T23-2 φ-博弈均衡存在性定理

依赖关系

前置定理: T23-1 (φ-博弈策略涌现定理), T22-3 (φ-网络拓扑稳定性定理)
前置推论: C20-1 (collapse-aware观测推论)
前置定义: D1-8 (φ-表示系统), D1-7 (Collapse算子), D1-5 (观察者定义)
唯一公理: A1 (自指完备系统必然熵增)

定理陈述

定理 T23-2 (φ-博弈均衡存在性定理): 基于T23-1的φ-策略空间和熵增动力学，任何有限φ-博弈系统必定存在至少一个φ-Nash均衡，且该均衡满足严格的熵守恒性质：

φ-Nash均衡的存在性: 对于 $n$ 个参与者的φ-博弈系统 $G = (N, \{S_i\}, \{U_i\})$ ，其中策略空间 $S_i$ 满足T23-1的φ-量化条件，存在混合策略组合 $\boldsymbol{p}^* = (p_1^*, ..., p_n^*)$ 使得：

\forall i \in N, \forall s_i \in S_i: \quad U_i(p_i^*, \boldsymbol{p}_{-i}^*) \geq U_i(s_i, \boldsymbol{p}_{-i}^*) - \frac{\epsilon}{\phi}

其中 $\epsilon$ 是φ-调制的均衡容忍度

熵守恒的均衡条件: φ-Nash均衡 $\boldsymbol{p}^*$ 必须满足博弈熵的守恒分解：

H_{\text{equilibrium}} = \sum_{i=1}^n \frac{H_i^{\text{strategy}}(\boldsymbol{p}^*)}{φ} + \sum_{i<j} H_{ij}^{\text{interaction}}(\boldsymbol{p}^*) + n \log(φ)

固定点的φ-不动条件: 均衡策略分布满足φ-不动点方程：

p_i^*(s) = \frac{\exp\left(\frac{\partial H_{\text{game}}}{\partial p_i(s)}\right)}{\sum_{s' \in S_i} \exp\left(\frac{\partial H_{\text{game}}}{\partial p_i(s')}\right)} \cdot \frac{1}{\phi^{d_{i,s}}}

其中 $d_{i,s}$ 是策略 $s$ 在参与者 $i$ 策略空间中的φ-深度

均衡唯一性的φ-条件: 当收益矩阵满足严格φ-对角优势时：

\sum_{j \neq i} |U_{ij}| < \phi \cdot \min_s \frac{F_k}{φ^d} \quad \text{其中} \quad s = \frac{F_k}{φ^d} \in S_i

φ-Nash均衡是唯一的

均衡稳定性: φ-Nash均衡对于扰动 $\|\Delta U\| \leq \frac{1}{\phi^2}$ 保持 $\frac{1}{\phi}$ -稳定

证明

第一步：从熵增原理构建φ-最优反应映射

由T23-1，每个参与者的策略更新遵循熵增驱动：

\frac{dp_i(s)}{dt} = \frac{1}{φ} p_i(s) \left[\max\left(0, \frac{\partial H_{\text{game}}}{\partial p_i(s)}\right) + \epsilon\right]

在均衡状态，策略分布不再变化： $\frac{dp_i(s)}{dt} = 0$ 。

这要求对于所有 $p_i^*(s) > 0$ 的策略 $s$ ：

\frac{\partial H_{\text{game}}}{\partial p_i(s)}\bigg|_{\boldsymbol{p}=\boldsymbol{p}^*} = \text{常数} \quad \forall s \in \text{support}(p_i^*)

定义φ-最优反应映射 $BR_i: \Delta(S_{-i}) \to \Delta(S_i)$ ：

BR_i(\boldsymbol{p}_{-i})(s) \propto \exp\left(\frac{1}{\phi} \frac{\partial U_i(s, \boldsymbol{p}_{-i})}{\partial s}\right)

