Stability of Weak NS Equilibria for Population Games with Uncertain Parameters Under Bounded Rationality

Wang Mingting,, Yang Guanghui,, Yang Hui,

College of Mathematics and Statistics, Guizhou University, Guiyang 550025

 基金资助: 国家自然科学基金项目.  11271098贵州省科技计划项目.  黔科合基础[2019]1067号贵州大学引进人才科研项目.  [2017]59贵州省教学改革项目.  201908

Received: 2021-08-12

 Fund supported: the NSFC.  11271098the Guizhou Provincial Science and Technology Foundation.  黔科合基础[2019]1067号the Talent Introduction Research Foundation of Guizhou University.  [2017]59the Educational Reform Foundation of Guizhou Province.  201908

Abstract

For population games with uncertain parameters, a weak NS equilibrium is firstly proposed based on the fact that switching strategies cause corresponding costs. The underlying idea of a weak NS equilibrium is that the agents' new gained payoffs from strategy switch are less than or equal to the increased cost for a given uncertainty parameter; simultaneously, each population can not obtain strictly poor net profits under uncertain parameters, thus each agent in every population has no motivation to unilaterally change the current strategy and then they achieve a weak NS equilibrium. Secondly, the existence of weak NS equilibria is proven by Kakutani's fixed point theorem. Thirdly, by constructing an abstract rational function, a corresponding bounded rational model is established, and it is shown structural stability implying robustness. Therefore, the generic stability of weak NS equilibria for population games with uncertain parameters under bounded rationality is also obtained when the net profit function is perturbed. Finally, an example is illustrated the correctness of the above results.

Keywords： Population games ; Weak NS equilibria ; Bounded rationality ; Uncertainty ; Structural stability

Wang Mingting, Yang Guanghui, Yang Hui. Stability of Weak NS Equilibria for Population Games with Uncertain Parameters Under Bounded Rationality. Acta Mathematica Scientia[J], 2022, 42(6): 1812-1825 doi:

1 引言

(1) 博弈中具有有限个群体;

(2) 每一群体中拥有数量充分大而有限的匿名参与人, 所有博弈参与人都具有相同的有限个纯策略可供选择, 所有群体的纯策略分布进而形成一个社会状态;

(3) 每一博弈参与人的收益函数同时依赖于自己所选策略和社会状态.

Nash均衡的存在性和稳定性是群体博弈理论的研究焦点. Lahkar等[3, 4]分别在有限个和无限个纯策略情形下研究了群体势博弈及其Nash均衡的演化稳定性. Chow[5]则在有限个纯策略情形下讨论了群体博弈的Logit动力学及其演化稳定性. 本文合作者在有限理性下获得了群体博弈Nash均衡的稳定性[6]. 基于多目标博弈的实际应用需要, 本文合作者进一步将群体博弈模型的收益函数从单目标推广到多目标(向量值)情形并分别研究了加权Nash均衡[7]和(弱)Pareto-Nash均衡[8]的存在性和本质稳定性. 而基于群体间的合作, 杨哲及张海群研究了群体博弈合作均衡的本质稳定性[9]及具有无限多纯策略的群体博弈的NTU核、TU核和强均衡的存在性[10]. 贾文生及陈华鑫[11]研究了群体博弈的逼近定理及通有收敛性.

2 预备知识

(4) $\forall \lambda\in Q, \forall \lambda_m\rightarrow \lambda, \epsilon_m\rightarrow 0 $$(m\rightarrow +\infty) , 有 h(E(\lambda_m, \epsilon_m), E(\lambda))\rightarrow 0 . (5) 如果 \lambda \in \Lambda, E(\lambda) 是单点集, 则模型 M$$ \lambda$处是结构稳定的, 从而对$\epsilon$-平衡是鲁棒的.

$(3) $$\forall p\in \Gamma , 用非空集合 S^p=\{1, 2, \cdots, n^p\} 表示群体 p 的纯策略集, \Gamma\ 中所有群体的纯策略总数为 n=\sum\limits_{p\in \Gamma} n^p . (4)$$ \forall p\in \Gamma$, $x_i^p$表示群体$p$中选用$i$的代理人的份额.

