What is a Universal? On the Explanatory Potential of Evolutionary Game Theory in Linguistics

Abstract

Natural languages are shaped by evolutionary processes, both in the sense of biological evolution of our species, and, on a much shorter time scale, by a form of cultural evolution. There are long research traditions in theoretical biology and economics (a) to model communication by means of game theory, and (b) to use game theory to study biological and cultural evolution. Drawing mostly on work by Huttegger (2007) and Pawlowitsch (2008), this chapter argues that results and methods from game theory are apt to formalize the intuitive notion of linguistic universals as emergent properties of communication.

Appendix

Numerical estimation of the size of the basin of attraction As mentioned in the text, the joint basin of attraction of the set of neutrally but not evolutionarily stable strategies is about 1.2 % of the entire strategy space. The result was obtained by using a Monte-Carlo method. A random initial state $s$ was picked out by a random generator according to the uniform distribution over the 729-dimensional simplex, the replicator dynamics was solved numerically for the initial condition $s$, and the asymptotic behavior was analyzed at $t=1000$ (by which time all time series have converged towards a rest point, within the limits of precision imposed by the numerical algorithm used). This procedure was repeated 5,000 times. It turned out that in 4,941 cases the solution converged towards an evolutionarily stable state, and in 59 cases to a neutrally but not evolutionarily stable state. This means that with a confidence of 95 %, the true value lies between 0.9 and 1.5 %, with a maximum likelihood estimation of 1.2 %.

Basins of attraction The utility matrix of the game in question is:

$$\begin{aligned} A = \left( \begin{array}{rrr} 2 &{} 2 &{} 2\\ 2 &{} 3 &{} -1\\ 2 &{}-1 &{} 0 \end{array} \right) \end{aligned}$$

This game is doubly symmetric, and it has two Nash equilibria: $e_1$ and $e_2$. Consider the linear manifold $E = \{x\in \mathrm{{int}}S_3: x_3 = 4x_2\}$. For points $x\in E$, we have

$$\begin{aligned} \dot{x}_2&= x_2(2x_1 + 3x_2 - x_3 - x\cdot Ax)\\&= x_2(2x_1 - x_2 - x\cdot Ax)\\ \dot{x}_3&= x_3(2x_1 - x_2 - x\cdot Ax)\\&= 4x_2(2x_1 - x_2 - x\cdot Ax), \end{aligned}$$

and thus $\dot{x}_3 = 4\dot{x}_2$. So $\dot{x}$ is always tangential to $E$, and $E$ is thus an invariant set. As $e_1$ is the only Nash equilibrium within the closure of this set, all points in $E$ converge towards $e_1$. Now consider the set $F = \{x\in \mathrm{{int}}S_3: x_3 > 4x_2\}$. Since $e_2$ is not in the closure of this set, a trajectory starting in $F$ and converging towards $e_2$ would have to leave $F$. This is impossible though because all interior points remain within the interior under the replicator dynamics—so no orbit can touch a boundary face—and a trajectory cannot cross $E$ due to uniqueness of solutions of autonomous differential equations. We thus conclude that all points in $F$ converge towards $e_1$. Since $F$ has a positive Lebesgue measure, $e_1$ has a basin of attraction that is not a null set. Nevertheless $e_1$ is not neutrally stable, because

$$\begin{aligned} e_2\cdot Ae_1&= e_1\cdot Ae_1\\ e_2\cdot Ae_2&> e_1\cdot Ae_2 \end{aligned}$$

In Fig. A.1 the phase portrait of the game is sketched. The bold line indicates the boundary between the basins of attraction of $e_1$ and $e_2$. It is easy to see that $e_1$ is not Lyapunov stable (because every open environment has a non-empty intersection with the basin of attraction of $e_2$), but nevertheless attracts a non-null set.

Fig. A.1

Phase portrait