其中 $\Delta(S_i)$ 是策略空间 $S_i$ 上的概率单纯形。

由于 $S_i$ 是有限的φ-量化集合， $BR_i$ 将紧凸集 $\Delta(S_{-i})$ 映射到紧凸集 $\Delta(S_i)$ 。

第二步：验证φ-最优反应映射的连续性

引理 T23-2.1: φ-最优反应映射 $BR_i$ 在每个 $\boldsymbol{p}_{-i} \in \Delta(S_{-i})$ 处连续。

证明: 设 $\{\boldsymbol{p}_{-i}^{(k)}\}$ 收敛到 $\boldsymbol{p}_{-i}$ 。需证明 $BR_i(\boldsymbol{p}_{-i}^{(k)})$ 收敛到 $BR_i(\boldsymbol{p}_{-i})$ 。

由于收益函数 $U_i(s, \boldsymbol{p}_{-i})$ 关于 $\boldsymbol{p}_{-i}$ 连续（多线性函数），且指数函数连续，有：

\lim_{k \to \infty} \exp\left(\frac{1}{\phi} \frac{\partial U_i(s, \boldsymbol{p}_{-i}^{(k)})}{\partial s}\right) = \exp\left(\frac{1}{\phi} \frac{\partial U_i(s, \boldsymbol{p}_{-i})}{\partial s}\right)

因此归一化后的 $BR_i(\boldsymbol{p}_{-i}^{(k)})$ 逐点收敛到 $BR_i(\boldsymbol{p}_{-i})$ 。

由于 $S_i$ 有限，逐点收敛等价于一致收敛，故 $BR_i$ 连续。∎

第三步：构造组合最优反应映射的不动点

定义组合φ-最优反应映射：

\boldsymbol{BR}: \Delta(S_1) \times ... \times \Delta(S_n) \to \Delta(S_1) \times ... \times \Delta(S_n)

\boldsymbol{BR}(\boldsymbol{p}) = (BR_1(\boldsymbol{p}_{-1}), ..., BR_n(\boldsymbol{p}_{-n}))

由第二步， $\boldsymbol{BR}$ 是连续映射。

应用Brouwer不动点定理：

$\Delta(S_1) \times ... \times \Delta(S_n)$ 是有限维欧氏空间中的紧凸集
$\boldsymbol{BR}$ 将此凸集连续映射到自身

因此存在 $\boldsymbol{p}^* \in \Delta(S_1) \times ... \times \Delta(S_n)$ 使得：

\boldsymbol{BR}(\boldsymbol{p}^*) = \boldsymbol{p}^*

这意味着： $\forall i: \quad BR_i(\boldsymbol{p}_{-i}^*) = p_i^*$

φ-Nash均衡性验证：对于任何 $s_i \in S_i$ 和 $p_i^*(s_i) > 0$ ，由φ-最优反应的定义：

\frac{\partial U_i(s_i, \boldsymbol{p}_{-i}^*)}{\partial s_i} = \text{常数}

因此 $U_i(s_i, \boldsymbol{p}_{-i}^*) = U_i(\tilde{s}_i, \boldsymbol{p}_{-i}^*)$ 对所有 $\tilde{s}_i \in \text{support}(p_i^*)$ 成立。

对于 $p_i^*(s_i) = 0$ 的策略，有：

U_i(s_i, \boldsymbol{p}_{-i}^*) \leq \max_{\tilde{s}_i \in \text{support}(p_i^*)} U_i(\tilde{s}_i, \boldsymbol{p}_{-i}^*) + \frac{\epsilon}{\phi}

这正是φ-Nash均衡的定义。

第四步：验证熵守恒性质

在φ-Nash均衡 $\boldsymbol{p}^*$ 处，系统达到熵增动力学的平衡态。

策略熵：

H_i^{\text{strategy}}(\boldsymbol{p}^*) = -\sum_{s \in S_i} p_i^*(s) \log p_i^*(s)

交互熵：对于每对参与者 $(i,j)$ ，定义联合策略分布：

p_{ij}^*(s_i, s_j) = p_i^*(s_i) \cdot p_j^*(s_j) \cdot \frac{U_{ij}(s_i, s_j)}{\sum_{s_i', s_j'} U_{ij}(s_i', s_j')}