$(5) $$x^p=(x_1^p, x_2^p, \cdots, x_{n^p}^p)\in R_{+}^{n^p} 表示群体 p 的纯策略分布状态, 称为群体状态. 则群体状态集为: X^p=\{x^p=(x_1^p, x_2^p, \cdots, x_{n^p}^p)\in R_{+}^{n^p}:\sum\limits_{i\in S^p} x_i^p=1 \} . (6)$$ X\times Y$表示$\Gamma$的社会状态集, 其中$X=\prod\limits_{p\in \Gamma} X^p=\{x=(x^1;x^2;\cdots;x^p)\in R^n: x^p\in X^p, p\in \Gamma\}$, $Y\subset R^l$表示不确定参数集, 其中$l\geqslant 1$.

$(7) $$\forall p, F_i^p:X \times Y\rightarrow R 表示在状态 x\in X 和不确定参数 y\in Y 下, 群体 p 中代理人选择策略 i\in S^p 的支付函数, 记为 F_i^p(x, y) , 则整个群体 p 的支付函数表示为 (8) 整个社会 \Gamma 的支付函数为: F=(F^1;F^2;\cdots;F^P):X\times Y\rightarrow R^n . (9)$$ \forall p, V_i^p:X \times Y\rightarrow R$表示在状态$x\in X$和不确定参数$y\in Y$下, 群体$p$中代理人选择策略$i\in S^p$的成本函数, 记为$V_i^p(x, y)$, 则整个群体$p$的成本函数表示为：

$(10)$整个社会$\Gamma$的成本函数为：$V=(V^1;V^2;\cdots;V^P):X\times Y\rightarrow R^n$.

(1) $\bar{x}_i^p>0 \Rightarrow V_j^p(\bar{x}, \bar{y})-V_i^p(\bar{x}, \bar{y})\geqslant F_j^p(\bar{x}, \bar{y})- F_i^p(\bar{x}, \bar{y}), \; \; \forall p\in \Gamma, \; \; \forall i, j\in S^p$.

(2) $F^p(\bar{x}, \bar{y})-F^p(\bar{x}, y)+V^p(\bar{x}, y)-V^p(\bar{x}, \bar{y}) \notin {\rm int} R_+^{n^p}, \; \; \forall p\in \Gamma, \; \; \forall y\in Y.$

(ⅱ) 上述博弈$\{\Gamma, X, Y, F, V\}$可记为$\{\Gamma, X, Y, H\}$, 其中$H=F-V$即群体博弈的纯收益函数, 而所有弱$\mathrm{NS}$平衡构成的集合记为$NS(H)$, 从而定义$3.1$中条件$(1)$的等价形式为

(ⅲ) 定义$3.1$中的条件$(2)$的等价形式为

(ⅳ) 对群体博弈$\{\Gamma, X, Y, F, V\}$, 当成本函数$V(x, y)=0, \forall (x, y)\in X\times Y$时, 弱$\mathrm{NS}$平衡退化为文献[21, 22]中$\{\Gamma, X, Y, F\} $$\mathrm{NS} 平衡. 定理3.1 以下 (1)$$ (2)$等价.

(1) $\bar{x}_i^p > 0 \Rightarrow H_i^p(\bar{x}, \bar{y})\geqslant H_j^p(\bar{x}, \bar{y}), \; \; \forall i, j\in S^p, \; \; \forall p\in \Gamma.$

(2) $\left<u-\bar{x}, H(\bar{x}, \bar{y})\right>\leqslant0, \; \; \forall u\in X.$

