Evolutionary Analysis and Computing of the Financial Safety Net

Abstract

Governments want to establish a financial safety net (FSN) to prevent financial crises from spreading. The FSN is a series of institutional arrangements to preserve financial stability. In the real FSN, it includes the central bank, deposit insurance institutions and their premium, commercial banks and their benefit rates, etc., and these parameters are interdependent and dynamic change. And thus, analysis and computing of the FSN is very challenging. Inspired by evolutionary game theory, in this paper, we first establish a network game model of the FSN to analyze the evolution of bank deposit insurance strategies, and further propose a method to measure the effectiveness of the FSN. Finally, we use computational experiments to simulate the operation of the FSN. In the experiments, an evolutionary computation method is employed to compute banks’ decisions to reduce computing time. Experimental results show that our evolutionary approach is suitable for the FSN, and is able to provide suggestions of macro policy for regulators.

Appendix

We give proofs of theorems in this paper as below.

Theorem 1:

We understand that the bank game is a routing game for funds seeking a better investment path. There are 2 alternative paths, and the funds amounts are H and (W − H), respectively. Therefore, the potential function of the bank game is

$$ \Phi \left( p \right) = \int_{0}^{H} {u(x)dx} + \int_{0}^{W - H} {v(x)dx} = \int_{0}^{H} {u(x)dx} + (W - H) \cdot v $$

∎

Theorem 2:

Since strategies are Nash equilibrium when potential function reaches extreme values, compute its first and second order derivative of Φ.

$\Phi ^{{\prime }} (H) = u(H) - v = a + b - c - b \cdot \left( {{H \mathord{\left/ {\vphantom {H W}} \right. \kern-0pt} W}} \right)^{e}$, $\Phi ^{{\prime \prime }} (H) = - b \cdot H^{e - 1} \cdot W^{ - e} < 0$. Thus, Φ reaches the maximum when $\Phi ^{{\prime }} (H) = 0$. And due to H = ∑ _i  p _i  · A _i , we get this theorem. ∎

Theorem 3:

Compute the first and second order derivative of social welfare function. $R\left( H \right) = u\left( H \right) \cdot H + v \cdot \left( {W - H} \right) = - b \cdot H \cdot \left( {{H \mathord{\left/ {\vphantom {H W}} \right. \kern-0pt} W}} \right)^{e} + \left( {a + b - c} \right) \cdot H + c \cdot W$, then $R^{\prime}\left( H \right) = \left( {a + b - c} \right) - b \cdot \left( {e + 1} \right) \cdot \left( {{H \mathord{\left/ {\vphantom {H W}} \right. \kern-0pt} W}} \right)^{e}$, $R^{\prime\prime}\left( H \right) = - b \cdot e \cdot \left( {e + 1} \right) \cdot H^{ - 1} \cdot \left( {{H \mathord{\left/ {\vphantom {H W}} \right. \kern-0pt} W}} \right)^{e} < 0$, R(H) reaches the maximum when $R^{\prime}\left( H \right) = 0$. Then, $R^{OPT} = \left( {\left( {{{\left( {a + b - c} \right)^{2} \cdot m} \mathord{\left/ {\vphantom {{\left( {a + b - c} \right)^{2} \cdot m} {2b \cdot \left( {m + 1} \right)}}} \right. \kern-0pt} {2b \cdot \left( {m + 1} \right)}}} \right) + c} \right) \cdot W$. In addition, $R\left( {p^{*} } \right) = c \cdot W$ from Theorem 2, together with the Eq. (12), we finally get the formula of this theorem. ∎

Theorem 4:

Let Lyapunov function be $L\left( p \right) =\Phi ^{ * } -\Phi \left( p \right)$, where Φ(p) is given by Eq. (9), Φ^* is the maximum of Φ(p). Thus, L(p) is positive definite.

