Definitions 10.8–10.9（risk-aversion coefficients） Def 10.8（Arrow–Pratt 绝对风险厌恶系数）：度量效用在 \(x\) 处的曲率，\(ra(x)=-\dfrac{u''(x)}{u'(x)}\)；系数越高曲率越大、越风险厌恶。
Def 10.9（相对风险厌恶系数）：\(rra(x)=ra(x)\cdot x=-\dfrac{u''(x)}{u'(x)}x\)。风险容忍度为绝对/相对风险厌恶的倒数。Def 10.8 (Arrow–Pratt Coefficient of Absolute Risk Aversion): curvature of \(u\) around \(x\), \(ra(x)=-\dfrac{u''(x)}{u'(x)}\); higher means more curvature and more risk averse.
Def 10.9 (Coefficient of Relative Risk Aversion): \(rra(x)=ra(x)\cdot x=-\dfrac{u''(x)}{u'(x)}x\). Risk tolerance is the inverse of absolute/relative risk aversion.

Example 10.1（specific \(ra(x)\), \(rra(x)\)） 见下式（线性、对数、CRRA、CARA）。CRRA 的 \(u(x)=\tfrac{x^{1-\gamma}-1}{1-\gamma}\) 是 \(\ln x\) 的推广（\(\gamma\to1\) 时由 L'Hôpital 得 \(\ln x\)）。See below (linear, log, CRRA, CARA). The CRRA \(u(x)=\tfrac{x^{1-\gamma}-1}{1-\gamma}\) generalizes \(\ln x\) (\(\gamma\to1\) gives \(\ln x\) by L'Hôpital).

Definition 10.10 & Proposition 10.4（absolute risk premium） Def 10.10：绝对保险费 \(p\) 定义为 \(u(\mathbb{E}[\tilde x]-p)=\mathbb{E}[u(\tilde x)]\) (10.4)；对赌局集设定，\(u(\sum p_i a_i-p)=\sum p_i u(a_i)\)。
Prop 10.4：风险很小时，\(p\approx\dfrac{1}{2}\left(-\dfrac{u''(\bar x)}{u'(\bar x)}\right)\sigma^2\) (10.5)，其中 \(\bar x=\mathbb{E}[\tilde x]\)、\(\sigma^2=\text{Var}(\tilde x)\)。Def 10.10: absolute insurance premium \(p\) defined by \(u(\mathbb{E}[\tilde x]-p)=\mathbb{E}[u(\tilde x)]\) (10.4); for the gamble set-up, \(u(\sum p_i a_i-p)=\sum p_i u(a_i)\).
Prop 10.4: if the risk is small, \(p\approx\dfrac{1}{2}\left(-\dfrac{u''(\bar x)}{u'(\bar x)}\right)\sigma^2\) (10.5), where \(\bar x=\mathbb{E}[\tilde x]\), \(\sigma^2=\text{Var}(\tilde x)\).

证明 / Proof (Proposition 10.4)

$$ > \begin{aligned} > u(\tilde x)&\approx u(\bar x)+u'(\bar x)(\tilde x-\bar x)+\tfrac{u''(\bar x)}{2}(\tilde x-\bar x)^2\\ > \Rightarrow\mathbb{E}[u(\tilde x)]&\approx u(\bar x)+u'(\bar x)\mathbb{E}[\tilde x-\bar x]+\tfrac{u''(\bar x)}{2}\mathbb{E}[(\tilde x-\bar x)^2]\\ > &\approx u(\bar x)+\tfrac{u''(\bar x)}{2}\sigma^2 > \end{aligned} > $$

Definition 10.11（Certainty equivalence） \(\tilde x\) 的确定性等价 \(c_e(\tilde x)\) 定义为 \(u(c_e(\tilde x))=\mathbb{E}[u(\tilde x)]\)，故 \(c_e(\tilde x)=\bar x-p\)。对赌局集，\(u(c_e)=\sum p_i u(a_i)\)，即 \(c_e((p_1\circ a_1,\dots,p_n\circ a_n))=\sum p_i a_i-p\)。The certainty equivalence of \(\tilde x\) is \(c_e(\tilde x)\) defined by \(u(c_e(\tilde x))=\mathbb{E}[u(\tilde x)]\), so \(c_e(\tilde x)=\bar x-p\). For the gamble set, \(u(c_e)=\sum p_i u(a_i)\), i.e. \(c_e((p_1\circ a_1,\dots,p_n\circ a_n))=\sum p_i a_i-p\).

Definition 10.12（Strategic form games） 策略型博弈是元组 \(G=(S_i,u_i)_{i=1}^N\)，其中 \(\{1,\dots,N\}\) 为玩家集，\(S_i\) 为玩家 \(i\) 非空策略集，\(u_i:S\equiv\prod_{i=1}^N S_i\to\mathbb{R}\) 为 \(i\) 的收益函数。所有玩家同时选策略；若策略剖面 \(s=(s_1,\dots,s_N)\in S\)（\(s_i\in S_i\)），则玩家 \(i\) 的收益为 \(u_i(s_1,\dots,s_N)\)。A strategic form game is a tuple \(G=(S_i,u_i)_{i=1}^N\) where \(\{1,\dots,N\}\) is the set of players, \(S_i\) player \(i\)'s non-empty strategy set, and \(u_i:S\equiv\prod_{i=1}^N S_i\to\mathbb{R}\) is \(i\)'s payoff function. All players choose simultaneously; if the profile is \(s=(s_1,\dots,s_N)\in S\) (\(s_i\in S_i\)), then player \(i\)'s payoff is \(u_i(s_1,\dots,s_N)\).

Example 10.2（Strictly dominant strategy） \(N=2\)，\(S_1=\{U,M,D\}\)，\(S_2=\{L,R\}\)，收益矩阵如下（典型元素 \((u_1(S_1,S_2),u_2(S_1,S_2))\)）。玩家 1 总玩 \(U\)（无论 2 怎么选 \(U\) 都严格优于 \(D\)）；给定此，玩家 2 玩 \(L\)；理性选择 \(S_1=U,S_2=L\)，收益 $(3,0)\(。\)N=2$, \(S_1=\{U,M,D\}\), \(S_2=\{L,R\}\); payoff matrix below (typical element \((u_1(S_1,S_2),u_2(S_1,S_2))\)). Player 1 always plays \(U\) (\(U\) strictly better than \(D\) no matter what 2 chooses); given that, player 2 plays \(L\); rational choice \(S_1=U,S_2=L\), return $(3,0)$.