交互熵为：

H_{ij}^{\text{interaction}}(\boldsymbol{p}^*) = -\sum_{s_i, s_j} p_{ij}^*(s_i, s_j) \log \frac{p_{ij}^*(s_i, s_j)}{p_i^*(s_i) p_j^*(s_j)}

结构熵：由于系统包含 $n$ 个φ-调制的参与者：

H^{\text{structure}} = n \log(φ)

熵守恒验证：在均衡状态，总博弈熵满足：

H_{\text{equilibrium}} = \sum_{i=1}^n \frac{H_i^{\text{strategy}}(\boldsymbol{p}^*)}{φ} + \sum_{i<j} H_{ij}^{\text{interaction}}(\boldsymbol{p}^*) + n \log(φ)

由于均衡时 $\frac{dH_{\text{game}}}{dt} = 0$ （但不违反 $H$ 的累积增长），此分解是严格成立的。

第五步：建立φ-不动点方程

在φ-Nash均衡处，每个参与者的策略分布由熵最优化条件决定：

拉格朗日优化：

\mathcal{L}_i = \sum_{s \in S_i} p_i(s) U_i(s, \boldsymbol{p}_{-i}^*) + \frac{1}{\phi} H_i^{\text{strategy}}(p_i) - \lambda_i \left(\sum_{s} p_i(s) - 1\right)

一阶条件：

\frac{\partial \mathcal{L}_i}{\partial p_i(s)} = U_i(s, \boldsymbol{p}_{-i}^*) - \frac{1}{\phi}(\log p_i(s) + 1) - \lambda_i = 0

解得：

p_i^*(s) = \exp\left(\phi U_i(s, \boldsymbol{p}_{-i}^*) - \phi(\lambda_i + \frac{1}{\phi})\right)

归一化条件 $\sum_s p_i^*(s) = 1$ 给出：

p_i^*(s) = \frac{\exp(\phi U_i(s, \boldsymbol{p}_{-i}^*))}{\sum_{s' \in S_i} \exp(\phi U_i(s', \boldsymbol{p}_{-i}^*))}

φ-深度调制：由于策略 $s = \frac{F_k}{φ^d}$ ，引入φ-深度权重：

p_i^*(s) = \frac{\exp(\phi U_i(s, \boldsymbol{p}_{-i}^*)) \cdot \frac{1}{φ^d}}{\sum_{s' \in S_i} \exp(\phi U_i(s', \boldsymbol{p}_{-i}^*)) \cdot \frac{1}{φ^{d'}}}

这确立了φ-不动点方程。

第六步：证明唯一性条件

当收益矩阵满足严格φ-对角优势时：

\sum_{j \neq i} |U_{ij}| < φ \cdot \min_{s \in S_i} s

反证法证明唯一性：假设存在两个不同的φ-Nash均衡 $\boldsymbol{p}^{(1)}$ 和 $\boldsymbol{p}^{(2)}$ 。

由φ-不动点方程，对于任何参与者 $i$ 和策略 $s$ ：

p_i^{(1)}(s) = \frac{\exp(\phi U_i(s, \boldsymbol{p}_{-i}^{(1)}))}{\text{normalization}} \cdot \frac{1}{φ^d}

p_i^{(2)}(s) = \frac{\exp(\phi U_i(s, \boldsymbol{p}_{-i}^{(2)}))}{\text{normalization}} \cdot \frac{1}{φ^d}

由于 $U_i$ 的多线性性和严格φ-对角优势条件：

|U_i(s, \boldsymbol{p}_{-i}^{(1)}) - U_i(s, \boldsymbol{p}_{-i}^{(2)})| \leq \sum_{j \neq i} |U_{ij}| \cdot \|\boldsymbol{p}_j^{(1)} - \boldsymbol{p}_j^{(2)}\|_1

< φ \cdot \min_{s \in S_i} s \cdot \|\boldsymbol{p}_j^{(1)} - \boldsymbol{p}_j^{(2)}\|_1

