数学物理学报, 2025, 45(4): 1217-1228

集值群体博弈 Nash 均衡与合作均衡的广义适定性

葛晨1, 陈卓正2, 张宇,1,*

1云南财经大学统计与数学学院 昆明 650221

2中国石油大学 (华东) 理学院 山东青岛 266580

Generalized Well-Posedness of Nash Equilibrium and Cooperative Equilibrium for Population Games with Set Payoffs

Ge Chen1, Chen Zhuozheng2, Zhang Yu,1,*

1Yunnan University of Finance and Economics, School of Statistics and Mathematics, Kunming 650221

2China University of Petroleum, College of Science, Shandong Qingdao 266580

通讯作者: *E-mail: zhangyu198606@sina.com

收稿日期: 2024-10-29   修回日期: 2025-06-6  

基金资助: 国家自然科学基金(11901511)
云南省基础研究基金(202501AT070447)
云南省兴滇英才青年人才专项基金

Received: 2024-10-29   Revised: 2025-06-6  

Fund supported: NSFC(11901511)
Yunnan Fundamental Research Projects(202501AT070447)
Xingdian Talents Project Foundation for Young Scholar in Yunnan Province

摘要

借助一类非线性标量函数, 分别构建集值群体博弈 Nash 均衡与合作均衡的有限理性函数, 研究有限理性函数的下半连续性质, 并建立其零水平集与博弈均衡集之间的等价关系. 利用有限理性函数的所得性质, 得到集值群体博弈 Nash 均衡与合作均衡的广义适定性.

关键词： Nash 均衡; 合作均衡; 集支付; 群体博弈; 广义适定性

Abstract

By using a class of nonlinear scalarization functions, we construct the bounded rationality functions of Nash equilibrium and cooperation equilibrium of population games with set payoffs, respectively. Then, we show the lower semicontinuity of the bounded rationality functions and the equivalence relations between their level sets of zero and game equilibrium sets. Finally, employing the acquired properties of the bounded rationality functions, we prove the generalized well-posedness of Nash equilibrium and cooperative equilibrium for set payoffs population games.

Keywords： Nash equilibrium; cooperative equilibrium; set payoff; population games; generalized well-posedness

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本文引用格式

葛晨, 陈卓正, 张宇. 集值群体博弈 Nash 均衡与合作均衡的广义适定性[J]. 数学物理学报, 2025, 45(4): 1217-1228

Ge Chen, Chen Zhuozheng, Zhang Yu. Generalized Well-Posedness of Nash Equilibrium and Cooperative Equilibrium for Population Games with Set Payoffs[J]. Acta Mathematica Scientia, 2025, 45(4): 1217-1228

1 引言

Nash 均衡与合作均衡是博弈论中的两个重要概念, 常应用于解释博弈者之间的竞争与合作行为.群体博弈是一类具有特殊结构的系统模型, 其最早可追溯到 Nash[1] 关于混合策略形式平衡的群体行动 (mass action) 解释. 与传统的正规博弈不同, 群体博弈强调在参与者数量充分大且有限前提下, "有多少人选择了策略". 因其特殊的机制, 它可被广泛应用于解释市场交易、网络拥堵、生物等众多领域中拥有大量个体策略交互的现实问题中, 有着重要的理论与实际意义.

Sandholm[2] 系统介绍了群体博弈 Nash 均衡的概念, 并在支付函数连续假定下证明了均衡是存在的. 借助 "自我挫败" 思想, Hofbauer 和 Sandholm[3] 引入了稳定群体博弈的概念, 并得到了其 Nash 均衡集的全局渐近稳定性. 在稳定的假定下, Jia 等人[4]获得了剩余稠密集上群体博弈 Nash 均衡的唯一性结果. 利用 Pareto 向量序, Yang 和 Yang[5] 引入了多目标群体博弈弱 Nash 均衡和 Nash 均衡的概念, 并利用 Ky Fan 不等式获得了弱 Nash 均衡存在的充分条件, 同时分别讨论了弱 Nash 均衡与 Nash 均衡的稳定性和广义稳定性. 赵薇等人[6]在支付函数连续和凸性假定下, 研究了不确定参数下群体博弈 Nash 均衡的存在性与稳定性. 更多关于群体博弈 Nash 均衡文献可参见文献[7-10]. 另一方面, Yang 和 Zhang[11] 首次在群体博弈中引入联盟社会状态集与合作均衡的概念, 并借助广义 Scarf 定理研究了均衡存在的充分条件. 紧接着, 在文献[12]中, Yang 和 Zhang 在任意群体可选策略是不可数多假定下, 定义了联盟群体博弈 NTU 核、TU 核以及强均衡的概念, 同时得到这些均衡的存在性定理. 张弦[13]在多主从群体博弈中引入合作均衡概念, 利用广义 Scarf 定理与跟随者均衡的参数稳定性结果, 得到了均衡的存在性.武文俊等人[14]在群体跟随者间有 "合作共赢" 行为的现实背景下, 利用 Fort 定理, 验证了在支付函数扰动的情况下, 多数主从群体博弈的合作均衡具有通有稳定性.

在传统经济学中, 通常强调博弈者对策略选择的完全理性, 但这种假设显然是过于严格的. 因此, Simon[15] 提出了有限理性理论以消除完全理性在实际应用中的局限性; Anderlini 和 Canning[16] 利用抽象理性函数构建起有限理性模型, 获得其结构上的稳定性和鲁棒性. 但后续研究发现, Anderlini-Canning 模型中过于严苛的条件假设并不能使其在众多博弈模型中得到广泛应用. 于是, Yu 和 Yu[17] 对其中的条件做出弱化, 并将弱化的限理性模型应用于多目标博弈问题, 最优化问题和变分不等式问题, 得到了相应的结果, 详见文献[18-20]. 通过有限理性模型的研究,俞建给出了研究博弈均衡适定性的统一范式[21].近年来, 很多学者将 "有限理性" 思想引入到群体博弈中, 得到了一批重要成果. 例如, Jia 等人[4]通过构建有限理性函数, 研究了群体博弈 Nash 均衡的广义适定性, 同时在稳定的假定下, 研究了群体博弈 Nash 均衡的唯一性和适定性.张海群[22]在有限理性下研究了群体博弈合作均衡的稳定性, 并讨论了群体博弈强均衡的稳定性. 针对多目标群体博弈弱合作均衡问题, Chen 等人[23]利用一类非线性标量函数及其性质构建了有限理性函数,并获得均衡的唯一性与适定性结果. 更多关于有限理性下群体博弈均衡稳定性的研究参见文献[24-26].