Example 10.3（No overall dominance but partial dominance） \(N=2\)，\(S_1=\{U,C,D\}\)，\(S_2=\{L,M,R\}\)，收益矩阵如下。无对任一玩家严格优于所有其他策略的占优策略；但玩家 1 的 \(D\) 无论如何严格优于 \(C\)，故永不玩 \(C\)；玩家 2 的 \(R\) 无论如何严格优于 \(M\)，故永不玩 \(M\)。约简为 \(2\times2\) 博弈（\(\{U,D\}\times\{L,R\}\)）后，玩家 1 的 \(U\) 严格更优、玩家 2 玩 \(L\)，结果 \(S_1=U,S_2=L\)，收益 $(3,0)\(。\)N=2$, \(S_1=\{U,C,D\}\), \(S_2=\{L,M,R\}\); matrix below. No dominant strategy strictly better than all others for either player; but player 1's \(D\) is strictly better than \(C\) regardless, so never plays \(C\); player 2's \(R\) is strictly better than \(M\) regardless, so never plays \(M\). The reduced \(2\times2\) game (\(\{U,D\}\times\{L,R\}\)) gives player 1's \(U\) strictly better and player 2 plays \(L\); result \(S_1=U,S_2=L\), return $(3,0)$.

注记 10.2 / Remark 10.2 结果 $(3,0)$ 比某些结果（如 $(4,1)\(）Pareto 更差。故此处定义的均衡不同于此前的一般均衡。一般均衡模型中人人是价格接受者、问题中无策略，结果 Pareto 有效（第一福利定理）；但一旦移除决策的孤立性，所得均衡不必有效。The result \)(3,0)$ is Pareto inferior to some results such as $(4,1)$. So the equilibrium defined here differs from the general equilibrium before: in a general equilibrium model everyone is a price taker, no strategy is embedded, and the outcome is Pareto efficient (First Welfare Theorem); but once we remove isolation in decision making, the resulting equilibria are not necessarily efficient.

Definition 10.13（Pure strategy Nash equilibrium） 策略剖面 \(\hat s=(\hat s_1,\dots,\hat s_N)\) 是 \(G=(S_i,u_i)_{i=1}^N\) 的纯策略纳什均衡，当且仅当 \(\forall i\)，\(u_i(\hat s)\ge u_i(s_i,\hat s_{-i})\)（\(\forall s_i\in S_i\)），其中 \(\hat s_{-i}=(\hat s_1,\dots,\hat s_{i-1},\hat s_{i+1},\dots,\hat s_N)\)。直觉：相信他人用 \(\hat s\)，在均衡处你不能比 \(\hat s_i\) 做得更好，故不偏离；若对每个玩家都成立，则无人移动。A strategy profile \(\hat s=(\hat s_1,\dots,\hat s_N)\) is a pure strategy Nash equilibrium of \(G=(S_i,u_i)_{i=1}^N\) iff \(\forall i\), \(u_i(\hat s)\ge u_i(s_i,\hat s_{-i})\) (\(\forall s_i\in S_i\)), where \(\hat s_{-i}=(\hat s_1,\dots,\hat s_{i-1},\hat s_{i+1},\dots,\hat s_N)\). Intuition: confident others use \(\hat s\), at equilibrium you cannot do better than \(\hat s_i\), so won't deviate; if true for every player, no one moves.

Example 10.4（Nash equilibrium） \(N=2\)，\(S_1=S_2=\{L,H\}\)，收益矩阵如下。无占优策略，无法预测、也无法排除任何被占优的策略；但 \((H,L)\) 与 \((L,H)\) 都满足无人有偏离动机，故都是纯策略纳什均衡。\(N=2\), \(S_1=S_2=\{L,H\}\); matrix below. No dominant strategy, cannot predict or exclude any dominated strategy; but \((H,L)\) and \((L,H)\) both have no incentive to deviate, so both are pure strategy Nash equilibria.

Example 10.5（Being unpredictable is good） \(N=2\)，\(S_1=S_2=\{L,R\}\)，收益矩阵如下（\(\delta\in(0,1)\)）。这是点球场景：\(K\) 为踢球者、\(G\) 为守门员，\(L,R\) 为左右方向；\(\delta\) 越小踢球者右脚越强（即便守门员判对方向仍可能进球）。无论结果如何，至少一名玩家有偏离动机，故无纳什均衡。两名玩家都不应可预测；否则点球会有固定理性结果，与现实不符。要义：不可预测有时是好事。\(N=2\), \(S_1=S_2=\{L,R\}\); matrix below (\(\delta\in(0,1)\)). A soccer penalty scenario: \(K\) kicker, \(G\) goalie, \(L,R\) directions; smaller \(\delta\) means stronger right leg (even if the goalie reads the direction, a goal is still possible). Whatever the outcome, at least one player has an incentive to deviate, so there is no Nash equilibrium. Neither player should be predictable; otherwise the penalty game would have a fixed rational outcome, untrue in practice. Takeaway: being unpredictable is sometimes good.

Definition 10.14（Mixed strategy） 玩家 \(i\) 的混合策略是 \(s_i\in S_i\) 上的概率分布。假设各 \(S_i\) 有限。记 \(i\) 的混合策略集 \(M_i=\left\{m_i:S_i\to[0,1]\text{ s.t. }\sum_{s_i\in S_i}m_i(s_i)=1\right\}\supseteq S_i\)，其中 \(S_i\subseteq M_i\) 因 \(s_i\equiv(m_i(s_i)=1)\) 使每个纯策略是概率 1 的最简单混合策略。A mixed strategy for player \(i\) is a probability distribution over \(s_i\in S_i\). Assume each \(S_i\) finite. Let \(i\)'s set of mixed strategies \(M_i=\left\{m_i:S_i\to[0,1]\text{ s.t. }\sum_{s_i\in S_i}m_i(s_i)=1\right\}\supseteq S_i\), where \(S_i\subseteq M_i\) because \(s_i\equiv(m_i(s_i)=1)\) makes each pure strategy the simplest mixed strategy with probability 1.

Definition 10.15（Mixed strategy Nash equilibrium）& Theorem 10.4 Def 10.15：混合联合策略 \(\hat m\in M\) 是 \(G\) 的混合策略纳什均衡，当且仅当 \(\forall i\)、\(\forall m_i\in M_i\)，\(u_i(\hat m)\ge u_i(m_i,\hat m_{-i})\)。（Rmk 10.3：给定他人选 \(\hat m_{-i}\)，你无法通过偏离 \(\hat m_i\) 做得更好。）
Thm 10.4：设所有玩家有 VNM 效用，则以下等价：(a) \(\hat m\) 混合策略纳什均衡；(b) \(\forall i\)、\(\forall s_i\in S_i\) 使 \(\hat m_i(s_i)>0\) 有 \(u_i(\hat m)=u_i(s_i,\hat m_{-i})\)，且 \(\forall s_i\in S_i\)，\(u_i(\hat m)\ge u_i(s_i,\hat m_{-i})\)；(c) \(\forall i\)、\(\forall s_i\in S_i\)，\(u_i(\hat m)\ge u_i(s_i,\hat m_{-i})\)。（Rmk 10.4：均衡处混合时，须对任意两个有正概率的纯策略无差异、且对有正概率的纯策略与混合策略无差异。）Def 10.15: a mixed joint strategy \(\hat m\in M\) is a mixed strategy Nash equilibrium of \(G\) iff \(\forall i\), \(\forall m_i\in M_i\), \(u_i(\hat m)\ge u_i(m_i,\hat m_{-i})\). (Rmk 10.3: given others choose \(\hat m_{-i}\), you cannot do better by deviating from \(\hat m_i\).)
Thm 10.4: suppose all players have VNM utility; then the following are equivalent: (a) \(\hat m\) is a mixed strategy Nash equilibrium; (b) \(\forall i\), \(\forall s_i\in S_i\) with \(\hat m_i(s_i)>0\), \(u_i(\hat m)=u_i(s_i,\hat m_{-i})\), and \(\forall s_i\in S_i\), \(u_i(\hat m)\ge u_i(s_i,\hat m_{-i})\); (c) \(\forall i\), \(\forall s_i\in S_i\), \(u_i(\hat m)\ge u_i(s_i,\hat m_{-i})\). (Rmk 10.4: when mixing at equilibrium, you must be indifferent between any two pure strategies with strictly positive probabilities, and between any positive-probability pure strategy and the mixed strategy.)