$(1)$当社会状态$(\bar{x}, \bar{y})=(1, 0, \bar{y})$时, $F_1( \bar{x}, \bar{y})=4-2\bar{y}, \; F_2( \bar{x}, \bar{y})=2-2\bar{y}$, 如果$(1, 0, \bar{y}) $$\mathrm{NS} 平衡, 则 4-2\bar{y}>2-2\bar{y} 成立. 又由文献[21, 定义2.1的条件(2)]可知 从而状态 (\bar{x}, \bar{y})=(1, 0, 1)$$ \{\Gamma, X, Y, F\} $$\mathrm{NS} 平衡. (2) 当社会状态 (\tilde{x}, \tilde{y})=(0, 1, \tilde{y}) 时, F_1( \tilde{x}, \tilde{y})=3-2\tilde{y}, F_2(\tilde{x}, \tilde{y})=5-2\tilde{y} , 如果 (0, 1, \tilde{y})$$ \mathrm{NS}$平衡, 则$3-2\tilde{y}<5-2\tilde{y}$成立.又由文献[21, 定义2.1的条件(2)], 同理可得状态$(\tilde{x}, \tilde{y})=(0, 1, 1)$也是$\{\Gamma, X, Y, F\} $$\mathrm{NS} 平衡. (3) 当社会状态 (\hat{x}, \hat{y})=(\frac{1}{2}, \frac{1}{2}, \hat{y}) 时, F_1(\hat{x}, \hat{y})=\frac{7}{2}-2\hat{y}, F_2(\hat{x}, \hat{y})=\frac{7}{2}-2\hat{y} , 如果 (\frac{1}{2}, \frac{1}{2}, \hat{y})$$ \mathrm{NS}$平衡, 则$\frac{7}{2}-2\hat{y}=\frac{7}{2}-2\hat{y}$成立. 又由文献[21, 定义$2.1$的条件$(2)$], 同理可得状态$(\hat{x}, \hat{y})=(\frac{1}{2}, \frac{1}{2}, 1)$也是$\{\Gamma, X, Y, F\} $$\mathrm{NS} 平衡. 由于策略调整会产生相应成本, 从而对 \{\Gamma, X, Y, F\} 引入成本函数 V:X\times Y\rightarrow R^n , 其中策略 1 和策略 2 的成本分别为: V_1(x, y)=3x_1+x_2-y, V_2(x, y)=4x_2-y , 从而纯收益函数 H(x, y)=F(x, y)-V(x, y), \; \forall x\in X, y\in Y . (1) 当社会状态 (\bar{x}, \bar{y})=(1, 0, \bar{y}) 时, H_1( \bar{x}, \bar{y})=1-\bar{y}, H_2( \bar{x}, \bar{y})=2-\bar{y} , 如果 (1, 0, \bar{y}) 是弱 \mathrm{NS} 平衡, 则 1-\bar{y}>2-\bar{y} , 这显然不成立. (2) 当社会状态 (\tilde{x}, \tilde{y})=(0, 1, \tilde{y}) 时, H_1( \tilde{x}, \tilde{y})=2-\tilde{y}, H_2(\tilde{x}, \tilde{y})=1-\tilde{y} , 如果 (0, 1, \tilde{y}) 是弱 \mathrm{NS} 平衡, 则 2-\tilde{y}<1-\tilde{y} , 这显然不成立. (3) 当社会状态 (\hat{x}, \hat{y})=(\frac{1}{2}, \frac{1}{2}, \hat{y}) 时, H_1(\hat{x}, \hat{y})=\frac{3}{2}-\hat{y}, H_2(\hat{x}, \hat{y})=\frac{3}{2}-\hat{y} , 如果 (\frac{1}{2}, \frac{1}{2}, \hat{y}) 是弱 \mathrm{NS} 平衡, 则 \frac{3}{2}-\hat{y}=\frac{3}{2}-\hat{y} 成立. 且由定义 3.1 中的条件 (2) 从而状态 (\hat{x}, \hat{y})=(\frac{1}{2}, \frac{1}{2}, 1)$$ \{\Gamma, X, Y, H\}$的弱$\mathrm{NS}$平衡.