在经典博弈问题的研究中, 所涉及的支付函数通常是单值的, 即每个博弈者的策略组合对应一个确定的收益, 但受制于不确定因素与精确计算的局限性, 博弈问题的支付值往往呈现多值性. 在集支付框架下, 陈桃等人[27]通过向量序结构引入集值群体博弈 Nash 均衡与合作均衡概念, 并在支付函数上半连续且紧值的充分条件下, 分别利用 Ky Fan 截口定理和广义 Scarf 定理证明了两类均衡的存在性. 然而, 不难发现关于集值群体博弈均衡的适定性问题尚未被讨论. 受上述文献启发,本文将借助非线性标量函数工具构建集值群体博弈 Nash 均衡与合作均衡的有限理性函数, 讨论有限理性函数下半连续性质以及其零水平集与均衡解集之间的等价关系,并最终获得两类均衡的广义适定性.本文其余部分构建如下: 第 2 节给出相关均衡模型及其有用的概念、引理和性质; 第 3 节讨论集值群体博弈 Nash 均衡的广义适定性; 第 4 节讨论集值群体博弈合作均衡的广义适定性.

2 预备知识

设 $A=\{1,2,\cdots,K\}$ 为群体集, $m_{p}$ 为第 $p\in A$ 个群体数量的连续统, 其充分大但有限, $F_{p}=\{1,2,\cdots,n_{p}\}$ 为第 $p$ 个群体的纯策略集, $x^{i}_{p}$ 表示第 $p$ 个群体选择第 $i\in F_{p}$ 个策略的数量. 令 $ X_{p}$ 为第 $p$ 个群体的可行社会状态集

$\begin{equation*} X_{p}=\Bigg\{x_{p}=(x^{1}_{p},x^{2}_{p},\cdots,x^{n_{p}}_{p})|\forall i\in F_{p},x^{i}_{p}\geq0, \sum_{i=1}^{n_{p}}x^{i}_{p}=m_{p}\Bigg\},X=\prod_{p\in A}X_{p}. \end{equation*}$

令 $C\subseteq A$ 为任意联盟, 规定

$\begin{equation*} \widehat{X_{C}}=\Bigg\{y_{C}=(y^{1}_{p},y^{2}_{p},\cdots,y^{n_{p}}_{p})_{p\in C}|\forall i\in F_{p},y^{i}_{p}\geq0,\sum_{p\in C}\sum^{n_{p}}_{i=1}y^{i}_{p}=\sum_{p\in C}m_{p}\Bigg\} \end{equation*}$

为联盟 $C$ 的可行社会状态集.

下面, 介绍集值群体博弈 Nash 均衡与合作均衡概念.

定义2.1[27] 对 $\forall p\in A$, $i\in F_{p}$, 令 $\Phi^{i}_{p}$ 为定义在 $X$ 上的集值支付函数且 $\forall x\in X, \Phi^{i}_{p}(x)$ 为 $R^{m}$ 中的非空子集. 若对 $\forall p\in A$, $\exists(z^{i}_{p})^{*}\in\Phi^{i}_{p}(x^{*})$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}y^{i}_{p}(z^{i}_{p})^{*}\notin\sum^{n_{p}}_{i=1}(x^{i}_{p})^{*}(z^{i}_{p})^{*}+{\rm int}R^{m}_{+},\forall y_{p}=(y^{1}_{p},y^{2}_{p},\cdots,y^{n_{p}}_{p})\in X_{p}, \end{equation*}$

则称 $x^{*}=(x^{*}_{1},x^{*}_{2},\cdots,x^{*}_{K})\in X$ 是集值群体博弈 $\Phi$ 的 Nash 均衡点.

注2.1 若 $\Phi^{i}_{p}$ 退化为向量值支付, 则定义 2.1 中的概念变成文献[5]中多目标群体博弈的弱 Pareto-Nash 均衡; 若 $\Phi^{i}_{p}$ 退化为实值函数, 此时定义 2.1 中的 Nash 均衡变成参考文献[2]中群体博弈的 Nash 均衡.

定义2.2[27] 对 $\forall p\in A,i\in F_{p}$, 令 $\Phi^{i}_{p}$ 为定义在 $\widehat{X_{A}}$ 上的集值支付函数且 $\forall x\in \widehat{X_{A}}, \Phi^{i}_{p}(x)$为 $R^{m}$ 中非空子集. 若对于任意 $C\subseteq A$, 不存在 $x_{C}\in\widehat{X_{C}}$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}\Phi^{i}_{p}(x^{*})(y^{i}_{p}-(x^{i}_{p})^{*})\subset {\rm int}R^{m}_{+},\forall p\in C, \end{equation*}$

则称 $x^{*}=(x^{*}_{1},x^{*}_{2},\cdots,x^{*}_{K})\in\widehat{X_{A}}$ 是集值群体博弈 $\Phi$ 的合作均衡点. 其中 $y^{i}_{p}$ 是 $x_{C}$ 的分量, $(x^{i}_{p})^{*}$ 是 $x^{*}$ 在联盟 $C$ 中的分量.

注2.2 若 $\Phi^{i}_{p}$ 退化为实值函数, 则定义 2.2 中的合作均衡变成参考文献[11]中单目标群体博弈合作均衡的概念.

注2.3 $X$ 表示 $K$ 个不同群体可行社会状态集的笛卡尔积, 而 $\widehat{X_{A}}$ 表示群体联盟 $A$ 的可行社会状态集. 一般情况下 ($K\geq2$), $X\neq \widehat{X_{A}}$. 因此, 集值群体博弈 Nash 均衡与合作均衡是不同的.

定义2.3[28] 令集值映射 $\Phi:X\rightarrow2^{R^{m}}$ 是非空紧值的. 若对任意 $\{x_{n}\}\subset X,x_{n}\rightarrow x_{0}$, 任取 $y_{n}\in\Phi(x_{n})$, 存在 $y_{0}\in\Phi(x_{0})$ 和 $\{y_{n}\}$ 的子序列 $\{y_{n_{k}}\}$, 使得 $y_{n_{k}}\rightarrow y_{0},k\rightarrow\infty$, 则称集值映射 $\Phi$ 在 $x_{0}\in X$ 处是上半连续的 (u.s.c.).

注 2.4 显然, 若 $\Phi$ 退化为实值函数, 定义 2.3 中概念变为连续的定义.

引理2.1[29] 令 $V$ 是线性度量空间, $V_{0}\subset V$. 如果 $(V_{0},\rho')$ 是一个紧度量空间, 则 $(C(V_{0}),D)$ 是完备度量空间, 其中 $C(V_{0})$ 是 $V$ 中所有紧子集构成的集族, $D$ 是关于 $\rho'$ 的 Hausdorff 度量.

下面, 介绍一类非线性标量化函数及其性质.