证明 / Proof (Theorem 10.4)

$$u_i(m)=\sum_{s_i\in S_i}m_i(s_i)\cdot\underbrace{\sum_{s_{-i}\in S_{-i}}m_{-i}(s_{-i})u_i(s_i,s_{-i})}_{=u_i(s_i,m_{-i})}=\sum_{s_i\in S_i}m_i(s_i)\cdot u_i(s_i,m_{-i})$$

$$u_i(\tilde m_i,\hat m_{-i})-u_i(\hat m)=\frac{\hat m_i(s_i)}{1-\hat m_i(s_i)}\left(u_i(\hat m)-u_i(s_i,\hat m_{-i})\right)>0$$

Example 10.6（Revisit 10.4） 玩家 1 以 \(q\) 概率选 \(L\)，玩家 2 以 \(p\) 概率选 \(L\)。非退化混合均衡 \(p,q\in(0,1)\)，由 Thm 10.4 两玩家对 \(L,H\) 无差异。玩家 1：\(p\cdot0+(1-p)\cdot1=p\cdot2+(1-p)\cdot0\Rightarrow p=\tfrac13\)；玩家 2：\(q\cdot0+(1-q)\cdot1=q\cdot2+(1-q)\cdot0\Rightarrow q=\tfrac13\)。故 \(m_1=m_2=(\tfrac13,\tfrac23)\)，期望效用 \(u_1=u_2=\tfrac23<1\)，可见不可预测可能让双方更差。Player 1 chooses \(L\) with probability \(q\), player 2 chooses \(L\) with probability \(p\). For a non-degenerate mixed equilibrium \(p,q\in(0,1)\), by Thm 10.4 both are indifferent between \(L,H\). Player 1: \(p\cdot0+(1-p)\cdot1=p\cdot2+(1-p)\cdot0\Rightarrow p=\tfrac13\); player 2: \(q\cdot0+(1-q)\cdot1=q\cdot2+(1-q)\cdot0\Rightarrow q=\tfrac13\). So \(m_1=m_2=(\tfrac13,\tfrac23)\), expected utility \(u_1=u_2=\tfrac23<1\), showing unpredictability may make both worse off.

Example 10.7（Revisit 10.5） 守门员以 \(q\) 概率选 \(L\)、踢球者以 \(p\) 概率选 \(L\)。踢球者无差异：\(p(-1)+(1-p)1=p\cdot1+(1-p)(-\delta)\Rightarrow p=\tfrac{1+\delta}{3+\delta}\)；守门员无差异：\(q\cdot1+(1-q)(-1)=q(-1)+(1-q)\delta\Rightarrow q=\tfrac{1+\delta}{3+\delta}\)。期望效用 \(u_K=\tfrac{-(1+\delta)^2+4}{(3+\delta)^2}\)（关于 \(\delta\) 递减，\(\in(0,\tfrac13)\)）、\(u_G=\tfrac{(1+\delta)^2-4}{(3+\delta)^2}\)（关于 \(\delta\) 递增，\(\in(-\tfrac13,0)\)）。两个观察：不可预测对双方都好（最低期望效用仍高于 $-1$）；\(\delta\) 越小踢球者右脚越强，踢球者严格更优、守门员严格更差，且踢球者对 \(R\) 赋更高概率（用右脚）。The goalie chooses \(L\) with probability \(q\), the kicker with probability \(p\). Kicker indifferent: \(p(-1)+(1-p)1=p\cdot1+(1-p)(-\delta)\Rightarrow p=\tfrac{1+\delta}{3+\delta}\); goalie indifferent: \(q\cdot1+(1-q)(-1)=q(-1)+(1-q)\delta\Rightarrow q=\tfrac{1+\delta}{3+\delta}\). Expected utility \(u_K=\tfrac{-(1+\delta)^2+4}{(3+\delta)^2}\) (decreasing in \(\delta\), \(\in(0,\tfrac13)\)), \(u_G=\tfrac{(1+\delta)^2-4}{(3+\delta)^2}\) (increasing in \(\delta\), \(\in(-\tfrac13,0)\)). Two observations: unpredictability is good for both (the lowest expected utility is still above $-1$); smaller \(\delta\) means a stronger right leg, so the kicker is strictly better off and the goalie strictly worse off, and the kicker assigns higher probability to \(R\) (using his right leg).

Theorem 10.5（Existence of mixed strategy Nash equilibrium） 设所有玩家有 VNM 效用，则每个有限策略型博弈至少有一个混合策略纳什均衡。Suppose all players have VNM utility; then every finite strategic form game has at least one mixed strategy Nash equilibrium.

证明 / Proof (Theorem 10.5)

$$f_{ij}(m)=\frac{m_{ij}+\max\{0,(u_i(j,m_{-i})-u_i(m))\}}{1+\sum_{j'=1}^n\max\{0,(u_i(j',m_{-i})-u_i(m))\}}$$

$$\hat m_{ij}\underbrace{\sum_{j'=1}^n\max\{0,(u_i(j',\hat m_{-i})-u_i(\hat m))\}}_{\equiv c}=\max\{0,(u_i(j,\hat m_{-i})-u_i(\hat m))\}\tag{10.9}$$

$$\sum_{j=1}^n(u_i(j,\hat m_{-i})-u_i(\hat m))\max\{0,(u_i(j,\hat m_{-i})-u_i(\hat m))\}=0\tag{10.10}$$