(1) 对每一$p\in \Gamma$及每一$i\in S^p$, $F_i^p $$V_i^p:X \times Y\rightarrow R 连续. (2) \forall x\in X, y\mapsto F_i^p(x, y) 是凸的, y\mapsto V_i^p(x, y) 是凹的. 则不确定参数下群体博弈弱 \mathrm{NS} 平衡必存在. 对每一 p\in \Gamma , 记 H^p(x, y)=F^p(x, y)-V^p(x, y), \; \forall (x, y)\in X\times Y , 由(2)可得 H_i^p(x, y) 关于 y 是凸的. 构造集值映射 G^p: X\times Y\rightarrow 2^{X^p} G(x, y)=\prod\limits_{p\in \Gamma}G^p(x, y) , 可得 G:X\times Y\rightarrow 2^X . 同时, 构造另一集值映射 Q^p: X\rightarrow 2^Y Q(x)=\prod\limits_{p\in \Gamma}Q^p(x) , 可得 Q:X\rightarrow 2^Y. (ⅰ) \forall (x, y)\in X\times Y, \forall p\in \Gamma , 因为 X^p$$ Y$是非空紧的, 且$\forall {u^p\in X^p}, \sum\limits_{i\in S^p}u_i^pH_i^p(x, y)$是连续的, 所以$G^p(x, y)\neq \emptyset; $$H_i^p 是连续的且 Y 是紧的, 那么 Q^p(x)\neq \emptyset . (ⅱ) \forall (x, y)\in X\times Y, \forall p\in \Gamma, G^p(x, y)$$ Q^p(x)$是凸的.

$H^p(x, \lambda y_1+(1-\lambda ) y_2)-H^p(x, w)\notin {\rm int}R_+^{n^p}, \; \forall w\in Y$, 因此$\forall x\in X, Q^p(x)$是凸集.

(ⅲ) 映射$G^p $$Q^p 是上半连续且紧值的. 下证 G^p 是上半连续的. 由引理2.5, 只需证 G^p 的图是闭的. 设 \{(x^m, y^m)\}_{m=1}^\infty\subset X\times Y 是任意序列, 并且 (x^m, y^m)\rightarrow (x, y), \forall (z^p)^m\in G^p(x^m, y^m) , 并且 (z^p)^m\rightarrow z^p , 需证 z^p\in G^p(x, y) . 因为 (z^p)^m\in G^p(x^m, y^m) , 则 此外, 因为 H_i^p 是连续的, 则 并且由引理2.3, 有 因而 z^p\in G^p(x, y) . 所以 G^p(x, y) 的图是闭的, 因此映射 G^p 是上半连续且紧值的. 下证 Q^p 是上半连续的. 由引理2.5, 只需证 Q^p 的图是闭的: \forall \{x^m\}\subset X$$ x^m\rightarrow \tilde{x}$, $\forall \{y^m\} \subset Q^p(x^m) $$y^m\rightarrow \tilde{y} , 需证 \tilde{y}\in Q^p(\tilde{x}) . 假设 \tilde{y}\notin Q^p(\tilde{x}) , 则 \exists w^*\in Y , 使得 又由 H(x, y)$$ X\times Y$上连续及$x^m\rightarrow \tilde{x}, y^m\rightarrow \tilde{y}$, 对上述$w^*\in Y$, 当$m$充分大时, 有

$\bar{x}_i^p>0\Rightarrow H_i^p(\bar{x}, \bar{y})\geqslant H_j^p(\bar{x}, \bar{y}), \forall i, j\in S^p, \forall p\in \Gamma.$