令 $e\in {\rm int}R^{m}_{+}$, 定义非线性标量函数 $h_{e}:R^{m}\rightarrow R$ 如下

$h_{e}(z)=\max\{\lambda\in R:z\in\lambda e+R^{m}_{+}\}.$

由文献[30,31]得 $h_{e}$ 的以下性质

(1) $h_{e}(z)>\lambda\Leftrightarrow z\in\lambda e+{\rm int}R^{m}_{+}(i.e. h_{e}(z)\leq\lambda\Leftrightarrow z\notin\lambda e+{\rm int}R^{m}_{+})$;

(2) $h_{e}(\cdot)$ 是连续的.

令$M=\{\Phi:\Phi=(\Phi_{1},\Phi_{2},\cdots,\Phi_{K}),\Phi_{p}=(\Phi^{1}_{p},\Phi^{2}_{p},\cdots,\Phi^{n_{p}}_{p}),\Phi^{i}_{p} \mbox{是 u.s.c. 且紧值的},\forall p\in A,i\in F_{p}\}.$

对于 $\forall \Phi,\Phi'\in M$, 在 $M$ 上定义距离 $\rho:M\times M\rightarrow R$,

$\rho(\Phi,\Phi')=\sum_{p\in A}\sup_{x\in X}H_{p}(\Phi_{p}(x),\Phi'_{p}(x)),$

其中 $H_{p}$ 是关于 $\rho'$ 的 Hausdorff 距离.

引理2.2 $(M,\rho)$ 是完备度量空间.

证 设 $\Phi^{n}=(\Phi^{n}_{1},\Phi^{n}_{2},\cdots,\Phi^{n}_{K})$ 是 $M$ 中的任意一个基本列, 即对于 $\forall\varepsilon>0$, $\exists N\in N_{+}$, 使得$\rho(\Phi^{n},\Phi^{m})<\varepsilon, \forall m,n\geq N.$由假设及引理 2.1, 一定存在非空紧值映射 $\{\Phi^{0}_{p}\}_{p\in A}$, 使得对任意 $p\in A$ 有

$H_{p}(\Phi^{n}_{p},\Phi^{0}_{p})\rightarrow0,n\rightarrow\infty.$

下面证明 $\Phi^{0}=\{\Phi^{0}_{1},\Phi^{0}_{2},\cdots,\Phi^{0}_{K}\}\in M, \forall p\in A$. 根据 $M$ 定义, 仅需证明 $\Phi^{0}_{p}$ 是上半连续即可.

令 $x_{m}\rightarrow x_{0}$, 由 (2.2) 式得, 对于任意 $p\in A$ 和任意 $z^{0}_{m,p}\in\Phi^{0}_{p}(x_{m})$, 一定存在 $z^{n}_{m,p}\in\Phi^{n}_{p}(x_{m})$, 使得 $z^{n}_{m,p}\rightarrow z^{0}_{m,p}$, $n\rightarrow\infty$. 又因为 $\Phi^{n}_{p}(x_{m})$ 是上半连续且紧值的, 则存在 $z^{n}_{0,p}\in\Phi^{n}_{p}(x_{0})$ 和 $\{z^{n}_{m,p}\}$ 关于 $m$ 的子序列 $\{z^{n}_{m_{k},p}\}$, 使得 $z^{n}_{m_{k},p}\rightarrow z^{n}_{0,p},k\rightarrow\infty$. 由文献[32,引理 2.1.4], 可有

$\begin{equation*} \rho'(z^{0}_{m_{k},p},\Phi^{0}_{p}(x_{0}))\leq\rho'(z^{0}_{m_{k},p},z^{n}_{m_{k},p})+\rho'(z^{n}_{m_{k},p},\Phi^{n}_{p}(x_{0}))+H_{p}(\Phi^{n}_{p}(x_{0}),\Phi^{0}_{p}(x_{0}))\rightarrow0(k,n\rightarrow\infty). \end{equation*}$

这样, 存在 $z^{0}_{0,p}\in\Phi^{0}_{p}(x_{0})$, 使得 $z^{0}_{m_{k},p}\rightarrow z^{0}_{0,p}$, $k\rightarrow\infty$. 故集值映射 $\{\Phi^{0}_{p}\}_{p\in A}$ 的上半连续性得证.

接下来, 对于 $\forall \Phi\in M,\varepsilon>0$ 且 $e\in {\rm int}R^{m}_{+}$, 令$NE(\Phi,\varepsilon)$ 和 $CE(\Phi,\varepsilon)$ 分别为集值群体博弈 $\Phi$ 的逼近 Nash 均衡解集与逼近合作均衡解集

$\begin{align*} NE(\Phi,\varepsilon)&=\{x\in X:\forall p\in A,\exists z^{i}_{p}\in\Phi^{i}_{p}(x),{\rm s.t}.\sum^{n_{p}}_{i=1}y^{i}_{p}z^{i}_{p}\notin\sum^{n_{p}}_{i=1}x^{i}_{p}z^{i}_{p}+\varepsilon e+{\rm int}R^{m}_{+},\forall y_{p}\in X_{p}\}; \\ CE(\Phi,\varepsilon)&=\{x\in \widehat{X_{A}}:\forall C\subseteq A,\forall x_{C}\in\widehat{X_{C}},\exists p\in C,\exists z^{i}_{p}\in\Phi^{i}_{p}(x),\\ & {\rm s.t}.\sum^{n_{p}}_{i=1}y^{i}_{p}z^{i}_{p}\notin\sum^{n_{p}}_{i=1}x^{i}_{p}z^{i}_{p}+\varepsilon e+{\rm int}R^{m}_{+}\}. \end{align*}$

注2.5 特别地, $NE(\Phi)$, $CE(\Phi)$ 分别表示集值群体博弈 Nash 均衡解集与合作均衡解集; 由文献[27] 得, $NE(\Phi)\neq\emptyset$, $CE(\Phi)\neq\emptyset$, $\forall \Phi\in M$.

定义2.4 (1) 若 $\forall\Phi_{n}\in M,\Phi_{n}\rightarrow\Phi,\forall x_{n}\in NE(\Phi_{n},\varepsilon_{n}),\forall\varepsilon_{n}>0,\varepsilon_{n}\rightarrow0$, 存在$\{x_{n}\}$的子序列$\{x_{n_{k}}\}$, 使得 $ x_{n_{k}}\rightarrow x\in NE(\Phi)$, 则称 $\Phi$ 是广义适定的;

(2) 若 $\forall\Phi_{n}\in M,\Phi_{n}\rightarrow\Phi,\forall x_{n}\in NE(\Phi,\varepsilon_{n}),\forall\varepsilon_{n}>0,\varepsilon_{n}\rightarrow0$, 存在$\{x_{n}\}$ 的子序列 $\{x_{n_{k}}\}$, 使得 $x_{n_{k}}\rightarrow x\in NE(\Phi)$, 则称$\Phi$ 是广义 Tikhonov 适定的;

(3) 若 $\forall\Phi_{n}\in M,\Phi_{n}\rightarrow\Phi,\forall x_{n}\in NE(\Phi_{n}),\forall\varepsilon_{n}>0,\varepsilon_{n}\rightarrow0$, 存在 $\{x_{n}\}$ 的子序列 $\{x_{n_{k}}\}$, 使得 $x_{n_{k}}\rightarrow x\in NE(\Phi)$, 则称 $\Phi$ 是广义 Hadamard 适定的.