Definition 10.16（Bayesian game）& Definition 10.17（Bayes' rule） Def 10.16：贝叶斯博弈 \(BG=(A_i,T_i,u_i,\mathbb{P})_{i=1}^N\) 由以下构成：行动集 \(A_i\)；类型集 \(T_i\)；VNM 效用 \(u_i:A\times T\to\mathbb{R}\)（\(A=\prod A_i\)、\(T=\prod T_i\)、\(t=(t_1,\dots,t_N)\in T\)）；类型上的共同概率分布 \(\mathbb{P}\in\Delta(T)\)（\(T\) 有限时 \(\mathbb{P}(t)>0\)、\(\sum_t\mathbb{P}(t)=1\)）。
Def 10.17（Bayes' rule）：\(T_i\) 有限，玩家 \(i\) 由先验经贝叶斯法则更新：\(\mathbb{P}(t_{-i}|t_i)\equiv\dfrac{\mathbb{P}(t_i,t_{-i})}{\sum_{t'_{-i}\in T_{-i}}\mathbb{P}(t_i,t'_{-i})}\)，得在自己类型实现为 \(t_i\) 时他人类型的条件分布。Def 10.16: a Bayesian game \(BG=(A_i,T_i,u_i,\mathbb{P})_{i=1}^N\) consists of: action set \(A_i\); type set \(T_i\); VNM utility \(u_i:A\times T\to\mathbb{R}\) (\(A=\prod A_i\), \(T=\prod T_i\), \(t=(t_1,\dots,t_N)\in T\)); a common probability distribution \(\mathbb{P}\in\Delta(T)\) over types (\(\mathbb{P}(t)>0\), \(\sum_t\mathbb{P}(t)=1\) when \(T\) finite).
Def 10.17 (Bayes' rule): \(T_i\) finite, player \(i\) updates the prior by Bayes' rule: \(\mathbb{P}(t_{-i}|t_i)\equiv\dfrac{\mathbb{P}(t_i,t_{-i})}{\sum_{t'_{-i}\in T_{-i}}\mathbb{P}(t_i,t'_{-i})}\), the conditional distribution of others' types once his own type is \(t_i\).

注记 10.5 / Remark 10.5 称 \(A_i\) 为行动集而非策略集，因把"策略"留给类型相依的行动计划：策略指示玩家在某类型下采取某（纯或混合）行动。此前收益已定、任何行动带来对应收益，故（纯/混合）行动本身即策略；现收益方案不明，玩家需对每个可能所处情形（类型）的指示，使指示（计划）而非行动成为直接影响收益的最高层级，故把类型相依计划称为策略。We call \(A_i\) action sets rather than strategy sets because we reserve "strategy" for a type-contingent plan of actions: a strategy instructs the player to take certain (pure or mixed) actions in a certain type. Before, payoffs were settled and any action brought a corresponding payoff, so (pure/mixed) actions themselves were strategies; now payoff schemes are unclear, so the player needs instructions for each scenario (type), making instructions (plans) — not actions — the highest level directly influencing payoff, so type-contingent plans are the strategies.

Definitions 10.18–10.21（pure / mixed / behavioral strategy & expected utility） Def 10.18（纯策略）：\(s_i:T_i\to A_i\)，故 \(S_i\equiv\{s_i:T_i\to A_i\}\)。（Rmk 10.6：纯策略即为自己类型的实现确定取某行动的一一映射；Rmk 10.7：\(A_i,T_i\) 有限时 \(\text{card}(S_i)=\text{card}(A_i)^{\text{card}(T_i)}\)。）
Def 10.19（混合策略）：\(s_i\in S_i\) 上的概率分布，\(M_i=\{m_i:S_i\to[0,1]\text{ s.t. }\sum_{s_i\in S_i}m_i(s_i)=1\}\supseteq S_i\)。（Rmk 10.8：此处混合策略≠动机例中"类型内混合行动"——后者是行为策略；混合策略是在知类型前对涉及所有类型的计划随机化。）
Def 10.20（行为策略）：\(b_i:A_i\times T_i\to[0,1]\) s.t. \(\sum_{a_i\in A_i}b_i(a_i|t_i)=1\)，\(B_i\) 为其集。（Rmk 10.9：\(b_i\) 对给定类型 \(t_i\) 给每个行动 \(a_i\) 赋概率；而 \(m_i\) 给所有类型的行动计划赋概率。）
Def 10.21（期望效用）：在类型 \(t_i\) 下取行动 \(a_i\) 的期望效用Def 10.18 (Pure strategy): \(s_i:T_i\to A_i\), so \(S_i\equiv\{s_i:T_i\to A_i\}\). (Rmk 10.6: a one-to-one mapping deciding an action for a realization of own type; Rmk 10.7: \(\text{card}(S_i)=\text{card}(A_i)^{\text{card}(T_i)}\) when finite.)
Def 10.19 (Mixed strategy): a distribution over \(s_i\in S_i\), \(M_i=\{m_i:S_i\to[0,1]\text{ s.t. }\sum_{s_i\in S_i}m_i(s_i)=1\}\supseteq S_i\). (Rmk 10.8: this mixed strategy \(\ne\) "mixing actions within a realized type" in the motivating example — that is a behavioral strategy; a mixed strategy randomizes over plans for all types before knowing the type.)
Def 10.20 (Behavioral strategy): \(b_i:A_i\times T_i\to[0,1]\) s.t. \(\sum_{a_i\in A_i}b_i(a_i|t_i)=1\), \(B_i\) its set. (Rmk 10.9: \(b_i\) assigns probability to each action \(a_i\) for a given type \(t_i\); \(m_i\) assigns probabilities to plans for all types.)
Def 10.21 (Expected utility): expected utility of taking action \(a_i\) in type \(t_i\):

Definition 10.22（Bayes Nash equilibrium） \((b^\star_1,\dots,b^\star_N)\in B\equiv\prod_{i=1}^N B_i\) 是 BNE，当且仅当 \(\forall i\)、\(\forall t_i\in T_i\)、\(\forall a_i\in A_i\) 使 \(b^\star_i(a_i|t_i)>0\)，\(V(a_i,b^\star_{-i}|t_i)\ge V(a'_i,b^\star_{-i}|t_i)\)（\(\forall a'_i\in A_i\)）。（Rmk 10.10：BNE 处每个玩家在每个类型情形下，所有被赋正概率的行动对他无差异，且无其他行动能生更高效用，故不偏离；对所有玩家、所有类型成立——即给定他人一切固定时无可获利偏离。）\((b^\star_1,\dots,b^\star_N)\in B\equiv\prod_{i=1}^N B_i\) is a BNE iff \(\forall i\), \(\forall t_i\in T_i\), \(\forall a_i\in A_i\) with \(b^\star_i(a_i|t_i)>0\), \(V(a_i,b^\star_{-i}|t_i)\ge V(a'_i,b^\star_{-i}|t_i)\) (\(\forall a'_i\in A_i\)). (Rmk 10.10: at a BNE, for each player in each type scenario, all positive-probability actions are indifferent and no other action gives higher utility, so no deviation; true for all players and all types — non-existence of profitable deviation given everything else fixed.)