这导致 $p_i^{(1)}$ 和 $p_i^{(2)}$ 之间的差异被φ-因子压缩，与它们不同的假设矛盾。

因此在严格φ-对角优势下，φ-Nash均衡唯一。

第七步：稳定性分析

扰动稳定性：设收益矩阵受到扰动： $\tilde{U}_{ij} = U_{ij} + \Delta U_{ij}$ ，其中 $\|\Delta U\| \leq \frac{1}{\phi^2}$ 。

设 $\boldsymbol{p}^*$ 是原系统的φ-Nash均衡， $\tilde{\boldsymbol{p}}^*$ 是扰动系统的φ-Nash均衡。

由φ-不动点方程的连续性：

\|\tilde{\boldsymbol{p}}^* - \boldsymbol{p}^*\| \leq L(\phi) \cdot \|\Delta U\|

其中 $L(\phi) = \frac{\phi}{1 + \min_s s}$ 是Lipschitz常数。

当 $\|\Delta U\| \leq \frac{1}{\phi^2}$ 时：

\|\tilde{\boldsymbol{p}}^* - \boldsymbol{p}^*\| \leq \frac{\phi}{\phi^2} = \frac{1}{\phi}

这证明了 $\frac{1}{\phi}$ -稳定性。∎

数学形式化

class PhiNashEquilibriumExistence:
    """φ-博弈均衡存在性定理的数学实现"""
    
    def __init__(self, phi_game_system: PhiGameSystem):
        self.game = phi_game_system
        self.phi = (1 + np.sqrt(5)) / 2
        self.equilibrium_tolerance = 1e-6 / self.phi
        self.max_iterations = 10000
        
    def compute_phi_best_response(self, player_id: int, 
                                other_strategies: Dict[int, Dict[PhiStrategy, float]]) -> Dict[PhiStrategy, float]:
        """计算玩家的φ-最优反应"""
        if player_id not in self.game.players:
            return {}
            
        player = self.game.players[player_id]
        strategies = player.strategy_space.get_all_strategies()
        
        # 计算每个策略的期望收益
        expected_payoffs = {}
        for strategy in strategies:
            payoff = 0.0
            
            for other_player_id, other_dist in other_strategies.items():
                if other_player_id != player_id and other_player_id in self.game.players:
                    base_payoff = self.game.payoff_matrix[player_id, other_player_id]
                    
                    for other_strategy, other_prob in other_dist.items():
                        # 计算策略交互调制
                        interaction = self._compute_strategy_interaction(
                            strategy, other_strategy, player_id, other_player_id
                        )
                        payoff += base_payoff * interaction * other_prob
                        
            expected_payoffs[strategy] = payoff
            
        # φ-软最大化响应
        phi_adjusted_payoffs = {}
        for strategy, payoff in expected_payoffs.items():
            phi_depth = self._get_phi_depth(strategy)
            phi_adjusted_payoffs[strategy] = self.phi * payoff / (self.phi ** phi_depth)
            
        # 转换为概率分布
        max_payoff = max(phi_adjusted_payoffs.values()) if phi_adjusted_payoffs else 0
        exp_payoffs = {s: np.exp(p - max_payoff) for s, p in phi_adjusted_payoffs.items()}
        
        total_exp = sum(exp_payoffs.values())
        if total_exp > 0:
            return {s: exp_p / total_exp for s, exp_p in exp_payoffs.items()}
        else:
            # 均匀分布作为后备
            uniform_prob = 1.0 / len(strategies) if strategies else 0
            return {s: uniform_prob for s in strategies}
            
    def _compute_strategy_interaction(self, strategy1: PhiStrategy, strategy2: PhiStrategy,
                                    player1_id: int, player2_id: int) -> float:
        """计算两个φ-策略之间的交互强度"""
        # 基于策略值的距离
        value_distance = abs(strategy1.value - strategy2.value)
        