$$ \begin{aligned} \dot{L}\left( p \right) & = - \dot{\Phi }\left( p \right) = - \left( {u\left( p \right) - v} \right) \cdot \dot{H} = - \left( {u\left( p \right) - v} \right) \cdot \sum\limits_{i} {A_{i} \cdot \dot{p}_{i} } \left( t \right) \\ & = - \left( {u\left( p \right) - v} \right) \cdot \sum\limits_{i} {A_{i} \cdot \lambda \left( t \right) \cdot p_{i} \left( t \right) \cdot \left( {1 - p_{i} (t)} \right) \cdot \left( {u(p) - v} \right)} \\ & = - \left( {u\left( p \right) - v} \right)^{2} \cdot \sum\limits_{i} {A_{i} \cdot \lambda \left( t \right) \cdot p_{i} \left( t \right) \cdot \left( {1 - p_{i} (t)} \right)} \le 0 \\ \end{aligned} $$

According to Lyapunov stability theorem, the solution ξ(p ₀, t) of Eq. (15) is asymptotic stable. Iff u(p) = v, $\dot{L}\left( p \right) = 0$ holds. Therefore, ξ(p ₀, t) converges to the Nash equilibrium as t → ∞. ∎

Theorem 5:

Let $\Psi \left( {p\left( t \right)} \right) =\Phi \left( {p\left( t \right)} \right) + R_{\hbox{min} }$, and its first order derivative with respect time t of $\Psi$ is $\dot{\Psi }\left( {p\left( t \right)} \right) = \dot{\Phi }\left( {p\left( t \right)} \right) \ge 0$. Iff $u\left( p \right) = v$, it holds $\dot{\Psi } = 0$, and $r_{i} = \bar{r}$ for each i.

According to Definition 2, if p(t) is not ε − δ equilibrium, then, it at least has ε · W part of funds amount, and its benefits rate is $\left( {1 - \delta } \right) \cdot \bar{r}$ at most. We use Jensen inequality in the following case. Suppose: the benefits rate of ε · W part is equal to $\left( {1 - \delta } \right) \cdot \bar{r}$, the rate of the rest (1 − ε) · W part is equal to $\hat{r}$. Therefore, $\bar{r}$ is given as below.

$$ \bar{r} = \varepsilon \cdot (1 - \delta )\bar{r} + (1 - \varepsilon ) \cdot \hat{r} $$

(16)

Jensen inequality is employed again, we have:

$$ \dot{\Psi }\left( {p\left( t \right)} \right) \ge \lambda \left( {\varepsilon W \cdot \left( {(1 - \delta )\bar{r}} \right)^{2} + (1 - \varepsilon )W \cdot \hat{r}^{2} - W \cdot \bar{r}^{2} } \right) $$

(17)

We get $\hat{r} = \bar{r} \cdot {{\left( {1 - \varepsilon + \varepsilon \delta } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \varepsilon + \varepsilon \delta } \right)} {1 - \varepsilon }}} \right. \kern-0pt} {1 - \varepsilon }}$ from Eq. (16), and substitute it into Eq. (17), then

$$ \dot{\Psi }\left( {p\left( t \right)} \right) \ge {{\lambda \varepsilon W\delta^{2} \bar{r}^{2} } \mathord{\left/ {\vphantom {{\lambda \varepsilon W\delta^{2} \bar{r}^{2} } {\left( {1 - \varepsilon } \right)}}} \right. \kern-0pt} {\left( {1 - \varepsilon } \right)}} = {{\varepsilon W\delta^{2} \bar{r}} \mathord{\left/ {\vphantom {{\varepsilon W\delta^{2} \bar{r}} {\left( {1 - \varepsilon } \right)}}} \right. \kern-0pt} {\left( {1 - \varepsilon } \right)}} \ge \varepsilon W\delta^{2} \bar{r} = \varepsilon \delta^{2} R $$