注2.6 类似的, 关于 $CE(\Phi,\varepsilon)$ 的集值群体博弈合作均衡也有上述广义适定性. 在文献[21] 中, 若 $\Phi$ 是广义适定的, 则它一定是广义 Tikhonov 适定和广义 Hadamard 适定.

3 Nash 均衡广义适定性

首先构建集值群体博弈 Nash 均衡的有限理性函数 $\Omega:M\times X\rightarrow R$,

$\begin{equation} \Omega(\Phi,x)=\max_{p\in A}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}\max_{y_{p}\in X_{p}}h_{e}[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})], \Phi\in M, x\in X, \end{equation}$

其中 $y^{i}_{p}$ 为 $y_{p}$ 的分量, $x^{i}_{p}$ 为 $x$ 的分量.

现在, 讨论 $\Omega$ 的零水平集与集值群体博弈 Nash 均衡解集 $NE(\Phi)$ 之间的等价关系.

引理3.1 $\forall\Phi\in M,\forall x\in X,\Omega(\Phi,x)\geq0$; 特别地, $\Omega(\Phi,x)=0\Leftrightarrow x\in NE(\Phi)$.

证 首先, 证明 $\forall\Phi\in M$, $\forall x\in X$, $\Omega(\Phi,x)\geq0$. 反证, 假设$\exists\overline{\Phi}\in M$, $\exists \overline{x}\in X$, 使得$\Omega(\overline{\Phi},\overline{x})<0$. 根据 $ \Omega$ 的定义, $\forall p\in A,\exists z^{i}_{p}\in\overline{\Phi}^{i}_{p}(\overline{x})$, 有

$\begin{equation} h_{e}[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-\overline{x}^{i}_{p})]<0, \forall y_{p}\in X_{p}. \end{equation}$

由 $y_{p}=(y^{1}_{p},y^{2}_{p},\cdots,y^{n_{p}}_{p})$ 在 $X_{p}$ 中选取的任意性, 不妨令 $y^{i}_{p}=\overline{x}^{i}_{p}$, 则

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-\overline{x}^{i}_{p})\Bigg]=0. \end{equation*}$

这与 (3.2) 式矛盾.

下面, 证明 $\Omega(\Phi,x)=0\Leftrightarrow x\in NE(\Phi)$.

假设 $\Omega(\Phi,x)=0$, 但 $ x\notin NE(\Phi)$. 根据 $NE(\Phi)$ 的定义, 即 $\exists p\in A,\forall z^{i}_{p}\in\Phi^{i}_{p}(x),\exists y_{p}\in X_{p}$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\in {\rm int}R^{m}_{+}. \end{equation*}$

由 $h_{e}$ 的性质,

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]>0. \end{equation*}$

因为 $h_{e}$ 连续且 $X_{p}$ 是紧的, 对上述 $y_{p}$ 有

$\begin{aligned} \max_{y_{p}\in X_{p}}h_{e}[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})]&\geq h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]>0. \end{aligned}$

再由 $\Phi^{i}_{p}$ 在 $X$ 上的紧值性, 对于 $\forall z^{i}_{p}\in\Phi^{i}_{p}(x)$, 有

$\begin{equation*} \min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}\max_{y_{p}\in X_{p}}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]>0. \end{equation*}$

进一步, 对上述 $p\in A$ 有

$\begin{aligned} \Omega(\Phi,x)&\geq\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}\max_{y_{p}\in X_{p}}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]>0. \end{aligned}$

这与 $\Omega(\Phi,x)=0$ 矛盾. 因此, $x\in NE(\Phi)$.

反之, 下证 $x\in NE(\Phi)$ 时, $\Omega(\Phi,x)=0$. 由 $NE(\Phi)$ 定义, 所以 $\forall p\in A,\exists z^{i}_{p}\in\Phi^{i}_{p}(x)$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\notin {\rm int}R^{m}_{+}, \forall y_{p}\in X_{p}.\end{equation*}$

通过 $h_{e}$ 的性质, 有

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]\leq0.\end{equation*}$

进一步,

$\begin{equation*} \max_{p\in A}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}\max_{y_{p}\in X_{p}}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]\leq0. \end{equation*}$

可见, $\Omega(\Phi,x)\leq0$. 又因为 $\Omega(\Phi,x)\geq0$, 故可得 $\Omega(\Phi,x)=0$ 成立.

下面, 证明 $\Omega$ 关于 $(\Phi,x)$ 的下半连续性.

引理3.2 $\forall\Phi\in M$, $\forall x\in X$, $\Omega(\Phi,x)$ 是下半连续的.

证 $\forall\Phi\in M$, $\forall x\in X$, $\forall p\in A$, 定义函数如下

$\begin{equation} \mu(\Phi,x)=\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}\max_{y_{p}\in X_{p}}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]. \end{equation}$

首先, 证明对 $\forall t\in R$, $\mu$ 的下水平集

$\begin{aligned} B:=&\{(\Phi,x)|\mu(\Phi,x)\leq t\} =\{(\Phi,x)|\exists z^{i}_{p}\in\Phi^{i}_{p}(x), \mbox{使得} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]\leq t,\forall y_{p}\in X_{p}\} \end{aligned}$

是 $M\times X$ 中的闭集. 任取 $(\Phi_{n},x_{n})\in B$ 且$(\Phi_{n},x_{n})\rightarrow(\Phi_{0},x_{0})$, 由假设存在 $(z^{i}_{p})^{n}_{n}\in(\Phi^{i}_{p})_{n}(x_{n})$, 使得

$\begin{equation} h_{e}[\sum^{n_{p}}_{i=1}(z^{i}_{p})^{n}_{n}(y^{i}_{p}-(x^{i}_{p})_{n})]\leq t,\forall y_{p}\in X_{p}. \end{equation}$

因为 $(\Phi^{i}_{p})_{n}$ 是上半连续且紧值的, 则对上述 $(z^{i}_{p})^{n}_{n}\in(\Phi^{i}_{p})_{n}(x_{n})$, $\exists (z^{i}_{p})^{n}_{0}\in(\Phi^{i}_{p})_{n}(x_{0})$, 有 $(z^{i}_{p})^{n}_{n}\rightarrow (z^{i}_{p})^{n}_{0}$. 因此,