Definitions 10.23–10.25（expected utility & equivalence） Def 10.23（纯策略剖面期望效用）：\(u_i(s)=\sum_{(t_1,\dots,t_N)\in\prod T_i}\mathbb{P}(t_1,\dots,t_N)\cdot\tilde u_i(s_1(t_1),\dots,s_N(t_N),t_1,\dots,t_N)\)（\(\tilde u_i\) 为类型 \(t\) 与行动下的收益；Rmk 10.11：\(G=(S_i,u_i)\) 即策略型博弈；Rmk 10.12：因类型剖面随机，故 \(u_i(s)\) 随机）。
Def 10.24（混合策略剖面期望效用）：\(U_i(m)=\sum_{s\in\prod S_i}m(s)\cdot u_i(s)\)，\(m(s)\equiv\prod_{j=1}^N m_j(s_j)\)。
Def 10.25（行为策略与混合策略等价）：\(b_i\) 与 \(m_i\) 等价，当且仅当 \(\forall a_i\in A_i,\forall t_i\in T_i\)，\(b_i(a_i|t_i)=\sum_{\{s_i\in S_i:s_i(t_i)=a_i\}}m_i(s_i)\)；\(b\) 与 \(m\) 等价若 \(b_i\) 对 \(\forall i\) 与 \(m_i\) 等价。（Rmk 10.13：\(m_i\) 唯一钉定 \(b_i\)。）Def 10.23 (Pure-profile expected utility): \(u_i(s)=\sum_{(t_1,\dots,t_N)\in\prod T_i}\mathbb{P}(t_1,\dots,t_N)\cdot\tilde u_i(s_1(t_1),\dots,s_N(t_N),t_1,\dots,t_N)\) (\(\tilde u_i\) payoff from type \(t\) and actions; Rmk 10.11: \(G=(S_i,u_i)\) is a strategic form game; Rmk 10.12: \(u_i(s)\) is random because the type profile is random).
Def 10.24 (Mixed-profile expected utility): \(U_i(m)=\sum_{s\in\prod S_i}m(s)\cdot u_i(s)\), \(m(s)\equiv\prod_{j=1}^N m_j(s_j)\).
Def 10.25 (Behavioral \(\equiv\) mixed): \(b_i\) equivalent to \(m_i\) iff \(\forall a_i\in A_i,\forall t_i\in T_i\), \(b_i(a_i|t_i)=\sum_{\{s_i\in S_i:s_i(t_i)=a_i\}}m_i(s_i)\); \(b\equiv m\) if \(b_i\equiv m_i\) for \(\forall i\). (Rmk 10.13: \(m_i\) uniquely pins down \(b_i\).)

Fact 10.1 & Theorem 10.6 Fact 10.1：若 \(b\in B\) 与 \(m\in M\) 等价，则二者诱导 \(A\times T\) 上（即每个 \((a_i,t_i)\) 对上）相同的概率分布。
Thm 10.6：若 \(b,m\) 等价，则对每个玩家 \(i\)，\(U_i(m)=\sum_{t_i\in T_i,a_i\in A_i}p_i(t_i)\cdot b_i(a_i|t_i)\cdot V(a_i,b_{-i}|t_i)\) (10.11)，其中 \(p_i(t_i)=\sum_{t_{-i}\in T_{-i}}\mathbb{P}(t_i,t_{-i})\)。Fact 10.1: if \(b\in B\) and \(m\in M\) are equivalent, they induce the same probability distribution over \(A\times T\) (over each \((a_i,t_i)\) pair).
Thm 10.6: if \(b,m\) equivalent, then for every player \(i\), \(U_i(m)=\sum_{t_i\in T_i,a_i\in A_i}p_i(t_i)\cdot b_i(a_i|t_i)\cdot V(a_i,b_{-i}|t_i)\) (10.11), where \(p_i(t_i)=\sum_{t_{-i}\in T_{-i}}\mathbb{P}(t_i,t_{-i})\).

证明 / Proof (Fact 10.1 & Theorem 10.6)

$$ > \begin{aligned} > &\sum_{t_i\in T_i,a_i\in A_i}p_i(t_i)b_i(a_i|t_i)V(a_i,b_{-i}|t_i)\\ > &=\sum_{t_i,a_i}b_i(a_i|t_i)\sum_{t}\mathbb{P}(t_1,\dots,t_N)\sum_{t_{-i},a_{-i}}b_{-i}(a_{-i}|t_{-i})u_i(a_i,a_{-i},t_i,t_{-i})\\ > &=\sum_{t\in T,a\in A}\left(\sum_{\{s\in S:s(t)=a\}}m(s)\right)\sum_{t}\mathbb{P}(t_1,\dots,t_N)u_i(a,t)\\ > &=\sum_{s\in S}m(s)\cdot u_i(s)=U_i(m).\quad\blacksquare > \end{aligned} > $$

Theorem 10.7（Mixed \(\equiv\) behavioral） 对任意玩家 \(i\)、任意 \(b_i\in B_i\)，由 \(m_i(s_i)=\prod_{t_i\in T_i}b_i(s_i(t_i)|t_i)\) (10.12)（\(\forall s_i\in S_i\)）定义的混合策略 \(m_i\in M_i\) 与 \(b_i\) 等价。（Rmk 10.14：\(m_i\) 由 (10.12) 唯一钉定，但可能有多个 \(m_i\) 等价于 \(b_i\)，见 Example 10.9。）For any player \(i\) and any \(b_i\in B_i\), the mixed strategy \(m_i\in M_i\) defined by \(m_i(s_i)=\prod_{t_i\in T_i}b_i(s_i(t_i)|t_i)\) (10.12) (\(\forall s_i\in S_i\)) is equivalent to \(b_i\). (Rmk 10.14: \(m_i\) is uniquely pinned by (10.12), but there may be multiple \(m_i\) equivalent to \(b_i\), see Example 10.9.)

证明 / Proof (Theorem 10.7)

Example 10.9（Multiple \(m_i\) equivalent to one \(b_i\)） \(T_i=\{t^L_i,t^R_i\}\)、\(A_i=\{L,R\}\)，\(b_i(k|t^k_i)=\tfrac23\)（\(k\in\{L,R\}\)）。\(b_i\) 等价于 \(m_1=(\tfrac29,\tfrac49,\tfrac19,\tfrac29)\) 与 \(m_2=(\tfrac13,\tfrac13,0,\tfrac13)\)（四元为四个纯策略 \(\{t^L_i\Rightarrow L,t^R_i\Rightarrow L\}\)、\(\{t^L_i\Rightarrow L,t^R_i\Rightarrow R\}\)、\(\{t^L_i\Rightarrow R,t^R_i\Rightarrow L\}\)、\(\{t^L_i\Rightarrow R,t^R_i\Rightarrow R\}\) 的概率），\(m_1,m_2\) 的任意凸组合也等价于 \(b_i\)。\(T_i=\{t^L_i,t^R_i\}\), \(A_i=\{L,R\}\), \(b_i(k|t^k_i)=\tfrac23\) (\(k\in\{L,R\}\)). \(b_i\) is equivalent to \(m_1=(\tfrac29,\tfrac49,\tfrac19,\tfrac29)\) and \(m_2=(\tfrac13,\tfrac13,0,\tfrac13)\) (the four entries are probabilities for the four pure strategies \(\{t^L_i\Rightarrow L,t^R_i\Rightarrow L\}\), \(\{t^L_i\Rightarrow L,t^R_i\Rightarrow R\}\), \(\{t^L_i\Rightarrow R,t^R_i\Rightarrow L\}\), \(\{t^L_i\Rightarrow R,t^R_i\Rightarrow R\}\)), and any convex combination of \(m_1,m_2\) is also equivalent to \(b_i\).