Next, $\Psi =\Phi + R_{\hbox{min} } \le {{\left( {2e + 3} \right)R} \mathord{\left/ {\vphantom {{\left( {2e + 3} \right)R} {\left( {e + 1} \right)}}} \right. \kern-0pt} {\left( {e + 1} \right)}}$, Substitute it into above inequality, we have

$$ \dot{\Psi }\left( {p\left( t \right)} \right) \ge {{\varepsilon \delta^{2} \cdot\Psi \left( {p\left( t \right)} \right) \cdot \left( {e + 1} \right)} \mathord{\left/ {\vphantom {{\varepsilon \delta^{2} \cdot\Psi \left( {p\left( t \right)} \right) \cdot \left( {e + 1} \right)} {\left( {2e + 3} \right)}}} \right. \kern-0pt} {\left( {2e + 3} \right)}} $$

Thus, if p(t) is not ε − δ equilibrium in period (t + Δt), it holds that

$$ \Psi \left( {p\left( {t +\Delta t} \right)} \right) \ge\Psi \left( {p\left( t \right)} \right) \cdot \exp \left( {{{\varepsilon \delta^{2} \cdot \left( {e + 1} \right)} \mathord{\left/ {\vphantom {{\varepsilon \delta^{2} \cdot \left( {e + 1} \right)} {\left( {2e + 3} \right)}}} \right. \kern-0pt} {\left( {2e + 3} \right)}} \cdot\Delta t} \right) $$

These periods include (t ₀ + Δt ₀), (t ₁ + Δt ₁), …, (t _m  + Δt _m ), …

Let T = ∑ _m Δt _m , When m goes to infinity, T includes the total periods when p does not reach equilibrium, we have

$$ \begin{aligned} & \quad exp\left( {\frac{e + 1}{2e + 3}\varepsilon \delta^{2} \cdot T} \right) \le \frac{{\Psi \left( {p(t_{i} + \Delta t_{i} )} \right)}}{{\Psi \left( {p(t_{0} )} \right)}} \le \frac{{\Psi _{\hbox{max} } }}{{\Psi _{\hbox{min} } }} \le \frac{{\Phi _{\hbox{max} } + R_{\hbox{max} } }}{{R_{\hbox{min} } }} \le \frac{2e + 3}{e + 1} \cdot \frac{{R_{\hbox{max} } }}{{R_{\hbox{min} } }} \\ & T \le {{\left( {e + 1} \right)\ln \left( {{{\left( {2e + 3} \right)R_{\hbox{max} } } \mathord{\left/ {\vphantom {{\left( {2e + 3} \right)R_{\hbox{max} } } {\left( {e + 1} \right)R_{\hbox{min} } }}} \right. \kern-0pt} {\left( {e + 1} \right)R_{\hbox{min} } }}} \right)} \mathord{\left/ {\vphantom {{\left( {e + 1} \right)\ln \left( {{{\left( {2e + 3} \right)R_{\hbox{max} } } \mathord{\left/ {\vphantom {{\left( {2e + 3} \right)R_{\hbox{max} } } {\left( {e + 1} \right)R_{\hbox{min} } }}} \right. \kern-0pt} {\left( {e + 1} \right)R_{\hbox{min} } }}} \right)} {\left( {2e + 3} \right)}}} \right. \kern-0pt} {\left( {2e + 3} \right)}}\varepsilon \delta^{2} \sim {\mathcal{O}}\left( {{{\ln \left( {{{R_{\hbox{max} } } \mathord{\left/ {\vphantom {{R_{\hbox{max} } } {R_{\hbox{min} } }}} \right. \kern-0pt} {R_{\hbox{min} } }}} \right)} \mathord{\left/ {\vphantom {{\ln \left( {{{R_{\hbox{max} } } \mathord{\left/ {\vphantom {{R_{\hbox{max} } } {R_{\hbox{min} } }}} \right. \kern-0pt} {R_{\hbox{min} } }}} \right)} {\varepsilon \delta^{2} }}} \right. \kern-0pt} {\varepsilon \delta^{2} }}} \right) \\ \end{aligned} $$

∎