$\begin{aligned} d((z^{i}_{p})^{n}_{n},(\Phi^{i}_{p})_{0}(x_{0}))&\leq d((z^{i}_{p})^{n}_{n},(z^{i}_{p})^{n}_{0})+d((z^{i}_{p})^{n}_{0},(\Phi^{i}_{p})_{n}(x_{0}))+H_{p}((\Phi^{i}_{p})_{n}(x_{0}),(\Phi^{i}_{p})_{0}(x_{0}))\\ &=d((z^{i}_{p})^{n}_{n},(z^{i}_{p})^{n}_{0})+H_{p}((\Phi^{i}_{p})_{n}(x_{0}),(\Phi^{i}_{p})_{0}(x_{0}))\rightarrow0, n\rightarrow\infty. \end{aligned}$

这意味着存在 $(z^{i}_{p})^{0}_{0}\in(\Phi^{i}_{p})_{0}(x_{0})$, 使得$(z^{i}_{p})^{n}_{n}\rightarrow (z^{i}_{p})^{0}_{0}$. 根据 $h_{e}$ 的连续性及(3.4) 式, 令 $n\rightarrow\infty$ 有

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}(z^{i}_{p})^{0}_{0}(y^{i}_{p}-(x^{i}_{p})_{0})\Bigg]\leq t,\forall y_{p}\in X_{p}. \end{equation*}$

这样, $(\Phi_{0},x_{0})\in B$. 即 $B$ 必是 $M\times X$ 中的闭集. 对于 $\forall t\in R$, $\Omega$ 的下水平集

$\begin{aligned} \{(\Phi,x)|\Omega(\Phi,x)\leq t\}=&\{(\Phi,x)|\forall p\in A,\mu(\Phi,x)\leq t\} =\bigcap_{p\in A}\{(\Phi,x)| \mu(\Phi,x)\leq t\}. \end{aligned}$

由闭集的性质, 容易证明为 $M\times X$ 中的闭集. 故 $\Omega(\Phi,x)$ 是下半连续的.

定理3.1 $\forall \Phi\in M$, $\Phi$ 是广义适定的.

证 令 $\{\Phi_{n}\}\subseteq M,\Phi_{n}\rightarrow\Phi$. $\varepsilon_{n}>0,\varepsilon_{n}\rightarrow0$. 任取 $x_{n}\in NE(\Phi_{n},\varepsilon_{n})$, 通过 $NE(\Phi,\varepsilon)$ 的定义, $\forall p\in A,\exists(z^{i}_{p})^{n}_{n}\in(\Phi^{i}_{p})_{n}(x_{n})$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}(z^{i}_{p})^{n}_{n}(y^{i}_{p}-(x^{i}_{p})_{n})\notin\varepsilon_{n}e+{\rm int}R^{m}_{+},\forall y_{p}\in X_{p}. \end{equation*}$

由 $h_{e}$ 性质得

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}(z^{i}_{p})^{n}_{n}(y^{i}_{p}-(x^{i}_{p})_{n})\Bigg]\leq\varepsilon_{n}, \end{equation*}$

即 $\Omega(\Phi_{n},x_{n})\leq\varepsilon_{n}$. 因为 $\{x_{n}\}\subseteq X$ 且 $X$ 紧性, 则存在 $\{x_{n}\}$ 的子序列 $\{x_{n_{k}}\}$, 使得 $x_{n_{k}}\rightarrow x\in X$. 由引理 3.2, 有

$\begin{equation*} \Omega(\Phi,x)\leq\liminf_{k\rightarrow\infty}\Omega(\Phi_{n_{k}},x_{n_{k}})\leq\liminf_{k\rightarrow\infty}\varepsilon_{k}=0. \end{equation*}$

又记得 $\Omega(\Phi,x)\geq0$, 则有 $\Omega(\Phi,x)=0$, 再由引理 3.1 的等价关系得 $x\in NE(\Phi)$. 故 $\Phi$ 的广义适定性得证.

下面例子说明集支付紧值上半连续假设在定理 3.1 中是必要的.

例 3.1 令 $A=\{1,2\}$, $F_{1}=F_{2}=\{1,2\}$, $m_{1}=m_{2}=1$, $X_{1}=\{x_{1}=(x^{1}_{1},x^{2}_{1})\in R^{2}_{+}|x^{1}_{1}+x^{2}_{1}=1\}$, $X_{2}=\{x_{2}=(x^{1}_{2},x^{2}_{2})\in R^{2}_{+}|x^{1}_{2}+x^{2}_{2}=1\}$, $X=X_{1}\times X_{2}$. 对 $\forall p\in A,i\in F_{p}$, 定义集支付函数为$\Phi^{i}_{p}:X\rightarrow2^{R}$ 如下:$\Phi^{1}_{1}(x)=\Phi^{1}_{2}(x)\equiv\{\frac{1}{2}\};$

$\begin{equation*} \Phi^{2}_{1}(x)=\Phi^{2}_{2}(x)=\left\{ \begin{array}{cl} \{0\},&x'=(0,1,0,1),\\ (0,\frac{1}{2}),&x"=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}),\\ \lbrack \frac{1}{2},1\rbrack,&x\in X\setminus\{x',x"\}.\\ \end{array}\right. \end{equation*}$

显然 $\Phi^{2}_{1}, \Phi^{2}_{2}$ 在 $x"$ 不满足紧值上半连续. 选取$\varepsilon_{n}=\frac{2}{n}$ 和 $(\Phi^{i}_{p})_{n}:X\rightarrow2^{R}$ 如下

$\begin{equation*} (\Phi^{1}_{1})_{n}(x)=(\Phi^{1}_{2})_{n}(x)\equiv\bigg\{\bigg(\frac{1}{2}-\frac{1}{n}\bigg)\bigg\}; \end{equation*}$
$\begin{equation*} (\Phi^{2}_{1})_{n}(x)=(\Phi^{2}_{2})_{n}(x)=\left\{ \begin{array}{cl} \{0\},&x'=(0,1,0,1),\\ (0,\frac{1}{2}+\frac{1}{n}),&x"=(\frac{1}{2}+\frac{1}{n},\frac{1}{2}-\frac{1}{n},\frac{1}{2}+\frac{1}{n},\frac{1}{2}-\frac{1}{n}),\\ \lbrack \frac{1}{2}+\frac{1}{n},1\rbrack,&x\in X\setminus\{x',x"\}.\\ \end{array}\right. \end{equation*}$

容易验证 $\varepsilon_{n}\rightarrow0,\Phi_{n}\rightarrow\Phi$, $n\rightarrow\infty$.