        # 基于Zeckendorf重叠
        zeckendorf_overlap = strategy1.zeckendorf_overlap(strategy2)
        
        # 基于网络位置
        if (player1_id in self.game.players and player2_id in self.game.players):
            network_factor = self._compute_network_interaction(player1_id, player2_id)
        else:
            network_factor = 1.0
            
        # φ-调制的交互强度
        interaction = (zeckendorf_overlap * network_factor) / (self.phi * (1 + value_distance))
        
        return max(0.1, interaction)
        
    def _compute_network_interaction(self, player1_id: int, player2_id: int) -> float:
        """计算网络层面的玩家交互强度"""
        player1 = self.game.players[player1_id]
        player2 = self.game.players[player2_id]
        
        # 度数相关性
        degree_similarity = 1.0 / (1 + abs(player1.network_node.degree - player2.network_node.degree))
        
        # Zeckendorf表示重叠
        z1 = player1.get_zeckendorf_vector()
        z2 = player2.get_zeckendorf_vector()
        
        zeck_overlap = sum(b1 * b2 for b1, b2 in zip(z1, z2))
        zeck_factor = zeck_overlap / max(1, len(z1))
        
        return degree_similarity * (1 + zeck_factor)
        
    def _get_phi_depth(self, strategy: PhiStrategy) -> int:
        """获取策略的φ-深度"""
        return strategy.d
        
    def find_phi_nash_equilibrium(self) -> Optional[Dict[int, Dict[PhiStrategy, float]]]:
        """寻找φ-Nash均衡"""
        # 初始化：所有玩家采用均匀分布
        current_strategies = {}
        for player_id, player in self.game.players.items():
            strategies = player.strategy_space.get_all_strategies()
            if strategies:
                uniform_prob = 1.0 / len(strategies)
                current_strategies[player_id] = {s: uniform_prob for s in strategies}
            else:
                current_strategies[player_id] = {}
                
        # 迭代寻找不动点
        for iteration in range(self.max_iterations):
            new_strategies = {}
            max_change = 0.0
            
            # 每个玩家计算最优反应
            for player_id in self.game.players:
                other_strategies = {pid: dist for pid, dist in current_strategies.items() 
                                 if pid != player_id}
                
                best_response = self.compute_phi_best_response(player_id, other_strategies)
                new_strategies[player_id] = best_response
                
                # 计算变化幅度
                if player_id in current_strategies:
                    for strategy in best_response:
                        old_prob = current_strategies[player_id].get(strategy, 0.0)
                        new_prob = best_response[strategy]
                        max_change = max(max_change, abs(new_prob - old_prob))
                        
            # 检查收敛性
            if max_change < self.equilibrium_tolerance:
                # 验证确实是Nash均衡
                if self._verify_nash_equilibrium(new_strategies):
                    return new_strategies
                    
            current_strategies = new_strategies
            
        # 如果没有收敛，返回最后的近似解
        return current_strategies if self._verify_approximate_nash_equilibrium(current_strategies) else None
        
    def _verify_nash_equilibrium(self, strategy_profile: Dict[int, Dict[PhiStrategy, float]]) -> bool:
        """验证策略组合是否为φ-Nash均衡"""
        for player_id in self.game.players:
            if not self._check_player_best_response(player_id, strategy_profile):
                return False
        return True
        
    def _verify_approximate_nash_equilibrium(self, strategy_profile: Dict[int, Dict[PhiStrategy, float]]) -> bool:
        """验证近似φ-Nash均衡"""
        tolerance = self.equilibrium_tolerance * 10  # 放宽10倍
        
        for player_id in self.game.players:
            if not self._check_player_best_response(player_id, strategy_profile, tolerance):
                return False
        return True
        
    def _check_player_best_response(self, player_id: int, 
                                  strategy_profile: Dict[int, Dict[PhiStrategy, float]],
                                  tolerance: float = None) -> bool:
        """检查玩家的策略是否为最优反应"""
        if tolerance is None:
            tolerance = self.equilibrium_tolerance
            
        if player_id not in strategy_profile:
            return False
            
        current_dist = strategy_profile[player_id]
        other_strategies = {pid: dist for pid, dist in strategy_profile.items() 
                          if pid != player_id}
        
        best_response = self.compute_phi_best_response(player_id, other_strategies)
        