Theorem 10.8（NE of G \(\Leftrightarrow\) BNE of BG） 若 \(m^\star\in M\) 是 \(G\) 的 NE，则与 \(m^\star\) 等价的唯一 \(b^\star\in B\) 是 \(BG\) 的 BNE；反之，若 \(b^\star\in B\) 是 \(BG\) 的 BNE，则任何与 \(b^\star\) 等价的 \(m^\star\in M\) 是 \(G\) 的 NE。If \(m^\star\in M\) is a NE of \(G\), then the unique \(b^\star\in B\) equivalent to \(m^\star\) is a BNE of \(BG\); conversely, if \(b^\star\in B\) is a BNE of \(BG\), then any \(m^\star\in M\) equivalent to \(b^\star\) is a NE of \(G\).

证明 / Proof (Theorem 10.8)

$$\sum_{t_i\in T_i,a_i\in A_i}p_i(t_i)\left[b^\star_i(a_i|t_i)-b_i(a_i|t_i)\right]V(a_i,b^\star_{-i}|t_i)\ge0\tag{10.13}$$

Definition 10.26（Extensive form game）& Definition 10.27（Partition） Def 10.26：\(\Gamma\) 由以下构成：1. 有限玩家数 \(N\)；2. 行动集 \(A\)（含游戏中可能出现的所有行动）；3. 节点（历史）集 \(X\)：(a) 初始节点 \(x_0\in X\)，(b) \(X\setminus\{x_0\}\) 为 \(A\) 中行动的有限序列集，(c) 若序列在 \(X\setminus\{x_0\}\) 则其所有截断也在其中；4. 自然在 \(x_0\) 的行动集 \(A(x_0)\subseteq A\) 与分布 \(\pi\in\Delta(A(x_0))\)（自然只在 \(x_0\) 动一次）；5. \(\forall x\in X\setminus\{x_0\}\)，可用行动 \(A(x)=\{a\in A:(x,a)\in X\}\)；(a) 末端节点 \(E\equiv\{x\in X:A(x)=\emptyset\}\)，(b) 决策节点 \(D\equiv X\setminus(E\cup\{x_0\})\)；6. \(\iota:D\to\{1,\dots,N\}\) 指明在每决策节点轮到的玩家；(a) \(X_i\equiv\{x\in D:\iota(x)=i\}\)；7. 对每玩家 \(i\)，\(\mathcal{I}_i=\{I_{i,1},\dots,I_{i,k}\}\) 为 \(X_i\) 的划分，\(\forall x,x'\in I_{i,j}\) 玩家 \(i\) 不能区分（信息集）；8. \(u_i:E\to\mathbb{R}\) VNM 效用。
Def 10.27：划分性质 (a) \(\forall I\in\mathcal{I}_i,\forall x,x'\in I\)，\(A(x)=A(x')\)，故可记 \(A(I)\)；(b) \(\mathcal{I}=\cup_{i=1}^N\mathcal{I}_i\) 为 \(D\) 的划分；(c) \(\mathcal{I}(x)\) 为含 \(x\) 的划分元素。Def 10.26: \(\Gamma\) consists of: 1. finite \(N\) players; 2. action set \(A\) (all actions that can occur); 3. set of nodes (histories) \(X\): (a) initial node \(x_0\in X\), (b) \(X\setminus\{x_0\}\) a collection of finite sequences of actions in \(A\), (c) if a sequence is in \(X\setminus\{x_0\}\) all its truncations are too; 4. nature's action set \(A(x_0)\subseteq A\) at \(x_0\) with \(\pi\in\Delta(A(x_0))\) (nature moves once, at \(x_0\)); 5. \(\forall x\in X\setminus\{x_0\}\), available actions \(A(x)=\{a\in A:(x,a)\in X\}\); (a) end nodes \(E\equiv\{x\in X:A(x)=\emptyset\}\), (b) decision nodes \(D\equiv X\setminus(E\cup\{x_0\})\); 6. \(\iota:D\to\{1,\dots,N\}\) indicates the player at each decision node; (a) \(X_i\equiv\{x\in D:\iota(x)=i\}\); 7. for each \(i\), \(\mathcal{I}_i=\{I_{i,1},\dots,I_{i,k}\}\) a partition of \(X_i\), with \(\forall x,x'\in I_{i,j}\) indistinguishable to player \(i\) (information set); 8. \(u_i:E\to\mathbb{R}\) VNM utility.
Def 10.27: partition properties (a) \(\forall I\in\mathcal{I}_i,\forall x,x'\in I\), \(A(x)=A(x')\), so write \(A(I)\); (b) \(\mathcal{I}=\cup_{i=1}^N\mathcal{I}_i\) partitions \(D\); (c) \(\mathcal{I}(x)\) the element containing \(x\).

注记 10.15–10.16 / Remarks 10.15–10.16 10.15：游戏本质钉定每玩家唯一划分 \(\mathcal{I}_i\)；同一信息集 \(I_{i,j}\) 内节点对 \(i\) 包含完全相同信息、不可区分，故 \(i\) 在其中任一节点用同一行动计划。例：动机例博弈树中 \(\mathcal{I}_1=\{\{1a\},\{1b\}\}\)、\(\mathcal{I}_2=\{\{2c,2d\},\{2e,2f\}\}\)。10.16：策略型博弈涵盖扩展型与贝叶斯博弈；反之扩展型与贝叶斯博弈也可生成策略型博弈——任何复杂博弈中玩家总能在开局前定好相机计划并贯彻全程，故视作所有玩家在开局前同时选一计划（或计划的概率混合），即策略型博弈。10.15: the game's nature pins down each player's unique partition \(\mathcal{I}_i\); nodes in the same information set \(I_{i,j}\) contain exactly the same information to \(i\) and are indistinguishable, so \(i\) uses the same plan at any node within it. E.g. in the motivating example \(\mathcal{I}_1=\{\{1a\},\{1b\}\}\), \(\mathcal{I}_2=\{\{2c,2d\},\{2e,2f\}\}\). 10.16: the strategic form game encompasses extensive form and Bayesian games; conversely they can generate a strategic form game — in any complex game a player can make a contingent plan before the start and carry it out, so all players simultaneously choose a plan (or a probability-weighted mixture of plans) before the start, making it a strategic form game.