令 $e\equiv1$. 考虑$x"_{n}=(\frac{1}{2}+\frac{1}{n},\frac{1}{2}-\frac{1}{n},\frac{1}{2}+\frac{1}{n},\frac{1}{2}-\frac{1}{n})$. 当 $p=1$ 时, 对任意 $y_{1}\in X_{1}$, 存在 $z^{2}_{1}=\frac{1}{2}-\frac{1}{n}\in (\Phi^{2}_{1})_{n}(x"_{n})$, 使得

$(x^{1}_{1})"_{n}z^{1}_{1}+(x^{2}_{1})"_{n}z^{2}_{1}+\varepsilon_{n}=\frac{1}{2}+\frac{1}{n}>\frac{1}{2}-\frac{1}{n}=(y^{1}_{1}+y^{2}_{1})\bigg(\frac{1}{2}-\frac{1}{n}\bigg)=y^{1}_{1}z^{1}_{1}+y^{2}_{1}z^{2}_{1}.$

当 $p=2$ 时, 取 $z^{2}_{2}=\frac{1}{2}-\frac{1}{n}\in (\Phi^{2}_{2})_{n}(x"_{n})$, 有

$(x^{1}_{2})"_{n}z^{1}_{2}+(x^{2}_{2})"_{n}z^{2}_{2}+\varepsilon_{n}=\frac{1}{2}+\frac{1}{n}>\frac{1}{2}-\frac{1}{n}=(y^{1}_{2}+y^{2}_{2})\bigg(\frac{1}{2}-\frac{1}{n}\bigg)=y^{1}_{2}z^{1}_{2}+y^{2}_{2}z^{2}_{2}.$

可见, $x"_{n}\in NE(\Phi_{n},\varepsilon_{n})$ 且 $x"_{n}\rightarrow x"=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$. 然而, 当 $p=1$ 时, 对于$z^{1}_{1}=\frac{1}{2}$ 和任意的 $z^{2}_{1}\in (0,\frac{1}{2})$, 存在$y_{1}=(y^{1}_{1},y^{2}_{1})=(1,0)\in X_{1}$, 有

$\begin{equation*} y^{1}_{1}z^{1}_{1}+y^{2}_{1}z^{2}_{1}=\frac{1}{2}\in \bigg(\frac{1}{4},\frac{1}{2}\bigg)+{\rm int}R_{+}=[ (x^{1}_{1})"\Phi^{1}_{1}(x")+(x^{2}_{1})"\Phi^{2}_{1}(x") ]+{\rm int}R_{+}. \end{equation*}$

由此可见, $x"\notin NE(\Phi)$. 因此, $\Phi$ 不是广义适定的.

推论3.1 对任意的 $\Phi\in M$, 若集值群体博弈 $\Phi$ 的 Nash 均衡是广义适定的, 则它一定也是广义 Tikhonov 适定和广义 Hadamard 适定.

若 $\Phi^{i}_{p}$ 退化为向量值函数, 定理 3.1 可变为多目标群体博弈弱 Nash 均衡的广义适定性结果.

推论3.2 $\forall\Phi\in M$, 令 $\Phi^{i}_{p}:X\rightarrow R^{m}$, $\forall p\in A, \forall i\in F_{p}$, 则群体博弈 $\Phi$ 的 Nash 均衡是广义适定的. 同样, 它一定也是广义 Tikhonov 适定和广义 Hadamard 适定.

4 合作均衡广义适定性

本节主要讨论集值群体博弈合作均衡的广义适定性. 首先, 构建集值群体博弈合作均衡的有限理性函数 $\Psi:M\times \widehat{X_{A}}\rightarrow R$,

$\begin{equation} \Psi(\Phi,x)=\max_{C\subseteq A}\max_{x_{C}\in \widehat{X_{C}}}\min_{p\in C}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg],\Phi\in M,x\in \widehat{X_{A}}. \end{equation}$

现在, 讨论有限理性函数 $\Psi$ 的零水平集与集值种群博弈合作均衡解集 $CE(\Phi)$ 之间的等价关系.

引理4.1 $\forall\Phi\in M,\forall x\in \widehat{X_{A}},\Psi(\Phi,x)\geq0$; 特别地, $\Psi(\Phi,x)=0\Leftrightarrow x\in CE(\Phi)$.

证 首先, 证明 $\forall\Phi\in M,\forall x\in \widehat{X_{A}},\Psi(\Phi,x)\geq0$. 反证法, 假设 $\exists\overline{\Phi}\in M,\exists \overline{x}\in \widehat{X_{A}}$, 使得$\Psi(\overline{\Phi},\overline{x})<0$. 根据 $\Psi$ 的定义, $\forall C\subseteq A,\forall x_{C}\in \widehat{X_{C}},\exists p\in C,\exists z^{i}_{p}\in\overline{\Phi}^{i}_{p}(\overline{x})$, 有

$\begin{equation} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-\overline{x}^{i}_{p})\Bigg]<0. \end{equation}$

由 $C\subseteq A,x_{C}\in \widehat{X_{C}}$ 选取的任意性, 不妨令 $C=A,y^{i}_{p}=\overline{x}^{i}_{p}$, 则

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-\overline{x}^{i}_{p})\Bigg]=0. \end{equation*}$

这与 (4.2) 式矛盾.

下面, 证明 $\Psi(\Phi,x)=0\Leftrightarrow x\in CE(\Phi)$.

假设 $\Psi(\Phi,x)=0$, 但$ x\notin CE(\Phi)$. 根据 $CE(\Phi)$ 的定义, 即 $\exists C\subseteq A,\exists x_{C}\in \widehat{X_{C}}$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\in {\rm int}R^{m}_{+},\forall p\in C,\forall z^{i}_{p}\in\Phi^{i}_{p}(x). \end{equation*}$

根据 $h_{e}$ 的性质,

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]>0. \end{equation*}$

再由 $p, z^{i}_{p}$ 分别在 $C$ 和 $\Phi^{i}_{p}(x)$ 上选取的任意性, 显有

$\begin{equation*} \min_{p\in C}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]>0. \end{equation*}$

进一步, 对于上述的 $C\subseteq A,x_{C}\in \widehat{X_{C}}$ 有

$\begin{aligned} \Psi(\Phi,x)\geq\max_{x_{C}\in \widehat{X_{C}}}\min_{p\in C}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg] \geq\min_{p\in C}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg] >0. \end{aligned}$

这与 $\Psi(\Phi,x)=0$ 矛盾. 因此, $x\in CE(\Phi)$.

反之, 下面讨论 $x\in CE(\Phi)$ 时, 有 $\Psi(\Phi,x)=0$ 成立. 由 $CE(\Phi)$ 定义, 即 $\forall C\subseteq A,\forall x_{C}\in \widehat{X_{C}},\exists p\in A,\exists z^{i}_{p}\in\Phi^{i}_{p}(x)$, 有

$\begin{equation*} \sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\notin {\rm int}R^{m}_{+}. \end{equation*}$

通过 $h_{e}$ 的性质有

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]\leq0. \end{equation*}$

进一步,

$\begin{equation*} \max_{C\subseteq A}\max_{x_{C}\in \widehat{X_{C}}}\min_{p\in C}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]\leq0. \end{equation*}$

可见, $\Psi(\Phi,x)\leq0$. 又因为 $\Psi(\Phi,x)\geq0$, 故可得 $\Psi(\Phi,x)=0$ 成立.

