Waiting times in evolutionary dynamics with time-decreasing noise

Abstract

Expected waiting times for equilibrium selection are exponentially increasing as the noise level goes to zero in evolutionary models with time-constant noise, raising questions about whether the history independent prediction of equilibrium selection is relevant in economic and social studies. However, by using the theoretical results on simulated annealing, we show that expected waiting times in models with time-decreasing noise need not tend toward infinity in the small noise limit. Our model thus describes conditions under which the waiting-time critique of the predictions of stochastic stability theory may have less force.

Appendix: Proof of Proposition 3.2

We denote $x\mapsto y$ if there is a directed link from state $x$ to $y$. For any subset $W \subset E$, we let $G(W)$ be the set of oriented graphs $g=\{ (x,y) \in E \times E\,\vert \, x\mapsto y\}$ such that $(i)$ every point $x \in \overline{W}=E{\setminus }W$ is the initial point of exactly one arrow (no arrow starts from $W$), and $(ii)$ there are no closed cycles in the graph. Let $G_{x,y}(W)$ denote the set of graphs $g \in G(W)$ which leads from $x \in E$ to $y \in W$, where, by convention, $G_{y,y}(W)=G(W)$ for $y \in W$.¹⁶ Catoni (1995, Lemma 3.4), based on Freidlin and Wenztel (1998, Lemma 6.3.4), gives a formula

$$\begin{aligned} E\left( \tau (\overline{C}) \,\vert \,X_0=x\right) =\left( \sum _{y \in C}\sum _{g \in G_{x,y}(\overline{C}\cup \{y\})}p(g)\right) \left( \sum _{g\in G(\overline{C})}p(g)\right) ^{-1} \quad \text{ for } \quad x \in C, \end{aligned}$$

(8)

where $p(g):=\prod _{(x,y)\in g}p_\eta (x,y)$ is the product of transition probabilities over the graph $g$. The transition probability (1) has a form

$$\begin{aligned} a\mathrm {e}^{-V(x,y)/\eta }\le p_\eta (x,y) \le a^{-1}\mathrm {e}^{-V(x,y)/\eta } \quad \text{ for } \text{ all } x \ne y \text{ and } \text{ some } a>0, \end{aligned}$$

where $V(x,y):=[U(y)-U(x)]_+$ if $q(x,y)>0$, and $V(x,y):=\infty$ otherwise. Let $V(g):=\sum _{(x,y)\in g}V(x,y)$ and $g^*$ the graph solving $\min _{y \in C}\min _{g \in G_{x,y}(\overline{C}\cup \{y\})}V(g)$. Then the numerator of (8) is bounded as follows:

$$\begin{aligned} a^{\left| g^*\right| }\mathrm {e}^{-\min _{y \in C}\min _{g \in G_{x,y}(\overline{C}\cup \{y\})}V(g)/\eta }&\le \sum _{y \in C}\sum _{g \in G_{x,y}(\overline{C}\cup \{y\})}a^{\left| g\right| }\mathrm {e}^{-V(g)/\eta }\\&\le \sum _{y \in C}\sum _{g \in G_{x,y}(\overline{C}\cup \{y\})}p(g)\\&\le \sum _{y \in C}\sum _{g \in G_{x,y}(\overline{C}\cup \{y\})}a^{-\left| g\right| }\mathrm {e}^{-V(g)/\eta }\\&\le B_1 a^{-\left| g^*\right| }\mathrm {e}^{-\min _{y \in C}\min _{g \in G_{x,y}(\overline{C}\cup \{y\})}V(g)/\eta } \end{aligned}$$

for some constant $B_1>0$, where $\left| g\right|$ is the number of elements in the graph $g$. The last inequality follows because the summation is over a finite number of terms, each of which is positive and bounded. Similarly, letting $g^{**} \in \mathrm{argmin}_{g\in G(\overline{C})}V(g)$, the denominator of (8) is bounded as follows:

$$\begin{aligned}&a^{\left| g^{**}\right| }\mathrm {e}^{-\min _{g\in G(\overline{C})}V(g)/\eta }\le \sum _{g \in G(\overline{C})}p(g)\le B_2 a^{-\left| g^{**}\right| }\mathrm {e}^{-\min _{g\in G(\overline{C})}V(g)/\eta }\\&\quad \text{ for } \text{ some } \text{ constant } B_2>0. \end{aligned}$$

Therefore, we conclude that there exists $b>0$ such that $b\mathrm {e}^{H_C(x)/\eta }\!\le \!E\left( \tau (\overline{C})\,\vert \,X_0\!=\!x\right) \le b^{-1}\mathrm {e}^{H_C(x)/\eta }$, where $H_C(x):=\min _{g\in G(\overline{C})}V(g)-\min _{y \in C}\min _{g \in G_{x,y}(\overline{C}\cup \{y\})}V(g)$. Now we are done if we show that $\max _{x \in C}H_C(x)=H(C)$. But this comes from Proposition 4.15 of Catoni (1995).$\square$