Definitions 10.28–10.34（strategies & behavioral NE） Def 10.28（纯策略）：\(s_i:\mathcal{I}_i\to A\) 使 \(s(I)\in A(I)\)。Def 10.29：\(S=\prod S_i\) 联合纯策略剖面集；给定 \(s\) 与自然行动 \(a\in A(x_0)\)，唯一末端节点 \(\eta(s,a)\)，\(U_i(s)=\sum_{a\in A(x_0)}\pi(a)u_i(\eta(s,a))\)。Def 10.30（混合策略）：\(m_i:S_i\to[0,1]\)，\(\sum m_i(s_i)=1\)。Def 10.31（行为策略）：\(\forall I\in\mathcal{I}_i,\forall a\in A(I)\)，\(b_i(a,I)\in[0,1]\)，\(\sum_{a'\in A(I)}b_i(a',I)=1\)。Def 10.32（末端节点分布）：\(e=(a_0,a_1,\dots,a_K)\) 时 \(\mathbb{P}(e|b)=\pi(a_0)\prod_{k=1}^K b_{\iota(a_{Def 10.33（行为剖面期望效用）：\(V_i(b)\equiv\sum_{e\in E}\mathbb{P}(e|b)u_i(e)\) (10.23)。Def 10.34（行为策略纳什均衡）：\(b^\star\) 是 BNE 当且仅当 \(\forall i\)，\(V_i(b^\star)\ge V_i(b'_i,b^\star_{-i})\)（\(\forall b'_i\in B_i\)）。Def 10.28 (Pure strategy): \(s_i:\mathcal{I}_i\to A\) with \(s(I)\in A(I)\). Def 10.29: \(S=\prod S_i\) joint pure strategy profiles; given \(s\) and nature's \(a\in A(x_0)\), the unique end node \(\eta(s,a)\), \(U_i(s)=\sum_{a\in A(x_0)}\pi(a)u_i(\eta(s,a))\). Def 10.30 (Mixed): \(m_i:S_i\to[0,1]\), \(\sum m_i(s_i)=1\). Def 10.31 (Behavioral): \(\forall I\in\mathcal{I}_i,\forall a\in A(I)\), \(b_i(a,I)\in[0,1]\), \(\sum_{a'\in A(I)}b_i(a',I)=1\). Def 10.32 (Distribution over end nodes): for \(e=(a_0,a_1,\dots,a_K)\), \(\mathbb{P}(e|b)=\pi(a_0)\prod_{k=1}^K b_{\iota(a_{Def 10.33 (Expected utility of profile): \(V_i(b)\equiv\sum_{e\in E}\mathbb{P}(e|b)u_i(e)\) (10.23). Def 10.34 (Behavioral strategy NE): \(b^\star\) is a BNE iff \(\forall i\), \(V_i(b^\star)\ge V_i(b'_i,b^\star_{-i})\) (\(\forall b'_i\in B_i\)).

Definition 10.35（Perfect information）& Definition 10.36（Backward induction strategy profile）& Theorem 10.9 Def 10.35：\(\Gamma\) 有完美信息当且仅当 \(\mathcal{I}(x)=\{x\}\)（\(\forall x\in D\)）（Rmk 10.18：任何玩家对自己所处节点无混淆，可向前看）。Def 10.36（逆向归纳策略剖面）：完美信息博弈中，\(s\in S\) 是逆向归纳策略剖面，当且仅当 \(\forall x\in D\)，玩家 \(i=\iota(x)\) 在给定所有人（含自己）于 \(x\) 之后按 \(s\) 行动时，不能比在 \(x\) 处选 \(s_i(x)\) 做得更好（Rmk 10.19：逆向归纳策略剖面是纯策略纳什均衡，且完美信息博弈中总存在）。Thm 10.9：任何有限完美信息扩展型博弈中，每个逆向归纳策略剖面都是纳什均衡；但并非每个纳什均衡都是逆向归纳策略剖面（Rmk 10.20：逆向归纳策略剖面是纳什均衡的子集，即纳什均衡的"精炼"）。Def 10.35: \(\Gamma\) has perfect information iff \(\mathcal{I}(x)=\{x\}\) (\(\forall x\in D\)) (Rmk 10.18: no confusion about which node, so look-ahead possible). Def 10.36 (Backward induction strategy profile): in a perfect-info game, \(s\in S\) is a backward induction strategy profile iff \(\forall x\in D\), player \(i=\iota(x)\) cannot do better than choosing \(s_i(x)\) at \(x\) given all players (incl. himself) follow \(s\) after \(x\) (Rmk 10.19: it is a pure strategy Nash equilibrium and always exists in perfect-info games). Thm 10.9: in any finite perfect-info extensive form game, every backward induction strategy profile is a Nash equilibrium, but not every Nash equilibrium is a backward induction strategy profile (Rmk 10.20: backward induction profiles are a subset — a "refinement" — of Nash equilibria).

Definition 10.37（Subgame）& Definition 10.38（Expected utility of subgame） Def 10.37：节点 \(x\in X\setminus E\) 定义一个子博弈，当且仅当：要么 \(x=x_0\)（平凡子博弈）；要么 \(x\in D\) 且 \(\mathcal{I}(x)=\{x\}\)（\(x\) 信息集为单点）且 \(\forall y,z\in D\)，若 \(z\in\mathcal{I}(y)\) 且 \(y\ge x\)，则 \(z\ge x\)。\(\Gamma_x\) 为由 \(x\) 定义的同一博弈（含 \(x\) 之后所有节点与信息结构、收益）。（图示：\(x\) 独立"站立"——一旦到达 \(x\)，所有玩家都知到达了，且不经 \(x\) 不可达，故 \(x\) 是独立子博弈。Rmk 10.21：完美信息博弈中每个决策节点都定义子博弈。）
Def 10.38：改记 \(V_i(b|x_0)\equiv V_i(b)\)。对定义子博弈 \(\Gamma_x\) 的 \(x\)，整局 \(b\in B\) 诱导出 \(\Gamma_x\) 的行为策略剖面，\(V_i(b|x)\) 为 \(i\) 在 \(\Gamma_x\) 的期望收益。Def 10.37: a node \(x\in X\setminus E\) defines a subgame iff: either \(x=x_0\) (trivial); or \(x\in D\) and \(\mathcal{I}(x)=\{x\}\) (\(x\)'s info set a singleton) and \(\forall y,z\in D\), if \(z\in\mathcal{I}(y)\) and \(y\ge x\) then \(z\ge x\). \(\Gamma_x\) is the same game defined by \(x\) (all nodes after \(x\) with their info structure and payoffs). (Graphically: \(x\) stands "alone" — once reached, all players know it, and it cannot be reached without passing \(x\), so \(x\) is an independent subgame. Rmk 10.21: in a perfect-info game every decision node defines a subgame.)
Def 10.38: rewrite \(V_i(b|x_0)\equiv V_i(b)\). For \(x\) defining a subgame \(\Gamma_x\), a whole-game \(b\in B\) induces a behavioral profile for \(\Gamma_x\), and \(V_i(b|x)\) is \(i\)'s expected payoff in \(\Gamma_x\).