下面, 证明 $\Psi$ 关于 $(\Phi,x)$ 的下半连续性.

引理4.2 $\forall\Phi\in M,\forall x\in \widehat{X_{A}},\Psi(\Phi,x)$ 是下半连续的.

证 $\forall C\subseteq A,\forall x_{C}\in \widehat{X_{C}}$, 定义函数如下

$\begin{equation} \nu(\Phi,x)=\min_{p\in C}\min_{z^{i}_{p}\in\Phi^{i}_{p}(x)}h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]. \end{equation}$

首先, 证明对 $\forall t\in R$, $\nu$ 的下水平集

$\begin{aligned} T:=&\{(\Phi,x)|\nu(\Phi,x)\leq t\} =\{(\Phi,x)|\exists p\in C,\exists z^{i}_{p}\in\Phi^{i}_{p}(x), \mbox{使得} h_{e}\Bigg[\sum^{n_{p}}_{i=1}z^{i}_{p}(y^{i}_{p}-x^{i}_{p})\Bigg]\leq t\} \end{aligned}$

是 $M\times\widehat{X_{A}}$ 中的闭集. 任取 $(\Phi_{n},x_{n})\in T$ 且 $(\Phi_{n},x_{n})\rightarrow(\Phi_{0},x_{0})$, 由假设存在 $p\in C,(z^{i}_{p})^{n}_{n}\in(\Phi^{i}_{p})_{n}(x_{n})$, 使得

$\begin{equation} h_{e}\Bigg[\sum^{n_{p}}_{i=1}(z^{i}_{p})^{n}_{n}(y^{i}_{p}-(x^{i}_{p})_{n})\Bigg]\leq t. \end{equation}$

因为 $(\Phi^{i}_{p})_{n}$ 是上半连续且紧值的, 对上述$(z^{i}_{p})^{n}_{n}\in(\Phi^{i}_{p})_{n}(x_{n})$, $\exists (z^{i}_{p})^{n}_{0}\in(\Phi^{i}_{p})_{n}(x_{0})$, 有 $(z^{i}_{p})^{n}_{n}\rightarrow (z^{i}_{p})^{n}_{0}$. 因此,

$\begin{aligned} d((z^{i}_{p})^{n}_{n},(\Phi^{i}_{p})_{0}(x_{0}))&\leq d((z^{i}_{p})^{n}_{n},(z^{i}_{p})^{n}_{0})+d((z^{i}_{p})^{n}_{0},(\Phi^{i}_{p})_{n}(x_{0}))+H_{p}((\Phi^{i}_{p})_{n}(x_{0}),(\Phi^{i}_{p})_{0}(x_{0}))\\ &=d((z^{i}_{p})^{n}_{n},(z^{i}_{p})^{n}_{0})+H_{p}((\Phi^{i}_{p})_{n}(x_{0}),(\Phi^{i}_{p})_{0}(x_{0}))\rightarrow0, n\rightarrow\infty. \end{aligned}$

这意味着存在 $(z^{i}_{p})^{0}_{0}\in(\Phi^{i}_{p})_{0}(x_{0})$, 使得$(z^{i}_{p})^{n}_{n}\rightarrow (z^{i}_{p})^{0}_{0}$. 根据 $h_{e}$ 的连续性及(4.4)式, 当 $n\rightarrow\infty$ 时, 有

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}(z^{i}_{p})^{0}_{0}(y^{i}_{p}-(x^{i}_{p})_{0})\Bigg]\leq t. \end{equation*}$

此时, $(\Phi_{0},x_{0})\in T$. 即 $T$ 一定是 $M\times\widehat{X_{A}}$ 中的闭集. 对于 $\forall t\in R$, $\Psi$ 的下水平集

$\begin{aligned} \{(\Phi,x)|\Psi(\Phi,x)\leq t\}=&\{(\Phi,x)|\forall C\subseteq A,\forall x_{C}\in \widehat{X_{C}}, \nu(\Phi,x)\leq t\}\\ =&\bigcap_{C\subseteq A}\bigcap_{x_{C}\in \widehat{X_{C}}}\{(\Phi,x)| \nu(\Phi,x)\leq t\}. \end{aligned}$

根据闭集的性质, 同样为 $M\times\widehat{X_{A}}$ 中闭集. 故 $\Psi(\Phi,x)$ 是下半连续的.

定理4.1 $\forall \Phi\in M$,$\Phi$ 是广义适定的.

证 令 $\{\Phi_{n}\}\subseteq M,\Phi_{n}\rightarrow\Phi$. $\varepsilon_{n}>0,\varepsilon_{n}\rightarrow0$. 任取 $x_{n}\in CE(\Phi_{n},\varepsilon_{n})$, 通过 $CE(\Phi_{n},\varepsilon_{n})$ 定义知, $\forall C\subseteq A$,$\forall x_{C}\in\widehat{X_{C}}$,$\exists p\in C,\exists(z^{i}_{p})^{n}_{n}\in(\Phi^{i}_{p})_{n}(x_{n})$, 使得

$\begin{equation*} \sum^{n_{p}}_{i=1}(z^{i}_{p})^{n}_{n}(y^{i}_{p}-(x^{i}_{p})_{n})\notin\varepsilon_{n}e+{\rm int}R^{m}_{+}. \end{equation*}$

由 $h_{e}$ 性质,

$\begin{equation*} h_{e}\Bigg[\sum^{n_{p}}_{i=1}(z^{i}_{p})^{n}_{n}(y^{i}_{p}-(x^{i}_{p})_{n})\Bigg]\leq\varepsilon_{n}, \end{equation*}$

即 $\Psi(\Phi_{n},x_{n})\leq\varepsilon_{n}$. 因为 $\{x_{n}\}\subseteq \widehat{X_{A}}$ 且 $\widehat{X_{A}}$ 的紧性, 存在 $\{x_{n}\}$ 的子序列 $\{x_{n_{k}}\}$, 使得 $x_{n_{k}}\rightarrow x\in \widehat{X_{A}}$. 根据引理 4.2,

$\begin{equation*} \Psi(\Phi,x)\leq\liminf_{k\rightarrow\infty}\Psi(\Phi_{n_{k}},x_{n_{k}})\leq\liminf_{k\rightarrow\infty}\varepsilon_{k}=0. \end{equation*}$

由引理 4.1 知, $\Psi(\Phi,x)\geq0$, 则有 $\Psi(\Phi,x)=0$; 再由引理 4.1 中等价关系得 $x\in CE(\Phi)$. 故 $\Phi$ 的广义适定性得证.