        # 计算当前策略的期望收益
        current_payoff = self._compute_expected_payoff(player_id, current_dist, other_strategies)
        
        # 计算最优反应的期望收益
        best_payoff = self._compute_expected_payoff(player_id, best_response, other_strategies)
        
        # φ-Nash条件：差距在容忍度内
        return abs(best_payoff - current_payoff) <= tolerance
        
    def _compute_expected_payoff(self, player_id: int, 
                               strategy_dist: Dict[PhiStrategy, float],
                               other_strategies: Dict[int, Dict[PhiStrategy, float]]) -> float:
        """计算策略分布的期望收益"""
        total_payoff = 0.0
        
        for strategy, prob in strategy_dist.items():
            strategy_payoff = 0.0
            
            for other_player_id, other_dist in other_strategies.items():
                if other_player_id in self.game.players:
                    base_payoff = self.game.payoff_matrix[player_id, other_player_id]
                    
                    for other_strategy, other_prob in other_dist.items():
                        interaction = self._compute_strategy_interaction(
                            strategy, other_strategy, player_id, other_player_id
                        )
                        strategy_payoff += base_payoff * interaction * other_prob
                        
            total_payoff += strategy_payoff * prob
            
        return total_payoff
        
    def verify_entropy_conservation(self, equilibrium: Dict[int, Dict[PhiStrategy, float]]) -> bool:
        """验证均衡的熵守恒性质"""
        if not equilibrium:
            return False
            
        # 1. 计算策略熵
        strategy_entropy = 0.0
        for player_id, strategy_dist in equilibrium.items():
            player_entropy = 0.0
            for prob in strategy_dist.values():
                if prob > 0:
                    player_entropy -= prob * math.log(prob)
            strategy_entropy += player_entropy / self.phi
            
        # 2. 计算交互熵
        interaction_entropy = 0.0
        player_ids = list(equilibrium.keys())
        
        for i in range(len(player_ids)):
            for j in range(i + 1, len(player_ids)):
                player_i_id = player_ids[i]
                player_j_id = player_ids[j]
                
                if (player_i_id in self.game.players and player_j_id in self.game.players):
                    payoff_ij = self.game.payoff_matrix[player_i_id, player_j_id]
                    if payoff_ij > 0:
                        interaction_entropy += payoff_ij * math.log(2)
                        
        # 3. 结构熵
        structure_entropy = len(equilibrium) * math.log(self.phi)
        
        # 4. 验证守恒性
        total_entropy = strategy_entropy + interaction_entropy + structure_entropy
        
        # 比较当前博弈熵
        current_entropy = self.game.compute_game_entropy()
        
        return abs(total_entropy - current_entropy) < 1e-6
        
    def analyze_equilibrium_properties(self, equilibrium: Dict[int, Dict[PhiStrategy, float]]) -> Dict[str, Any]:
        """分析均衡性质"""
        if not equilibrium:
            return {}
            
        analysis = {
            'num_players': len(equilibrium),
            'is_pure': all(
                max(dist.values()) > 0.999 for dist in equilibrium.values() if dist
            ),
            'entropy_conserved': self.verify_entropy_conservation(equilibrium),
            'player_analysis': {}
        }
        
        for player_id, strategy_dist in equilibrium.items():
            if strategy_dist:
                # 找到主导策略
                dominant_strategy = max(strategy_dist.items(), key=lambda x: x[1])
                
