| Russian Mathematical Surveys |
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This article is cited in 2 scientific papers (total in 2 papers) Brief communications Stable and historic behavior in replicator equations given by nonlinear mappings M. Saburov College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait
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     1.While studying the evolution of a ‘binary reaction system’ of three species which is a zero-sum game, Ulam conjectured the mean ergodicity of quadratic stochastic operators acting on the finite-dimensional simplex (see [1]). However, Zakharevich showed that Ulam’s conjecture is in general false (see [2]). Subsequently, the complicated dynamics of ‘binary reaction systems’ which is known as the historic behaviour in the literature (see [3] and [4]) was studied in great detail (see [5] and [6]). However, this is against the common belief (see [7] and [8]) that all ‘reasonable’ replicator equations satisfy the ‘Folk Theorem’ of evolutionary game theory. In this paper we propose two distinct classes of replicator equations which exhibit ‘stable as well as ‘mean historic’ behaviour. In the latter case, the time averages of the orbit oscillate slowly during the evolution and do not converge to any limit. This will eventually cause the divergence of higher-order repeated time averages. Suppose $\mathbb{R}^m$ is equipped with the $l_1$-norm $\|\mathbf{x}\|_1:=\sum_{k=1}^{m}|x_k|$, where $\mathbf{x}:=(x_1,\dots,x_m) \in \mathbb{R}^m$. Let $\mathbb{R}_{+}^m:=\{\mathbf{x}\in\mathbb{R}^{m}\colon \mathbf{x}\geqslant 0\}$ and $\mathbb{B}_{+}^m:=\{\mathbf{x}\in\mathbb{R}_{+}^{m}\colon \|\mathbf{x}\|_1\leqslant 1\}$, and let $\mathbb{S}^{m-1}:=\{\mathbf{x}\in \mathbb{R}_{+}^{m}\colon \|\mathbf{x}\|_1=1\}$ be the simplex. We set $\langle\mathbf{x},\mathbf{y}\rangle:=\sum_{k=1}^{m}x_ky_k$ for two vectors $\mathbf{x},\mathbf{y}\in\mathbb{R}^{m}$. Two vectors $\mathbf{x},\mathbf{y}\in\mathbb{R}_{+}^{m}$ are called similarly ordered (denoted by $\mathbf{x}\approx\mathbf{y}$) if one has $x_i\gtreqqless x_j$ if and only if $y_i\gtreqqless y_j$ for all $1\leqslant i,j\leqslant m$. For a vector $\mathbf{x}\in\mathbb{R}_{+}^{m}$ the set ${\mathbf{SOC}}[\mathbf{x}]=\{\mathbf{y}\in\mathbb{R}_{+}^{m}\colon \mathbf{x}\approx\mathbf{y}\}$ of all vectors $\mathbf{y}\in\mathbb{R}_{+}^{m}$ for which $\mathbf{x}$ and $\mathbf{y}$ are similarly oriented is obviously a convex cone. A continuous mapping $\mathbf{F}\colon\mathbb{R}^m_{+}\to\mathbb{R}^m_{+}$ is said to be similar-order preserving if $\mathbf{F}(\mathbf{x})\in{\mathbf{SOC}}[\mathbf{x}]$, that is, $\mathbf{F}(\mathbf{x})\approx\mathbf{x}$, for all $\mathbf{x}\in\mathbb{R}_{+}^{m}$. It is well known (see [9]) that for any continuously differentiable, strictly increasing and strictly Schur-convex function $\varphi\colon\mathbb{R}_{+}^m\to\mathbb{R}_{+}$ the gradient vector field $\nabla\varphi\colon\mathbb{R}_{+}^m\to\mathbb{R}_{+}^m$ is similar-order preserving. Throughout this paper we always assume that a similar-order preserving mapping $\mathbf{F}\colon\mathbb{B}^m_{+}\to\mathbb{R}^m_{+}$, $\mathbf{F}(\mathbf{x}):=(f_1(\mathbf{x}),\dots,f_m(\mathbf{x}))$, is continuously differentiable and $0<\mathbf{F}(\mathbf{x})\leqslant \mathbf{1}$ for all $\mathbf{x}\in\mathbb{B}_{+}^{m}$ (here $\mathbf{1}=(1,\dots,1)$). 2.Consider the replicator equation $\mathcal{R}_{S}\colon\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$ associated with a similar- order preserving mapping $\mathbf{F}\colon\mathbb{B}^m_{+}\to\mathbb{R}^m_{+}$, $\mathbf{F}(\mathbf{x}):=(f_1(\mathbf{x}),\dots,f_m(\mathbf{x})\kern-0.5pt)$:
$$
\begin{equation}
(\mathcal{R}_{S}(\mathbf{x}))_k=x_k\biggl(1+f_k(\mathbf{x})- \sum_{i=1}^mx_if_i(\mathbf{x})\biggr), \qquad 1 \leqslant k\leqslant m.