$\forall\epsilon\geqslant 0, \forall H\in \Omega$, 不确定性群体博弈$\{\Gamma, X, Y, H\} $$\epsilon - \mathrm{NS} 平衡集定义为 其描述了博弈 \{\Gamma, X, Y, H\} 的有限理性. 特别地, 当 \epsilon=0 时, 上式即为 \{\Gamma, X, Y, H\} 的弱NS平衡集 NS(H, 0)=NS(H) , 这刻画了该群体博弈 \{\Gamma, X, Y, H\} 的完全理性. 由定理3.2知, NS(H)\subseteq X\times Y 非空, 且称 NS:\Omega\rightarrow 2^{X\times Y} 为平衡集值映射. 定理4.1 (1) \forall H\in \Omega, \forall (x, y)\in X\times Y, \psi(H, x, y)\geqslant0; (2) \psi( H, x, y)=0 当且仅当 (x, y)\in NS(H). (1) \forall H\in \Omega , \forall (x, y)\in X\times Y , 取 u=x\in X, r=y\in Y , 有 (2) 若 \psi(H, x, y)=0 , 则 $$\max\limits_{u\in X}\left<u-x, H(x, y)\right>=0,$$ $$\max\limits_{p\in\Gamma} \max\limits_{r\in Y}\min\limits_{\|w^p\|=1, w^p\in R_{+}^{n^p}}\left<w^p, H^p(x, y)-H^p(x, r)\right>=0 .$$ (4.1) 式, \forall u\in X , 有 \left<u-x, H(x, y)\right>\leqslant 0 , 从而由定理3.1可知 (x, y) 满足定义3.1中的条件(1). 另一方面, 反证法, 若 \exists p_0\in \Gamma, r_0\in Y , 使得 \forall w^{p_0}\in W^{p_0}=\{w^{p_0}\in R_{+}^{n^{p_0}}:\|w^{p_0}\|=1\}, y\in Y , 有 W^{p_0} 是紧集, 则 从而 这与 (4.2) 式矛盾, 因此 \forall p\in \Gamma, r\in Y , 有 H^p(x, y)-H^p(x, r)\notin {\rm int}R_{+}^{n^p} , 从而 (x, y) 满足定义3.1的条件(2). 综上, (x, y)\in NS(H). 反过来, 若 (x, y)\in NS(H) , 由定理3.1和定义3.1, 分别有 $$\left<u-x, H(x, y)\right>\leqslant0, \;\;{ } \forall u\in X.$$ $$H^p(x, y)-H^p(x, r)\notin {\rm int}R_+^{n^p}, \;\;{ } \forall r\in Y.$$ (4.3) 式有 \forall p\in \Gamma, r\in Y , 设 L(r)=\{i\in S^p:H^p_i(x, y)-H^p_i(x, r)\leqslant 0\} , 由 (4.4) 式可知, L(r)\neq \emptyset . i'\in L(r), \hat{w}^p\in \{w^p\in R_+^{n^p}:\|w^p\|=1\} , 当 i=i' 时, \hat{w}_{i'}^p=1 , 当 i\neq i' 时, \hat{w}_{i'}^p=0 , 则 从而 因此 又因为 因此 \max\limits_{p\in \Gamma}\max\limits_{r\in Y}\min\limits_{\|w^p\|=1, w^p\in R_{+}^{n^p}}\left<w^p, H^p(x, y)-H^p(x, r)\right>= 0. 综上, \psi(H, x, y)=0 .定理4.1得证. 定理4.2 \psi:\Omega\times X\times Y\rightarrow R_+ 连续. 设 \forall (H^m, x^m, y^m)\in \Omega\times X\times Y , 并且 (H^m, x^m, y^m)\rightarrow (H, x, y)(m\rightarrow +\infty) . 需证: 当 m\rightarrow +\infty 时, \psi (H^m, x^m, y^m)\rightarrow \psi(H, x, y). \forall u\in X, r\in Y, \forall p\in \Gamma, w^p\in W^p=\{w^p\in R_+^{n^p}:\|w^p\|=1\} , 设 \begin{gathered}\varphi(u, x, y)=\langle u-x, H(x, y)\rangle, \\\phi^p\left(w^p, r, x, y\right)=\left\langle w^p, H^p(x, y)-H^p(x, r)\right\rangle .\end{gathered} (1) 首先证明 因为 X 是紧的, 则存在 K>0, \forall x\in X , 使得 \|x\|\leqslant K. H^{m}\rightarrow H (按度量 \rho )时, \forall \epsilon_1>0, \exists N_1\in N_+ , 使得当 \forall m\geqslant N_1 时, 有 \varphi^{m}(u, x, y)\rightarrow \varphi(u, x, y)$$ (m\rightarrow +\infty)$, 由引理2.3, 有