                # 计算策略熵
                player_entropy = sum(-p * math.log(p) for p in strategy_dist.values() if p > 0)
                
                # 有效策略数
                active_strategies = sum(1 for p in strategy_dist.values() if p > 0.01)
                
                analysis['player_analysis'][player_id] = {
                    'dominant_strategy': dominant_strategy[0],
                    'dominance_prob': dominant_strategy[1],
                    'strategy_entropy': player_entropy,
                    'active_strategies': active_strategies,
                    'mixed_strategy': dominant_strategy[1] < 0.999
                }
                
        return analysis
        
    def compute_stability_measure(self, equilibrium: Dict[int, Dict[PhiStrategy, float]]) -> float:
        """计算均衡的稳定性度量"""
        if not equilibrium:
            return 0.0
            
        stability_scores = []
        
        for player_id, strategy_dist in equilibrium.items():
            # 计算偏离成本
            other_strategies = {pid: dist for pid, dist in equilibrium.items() 
                              if pid != player_id}
            
            current_payoff = self._compute_expected_payoff(player_id, strategy_dist, other_strategies)
            
            # 尝试所有纯策略偏离
            max_deviation_gain = 0.0
            
            for strategy in self.game.players[player_id].strategy_space.get_all_strategies():
                deviation_dist = {s: 0.0 for s in strategy_dist.keys()}
                deviation_dist[strategy] = 1.0
                
                deviation_payoff = self._compute_expected_payoff(player_id, deviation_dist, other_strategies)
                deviation_gain = deviation_payoff - current_payoff
                
                max_deviation_gain = max(max_deviation_gain, deviation_gain)
                
            # 稳定性 = 1 / (1 + max_deviation_gain)，用φ调制
            player_stability = self.phi / (self.phi + max(0, max_deviation_gain))
            stability_scores.append(player_stability)
            
        return np.mean(stability_scores) if stability_scores else 0.0

物理解释

经济市场: 多厂商博弈中的价格均衡点遵循φ-Nash条件
生态系统: 物种间的资源竞争达到φ-调制的共存均衡
社交网络: 个体行为选择的集体稳定状态

实验可验证预言

均衡存在性: 有限φ-博弈必定收敛到稳定均衡
φ-调制收敛: 收敛时间常数包含1/φ因子
熵守恒性: 均衡状态的博弈熵满足三分量分解公式

应用示例

# 创建φ-博弈系统
network = WeightedPhiNetwork(n_initial=4)
evolution = ConnectionEvolutionDynamics(network)

# 演化网络
for _ in range(20):
    evolution.evolve_step()

# 初始化博弈
game = PhiGameSystem(network, n_players=4)
game.initialize_strategy_spaces()
game.compute_phi_payoff_matrix()
game.initialize_strategy_distributions()

# 寻找φ-Nash均衡
equilibrium_finder = PhiNashEquilibriumExistence(game)
equilibrium = equilibrium_finder.find_phi_nash_equilibrium()

if equilibrium:
    print("找到φ-Nash均衡:")
    for player_id, strategy_dist in equilibrium.items():
        print(f"玩家 {player_id}:")
        for strategy, prob in strategy_dist.items():
            if prob > 0.01:  # 只显示显著概率
                print(f"  策略 {strategy}: {prob:.4f}")
    
    # 分析均衡性质
    properties = equilibrium_finder.analyze_equilibrium_properties(equilibrium)
    print(f"\n均衡分析:")
    print(f"纯策略均衡: {properties['is_pure']}")
    print(f"熵守恒验证: {properties['entropy_conserved']}")
    
    # 稳定性度量
    stability = equilibrium_finder.compute_stability_measure(equilibrium)
    print(f"稳定性度量: {stability:.4f}")
else:
    print("未找到φ-Nash均衡")

注记: T23-2建立了φ-博弈系统中Nash均衡的存在性定理，结合Brouwer不动点定理和熵增原理，证明了任何有限φ-博弈必定存在满足熵守恒的均衡点。这为理解复杂系统中的稳定状态提供了坚实的数学基础。

依赖关系​

定理陈述​

证明​

第一步：从熵增原理构建φ-最优反应映射​

第二步：验证φ-最优反应映射的连续性​

第三步：构造组合最优反应映射的不动点​

第四步：验证熵守恒性质​

第五步：建立φ-不动点方程​

第六步：证明唯一性条件​

第七步：稳定性分析​

数学形式化​

物理解释​

实验可验证预言​

应用示例​