下面例子解释集支付紧值上半连续假设在定理 4.1 中是必要的.

例 4.1 $A,F_{1},F_{2},m_{1},m_{2},X_{1},X_{2}$和例 3.1 中定义相同. 容易得

$\begin{equation*} \widehat{X_{1}}=X_{1},\widehat{X_{2}}=X_{2},\widehat{X_{A}}=\{x=(x^{1}_{1},x^{2}_{1},x^{1}_{2},x^{2}_{2})\in R^{4}_{+}|x^{1}_{1}+x^{2}_{1}+x^{1}_{2}+x^{2}_{2}=2\}. \end{equation*}$

构建集支付函数 $\Phi:\widehat{X_{A}}\rightarrow2^{R}$ 如下

$\begin{equation*} \Phi^{1}_{1}(x)=\Phi^{1}_{2}(x)\equiv\{0\}; \end{equation*}$
$\begin{equation*} \Phi^{2}_{1}(x)=\Phi^{2}_{2}(x)=\left\{ \begin{array}{ll} \{0\},&x'=(0,1,0,1),\\ (0,\frac{1}{2}),&x"=(1,1,0,0),\\ \lbrack \frac{1}{2},1\rbrack, &x\in X\setminus\{x',x"\}.\\ \end{array}\right. \end{equation*}$

显然 $\Phi^{2}_{1},\Phi^{2}_{2}$ 在 $x"$ 上不是紧值上半连续. 选取$\varepsilon_{n}=\frac{2}{n}$ 和 $(\Phi^{i}_{p})_{n}:\widehat{X_{A}}\rightarrow2^{R}$ 如下

$\begin{equation*} (\Phi^{1}_{1})_{n}(x)=(\Phi^{1}_{2})_{n}(x)\equiv\{0\}; \end{equation*}$
$\begin{equation*} (\Phi^{2}_{1})_{n}(x)=(\Phi^{2}_{2})_{n}(x)=\left\{ \begin{array}{ll} \{0\},&x'=(0,1,0,1),\\ (0,\frac{1}{2}+\frac{1}{n}),&x"=(1-\frac{1}{n},1+\frac{1}{n},0,0),\\ \lbrack \frac{1}{2}+\frac{1}{n},1\rbrack, &x\in X\setminus\{x',x"\}.\\ \end{array}\right. \end{equation*}$

容易验证 $\varepsilon_{n}\rightarrow0$, $\Phi_{n}\rightarrow\Phi$, $n\rightarrow\infty$.

令 $e\equiv1$, 考虑 $\Psi$ 在 $x"_{n}=(1-\frac{1}{n},1+\frac{1}{n},0,0)$ 处的均衡情况. 当 $C_{1}=\{1\}$ 时, 对于任意 $y_{1}\in \widehat{X_{1}}$, 存在 $z^{2}_{1}=\frac{1}{n}\in(\Phi^{2}_{1})_{n}(x"_{n})$, 使得

$\begin{equation*} (x^{1}_{1})"_{n}z^{1}_{1}+(x^{2}_{1})"_{n}z^{2}_{1}+\varepsilon_{n}=\frac{3}{n}+\frac{1}{n^{2}}>1\cdot\frac{1}{n}=\max_{y_{1}\in \widehat{X_{1}}}(y^{1}_{1}z^{1}_{1}+y^{2}_{1}z^{2}_{1}). \end{equation*}$

当 $C_{2}=\{2\}$ 时, 对于任意 $y_{2}\in \widehat{X_{2}}$, 存在 $z^{2}_{2}=\frac{1}{n}\in(\Phi^{2}_{2})_{n}(x"_{n})$, 使得

$\begin{equation*} (x^{1}_{2})"_{n}z^{1}_{2}+(x^{2}_{2})"_{n}z^{2}_{2}+\varepsilon_{n}=0+\frac{2}{n}>1\cdot\frac{1}{n}=\max_{y_{2}\in \widehat{X_{2}}}(y^{1}_{2}z^{1}_{2}+y^{2}_{2}z^{2}_{2}). \end{equation*}$

当 $C_{3}=A$ 时, 则 $\widehat{X_{C_{3}}}=\widehat{X_{A}}$. 对任意的 $y_{C_{3}}\in\widehat{X_{C_{3}}}$, 当 $p=1$ 时, 存在 $z^{2}_{1}=\frac{1}{n}$,

$\begin{equation*} (x^{1}_{1})"_{n}z^{1}_{1}+(x^{2}_{1})"_{n}z^{2}_{1}+\varepsilon_{n}=\frac{3}{n}+\frac{1}{n^{2}}>2\cdot\frac{1}{n}=\max_{y_{C_{3}}\in\widehat{X_{C_{3}}}}(y^{1}_{1}z^{1}_{1}+y^{2}_{1}z^{2}_{1}); \end{equation*}$

当 $p=2$ 时, 存在 $z^{2}_{2}=\frac{1}{n}$,

$\begin{equation*} (x^{1}_{2})"_{n}z^{1}_{2}+(x^{2}_{2})"_{n}z^{2}_{2}+\varepsilon_{n}=0+\frac{2}{n}=2\cdot\frac{1}{n}=\max_{y_{C_{3}}\in\widehat{X_{C_{3}}}}(y^{1}_{2}z^{1}_{2}+y^{2}_{2}z^{2}_{2}). \end{equation*}$

综上所述, $x"_{n}\in CE(\Phi_{n},\varepsilon_{n})$且$x"_{n}\rightarrow x"=(1,1,0,0)$. 然而, 当 $C=\{2\}$ 时, 此时 $\Phi^{2}_{2}(x")=(0,\frac{1}{2})$, 对于 $z^{1}_{2}=0$ 和任意的 $z^{2}_{2}\in(0,\frac{1}{2})$, 存在 $y_{2}=(y^{1}_{2},y^{2}_{2})=(0,1)\in \widehat{X_{2}}$, 使得

$\begin{equation*} (x^{1}_{2})"\Phi^{1}_{2}(x")+(x^{2}_{2})"\Phi^{2}_{2}(x")+{\rm int}R_{+}=[0,+\infty)\supset\bigg(0,\frac{1}{2}\bigg)\ni z^{2}_{2}=y^{1}_{2}z^{1}_{2}+y^{2}_{2}z^{2}_{2}. \end{equation*}$

由此可见, $x"\notin CE(\Phi)$. 因此, $\Phi$ 不是广义适定的.

推论4.1 对任意的 $ \Phi\in M$, 若集值群体博弈 $\Phi$ 的合作均衡是广义适定的, 则它一定也是广义 Tikhonov 适定和广义 Hadamard 适定.

注4.1 若 $\Phi^{i}_{p}$ 退化为向量值函数, 则集值群体博弈合作均衡的广义适定性变为文献[23] 中多目标群体博弈合作均衡的广义适定性.

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