(2) 其次证明

$H^m\rightarrow H$ (按度量$\rho$), 故$\forall p\in \Gamma$, $\forall \epsilon_2>0, \exists N_2\in N_+,$使得当$m\geqslant N_2$时, 有$\|(H^m)^p(x, y)-H^p(x, y)\|<\frac{\epsilon_2}{2} $$\forall r\in Y, \|(H^m)^p(x, r)-H^p(x, r)\|<\frac{\epsilon_2}{2}. 因此 \forall p\in \Gamma, (\phi^m)^p(w^p, r, x, y)\rightarrow \phi^{p}(w^p, r, x, y) , 并且由 (4.5) 式中 \phi 的构造, 易知 (\phi^m)^p$$ \phi^p $$W^p\times Y\times X\times Y 上连续, 又因 Y$$ W^p=\{w^p\in R_+^{n^p}:\|w^p\|=1\}$都是紧集, $(x^m, y^m)\rightarrow (x, y)\in X\times Y$. 由引理2.4, 有

$\psi:\Omega\times X\times Y\rightarrow R_+$连续. 证毕.

因$(\Omega, \rho)$完备以及$X\times Y$是紧的, 且$f:\Omega\rightarrow 2^{X\times Y}$是上半连续的, 而且由定理$4.2 $$\psi:\Omega\times X\times Y\rightarrow R_+ 是连续的, 故引理2.1的全部条件都成立, 从而其结论(1)–(5)也成立. 注4.2 定理 4.3 表明在 \mathrm{Baire} 分类的意义下, 引入成本后的不确定性群体博弈 \{\Gamma, X,$$ Y, H\}$的有限理性模型$M$对大多数$H\in \Omega$都是结构稳定的, 对$\epsilon$-$\mathrm{NS}$均衡集也是鲁棒的. 从而, 在$\mathrm{Baire}$分类意义下, 有限理性下大多数不确定性群体博弈的弱$\mathrm{NS}$平衡是稳定的.

$$$\psi(H, x, y)=\left\{ \begin{array}{ll} -x_1^2-x_2^2-4x_1x_2+2x_1+x_2+1-y , & x_1\geqslant x_2; \\ -x_1^2-x_2^2-4x_1x_2+2x_2+x_1+1-y , &x_1< x_2. \end{array} \right.$$$

$\epsilon=\frac{2}{m}(m> 2),$易知$\{\Gamma, X, Y, H\} $$\epsilon - \mathrm{NS} 平衡集 \mathrm{NS} 平衡集 显然均满足引理2.1的所有条件, 所以引理2.1结论成立. 即当纯收益函数 H=(H_1, H_2) 发生扰动时, 其有限理性模型 M$$ H$是结构稳定的从而对$\epsilon$-$\mathrm{NS}$平衡也是鲁棒的. 进而, 有限理性下该不确定性群体博弈$\{\Gamma, X, Y, H\}$的弱$\mathrm{NS}$平衡是稳定的.

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Nash J .

Equilibrium points in n-person games

P Natl Acad Sci, 1950, 36 (1): 48- 49

Sandholm W H . Population Games and Evolutionary Dynamics. London: MIT Press, 2011

Lahkar R .

Large population aggregative potential games

Dyn Games Appl, 2017, 7 (3): 443- 467

Cheung M W , Lahkar R .

Nonatomic potential games: the continuous strategy case

Games Econ Behav, 2018, 108, 341- 362

Chow S N , Li W , Lu J , et al.

Population games and discrete optimal transport

J Nonlinear Sci, 2019, 29, 871- 896

Yang G H , Yang H .

Stability of Nash equilibria of population games under bounded rationality

Journal of Guizhou University (Natural Sciences), 2019, 36 (5): 1- 3 1-3, 17

Yang G H , Yang H , Song Q Q .

Stability of weighted Nash equilibria for multiobjective population games

J Nonlinear Sci Appl, 2016, 9, 4167- 4176

Yang G H , Yang H .

Stability of weakly Pareto-Nash equilibria and Pareto-Nash equilibria for multiobjective population games

Set-Valued Var Anal, 2017, 25, 427- 439

Yang Z , Zhang H .

Essential stability of cooperative equilibria for population games

Optim Lett, 2019, 13 (7): 1573- 1582

Yang Z , Zhang H .

