of the strategies of the other members of that player's own community, because those strategies cannot affect the subset of the world population sharing at least one language with the player. But a player can affect the payoff of another player in the same community indirectly, by affecting the choices of players in other communities.

We show further that a strategy which is payoff-maximizing for two players in a community (given the combination of all other communities' players' strategies) is also payoff-maximizing for all other players in that community whose personal factors lie between those two players'. The corresponding empirical prediction is that those native speakers of language X who learn the same set of foreign languages will tend to be more similar in (dis)inclination and (in)ability to learn languages than fellow community members in general.

We further show that in any community-wide combination of payoff-maximizing strategies ("group best reply") the two factors of the learning cost are negatively associated, with players learning more expensive sets of languages being those who incur lower learning costs. The greater the linguistic factor, the greater the communicative benefit. Moreover, and perhaps counterintuitively, the greater the linguistic factor, the greater the payoff. A corresponding empirical prediction is that those who learn more difficult sets of languages get higher profits, despite the difficulty of the languages they learn.

We move on to prove, by examining aggregated games, that it is possible for every player to adopt a best reply at the same time, i.e. that at least one equilibrium point exists.

We discuss further empirical insights in light of numerical examples of our model. The examples illustrate tendencies for those who learn non-native languages to choose the most widely spoken ones, for members of small language communities to learn foreign languages more than members of large communities, for few persons to learn auxiliary languages (those without native speakers), even if they have relatively low learning costs, and for language communities with chronically poor second-language-learning abilities to enjoy above-average welfare levels.


language birth
language death
language growth
language shrinkage
language diversity
linguistic diversity
intergenerational transmission of language
language learning
second language learning
language learning cost
language learning incentives
language learning motivations
language learnability
language loss
auxiliary languages
artificial languages
artificial auxiliary languages
economics of language
language economics
language game models
linguistic equilibria
noncooperative games
continuous players
aggregated games
group strategies

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