\end{equation}
\tag{1}
$$
Recall (see [7] and [8]) that $\mathbf{x}\in\mathbb{S}^{m-1}$ is called a Nash equilibrium if $\langle\mathbf{x},\mathbf{F}(\mathbf{x})\rangle\geqslant \langle\mathbf{y},\mathbf{F}(\mathbf{x})\rangle$ for all $\mathbf{y}\in\mathbb{S}^{m-1}$. Moreover, $\mathbf{x}\in\mathbb{S}^{m-1}$ is called a strictly Nash equilibrium if $\langle\mathbf{x},\mathbf{F}(\mathbf{x})\rangle > \langle\mathbf{y},\mathbf{F}(\mathbf{x})\rangle$ for all $\mathbf{y}\in\mathbb{S}^{m-1}$ such that $\mathbf{y}\ne\mathbf{x}$. A point $\mathbf{x}\in\mathbb{S}^{m-1}$ is called a fixed (or rest) point if $\mathcal{R}_{S}(\mathbf{x})=\mathbf{x}$. Theorem 1. The following statements are true for the replicator equation (1): (a) a Nash equilibrium is a fixed point; (b) a stable fixed point is a Nash equilibrium; (c) a strictly Nash equilibrium is asymptotically stable; (d) any convergent orbit that belongs to the interior of the simplex evolves to a Nash equilibrium. We now present a replicator equation of a zero-sum game that exhibits a mean historic behaviour. We consider either of the following replicator equations $\mathcal{R}_{H}\colon \mathbb{S}^{2}\to\mathbb{S}^{2}$ of a zero-sum game associated with a similar-order preserving mapping $\mathbf{F}\colon\mathbb{B}^3_{+}\to\mathbb{R}^3_{+}$, $\mathbf{F}(\mathbf{x}):=(f_1(\mathbf{x}),f_2(\mathbf{x}),f_3(\mathbf{x}))$:
$$
\begin{equation}
{\small \!\! \begin{cases} (\mathcal{R}_{H}(\mathbf{x}))_1=x_1(1+x_2f_1(\mathbf{x})-x_3f_3(\mathbf{x})), \\ (\mathcal{R}_{H}(\mathbf{x}))_2=x_2(1+x_3f_2(\mathbf{x})-x_1f_1(\mathbf{x})), \\ (\mathcal{R}_{H}(\mathbf{x}))_3=x_3(1+x_1f_3(\mathbf{x})-x_2f_2(\mathbf{x})) \end{cases}\ \ \text{or }\ \begin{cases} (\mathcal{R}_{H}(\mathbf{x}))_1=x_1(1+x_3f_1(\mathbf{x})-x_2f_2(\mathbf{x})), \\ (\mathcal{R}_{H}(\mathbf{x}))_2=x_2(1+x_1f_2(\mathbf{x})-x_3f_3(\mathbf{x})), \\ (\mathcal{R}_{H}(\mathbf{x}))_3=x_3(1+x_2f_3(\mathbf{x})-x_1f_1(\mathbf{x})). \end{cases} }
\end{equation}
\tag{2}
$$
We say that the replicator equation (2) has a mean historic behavior if the set of initial points $\mathbf{x}\in\mathbb{S}^{2}$ which give rise to orbits with divergent time averages
$$
\begin{equation*}
\frac1n\sum_{k=0}^{n-1}\mathcal{R}_{H}^{(k)}(\mathbf{x})
\end{equation*}
\notag
$$
has positive Lebesgue measure (see [10]). We define the $s$th-order ($s\in\mathbb{N}$) repeated time averages $\{\mathcal{A}^{(s)}_{n}(\mathbf{x})\}_{n=1}^\infty$ by
$$
\begin{equation*}
\mathcal{A}^{(s)}_{n}(\mathbf{x}):=\frac1n\sum_{k=1}^{n}\mathcal{A}^{(s-1)}_{k}(\mathbf{x}) \quad\text{for}\quad s\geqslant 2
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{A}^{(1)}_{n}(\mathbf{x}):=\frac1n\sum_{k=0}^{n-1}\mathcal{R}_{H}^{(k)}(\mathbf{x}).
\end{equation*}
\notag
$$
Theorem 2. The replicator equation (2) has a mean historic behavior. Moreover, for any interior initial point $\mathbf{x}\ne(1/3,1/3,1/3)$ the $s$th-order repeated time averages $\{\mathcal{A}^{(s)}_{n}(\mathbf{x})\}_{n=1}^\infty$ do not converge for any $s\in\mathbb{N}$. 
Citation in format AMSBIB
\Bibitem{Sab23} |
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