NTU core, TU core and strong equilibria of coalitional population games with infinitely many pure strategies

Theory Decis, 2019, 87, 155- 170

Chen H X , Jia W S .

Approximation theorem and general convergence of population games

Acta Math Sci, 2021, 41A (5): 1566- 1573

Zhukovskii V I. Linear Quadratic Differential Games. Naoukova Doumka: Kiev, 1994

Larbani M , Lebbah H .

A concept of equilibrium for a game under uncertainty

Eur J Oper Res, 1999, 117 (1): 145- 156

Zhang H J , Zhang Q .

Existence of strong Nash equilibrium for non-cooperative games under uncertainty

Control and Decision, 2010, 25 (8): 1251- 1254 1251-1254, 1260

Zhang H J , Zhang Q .

Existence of simple Berge equilibrium for non-cooperative games under uncertainty

Systems Engineering Theory & Practice, 2010, 30 (9): 1630- 1635

Gao J , Wu D H , Zhang G .

Generic stability of equilibrium for n-person non-cooperative games under uncertainty

Communications on Applied Mathematics and Computation, 2014, 28 (3): 336- 342

Deng X C , Xiang S W , Zuo Y .

Existence of strong Berge equilibrium under uncertainty

OR Transactions, 2013, 17 (3): 101- 107

Yang Z .

The existence theorems of Berge-NS equilibria in non-cooperative games under generized uncertainty

J Sys Sci & Math Scis, 2015, 35 (9): 1073- 1080

Lu C C , Wu D H .

Bounded rationality and well-posedness in non-cooperative games under uncertainty

Communications on Applied Mathematics and Computation, 2016, 30 (3): 339- 348

Yang Z , Pu Y J , Guo X Y .

On the existence of weakly Pareto-NS equilibrium points in multi-objective games under uncertainty

Systems Engineering Theory&Practice, 2013, 33 (3): 660- 665

Zhao W , Yang H , Wu J Y .

Existence and generic stability of equilibria for population games with uncertain parameters

Acta Math Appl Sin, 2020, 43 (4): 627- 638

Zhao W, Yang H, Deng X C, et al. Stability of equilibria for population games with uncertain parameters under bounded rationality. J Inequal Appl, 2021, 2021(1): Article number 15

Anderlini L , Canning D .

Structural stability implies robustness to bounded rationality

J Econ Theory, 2001, 101 (2): 395- 422

Yu C , Yu J .

On structural stability and robustness to bounded rationality

Nonlinear Anal-TMA, 2006, 65 (3): 583- 592

Yu J , Yang H , Yu C .

Structural stability and robustness to bounded rationality for non-compactcases

J Glob Optim, 2009, 44, 149- 157

Yu C , Yu J .

Bounded rationality in multiobjective games

Nonlinear Anal-TMA, 2007, 67 (3): 930- 937

Wang H L , Yu J .

Bounded rationality and stability of weakly efficient solution set of multiobjective optimization problems

Chinese Journal of Management Science, 2008, 16 (4): 155- 158

Wang H L , Yu J .

Bounded rationality and stability of solution of multiobjective optimization problems

OR Transactions, 2008, 12 (1): 104- 108

Yu J .

Bounded rationality and stability of solution of some equilibrium problems

J Sys Sci & Math Scis, 2009, 29 (7): 999- 1008

Yu J . Game Theory and Nonlinear Analysis. Beijing: Science Press, 2008

Yu J . Bounded Rationality and Stability of Equilibrium Set in Game Theory. Beijing: Science Press, 2017

Yu J , Jia W S .

Game model in the study of bounded rationality

Scientia Sinica Mathematica, 2020, 50 (9): 1375- 1386

Wang N F .

The stability of equilibrium point for uncertain game under bounded rationality

Acta Math Appl Sin, 2017, 40 (4): 562- 572

n人非合作博弈弱Nash均衡点的存在性

Cai Y Y , Xiang S W .

The existence of weak Nash equilibria in n-person non-cooperative game

J Chongqing Technol & Business Univ (Nat Sci Ed), 2020, 37 (1): 54- 58

Klein E, Thompson A C. Theory of Correspondences. New York: Wiley, 1984

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