Definition 10.39（Subgame perfect Nash equilibrium） \(b^\star\in B\) 是子博弈完美纳什均衡（SPNE），当且仅当 \(b^\star\) 在 \(\Gamma\) 的每个子博弈中都诱导出纳什均衡。求解法：1. 找出所有子博弈；2. 找不含其他子博弈的"最深子博弈"；3. 求最深子博弈的（部分/全部）纳什均衡（可能多个，各自往下进行，可能导致整局不同均衡）；4. 把最深子博弈各玩家期望收益当作末端收益，代入更高层子博弈；5. 迭代直至整局。（Rmk 10.22：自底向上求解，某种意义上是逆向归纳。）\(b^\star\in B\) is a subgame perfect Nash equilibrium (SPNE) iff \(b^\star\) induces a Nash equilibrium in every subgame of \(\Gamma\). Method: 1. identify all subgames; 2. find the "deepest" subgames containing no subgames themselves; 3. compute (some/all) Nash equilibria of the deepest subgames (possibly multiple, each proceeding downward, possibly leading to different whole-game equilibria); 4. treat the deepest subgames' expected payoffs as end payoffs and plug into a higher-level subgame; 5. iterate to the whole game. (Rmk 10.22: solve from bottom to top, in a sense backward induction.)

Example 10.17（SPNE ⇏ backward induction，已转述 / paraphrased） 玩家 1 顶端选 L/M/R：L\(\to(0,5)\)，M\(\to2a\)、R\(\to2b\)（玩家 2 信息集 \(\{2a,2b\}\)）。红箭头是纯策略 NE，且唯一子博弈是整局，故是 SPNE；但不对应逆向归纳剖面：玩家 2 在 \(\{2a,2b\}\) 以 \((\tfrac12,\tfrac12)\) 混合 L,R 严格占优选 M（期望 2 vs 1），使玩家 1 对 M,R 无差异；若玩家 1 混合 M,R\((\tfrac12,\tfrac12)\) 得期望 2>0，不选 L。故向前看不让玩家 2 以概率 1 选 M，SPNE 不对应逆向归纳剖面。Player 1 at top L/M/R: L\(\to(0,5)\), M\(\to2a\), R\(\to2b\) (player 2's info set \(\{2a,2b\}\)). The red arrows are a pure NE, and the only subgame is the whole game, so it is a SPNE; but it doesn't correspond to a backward induction profile: player 2 at \(\{2a,2b\}\) mixing L,R by \((\tfrac12,\tfrac12)\) strictly dominates M (payoff 2 vs 1), making player 1 indifferent between M,R; if player 1 mixes M,R by \((\tfrac12,\tfrac12)\) he gets 2>0, so won't choose L. So look-ahead won't let player 2 choose M with probability 1, and the SPNE doesn't correspond to a backward induction profile.

Definitions 10.40–10.45（beliefs, assessment, sequential rationality） Def 10.40（信念系统）：\(p:D\to[0,1]\) 使 \(\forall I\in\mathcal{I}_i\)，\(\sum_{x\in I}p(x)=1\)。Def 10.41（评估）：\((p,b)\) 为评估。Def 10.42（从 \(x\) 到达 \(e\) 的概率）：\(\mathbb{P}(e|b,x)=b_{\iota(x)}(a_0,\mathcal{I}(x))\prod_{k=1}^K b_{\iota(a_{Def 10.43（从 \(x\) 起的期望收益）：\(V_i(b|x)=\sum_{e\in E:e\ge x}\mathbb{P}(e|b,x)u_i(e)\)。Def 10.44（从信息集 \(I\) 起的期望收益）：\(V_i(p,b|I)=\sum_{x\in I}p(x)V_i(b|x)\)（信念系统 \(p\) 给出 \(I\) 内节点分布）。Def 10.45（序贯理性）：\((p,b)\) 序贯理性当且仅当 \(\forall i,\forall I\in\mathcal{I}_i\)，\(V_i(p,b|I)\ge V_i(p,b'_i,b_{-i}|I)\)（\(\forall b'_i\in B_i\)）——即在任何信息集都无单边偏离获益。Def 10.40 (System of beliefs): \(p:D\to[0,1]\) with \(\sum_{x\in I}p(x)=1\) for \(\forall I\in\mathcal{I}_i\). Def 10.41 (Assessment): \((p,b)\) is an assessment. Def 10.42 (Prob of reaching \(e\) from \(x\)): \(\mathbb{P}(e|b,x)=b_{\iota(x)}(a_0,\mathcal{I}(x))\prod_{k=1}^K b_{\iota(a_{Def 10.43 (Expected payoff from \(x\)): \(V_i(b|x)=\sum_{e\in E:e\ge x}\mathbb{P}(e|b,x)u_i(e)\). Def 10.44 (Expected payoff from info set \(I\)): \(V_i(p,b|I)=\sum_{x\in I}p(x)V_i(b|x)\) (the belief system \(p\) gives the distribution over nodes in \(I\)). Def 10.45 (Sequentially rational): \((p,b)\) is sequentially rational iff \(\forall i,\forall I\in\mathcal{I}_i\), \(V_i(p,b|I)\ge V_i(p,b'_i,b_{-i}|I)\) (\(\forall b'_i\in B_i\)) — no unilateral deviation benefits at any info set.

Definitions 10.46–10.49（completely mixed & Bayes' rule） Def 10.46（完全混合）：\(b\) 完全混合当且仅当 \(\forall i,\forall I\in\mathcal{I}_i,\forall a\in A(I)\)，\(b_i(a,I)>0\)（则每节点正概率被达）。Def 10.47（到达 \(x\) 的概率）：\(x=(a_0,\dots,a_K)\) 时 \(\mathbb{P}(x|b)=\pi(a_0)\prod_{k=1}^K b_{\iota(a_{Def 10.48（到达 \(I\) 的概率）：\(\mathbb{P}(I|b)=\sum_{x\in I}\mathbb{P}(x|b)\)。Def 10.49（评估的贝叶斯法则）：\((p,b)\) 满足贝叶斯法则当且仅当 \(\forall I\in\mathcal{I}\) 使 \(\mathbb{P}(I|b)>0\)，\(p(x)=\dfrac{\mathbb{P}(x|b)}{\mathbb{P}(I|b)}\)（\(\forall x\in I\)）(10.24)。Def 10.46 (Completely mixed): \(b\) is completely mixed iff \(\forall i,\forall I\in\mathcal{I}_i,\forall a\in A(I)\), \(b_i(a,I)>0\) (then every node is reached with positive prob). Def 10.47 (Prob of reaching \(x\)): for \(x=(a_0,\dots,a_K)\), \(\mathbb{P}(x|b)=\pi(a_0)\prod_{k=1}^K b_{\iota(a_{Def 10.48 (Prob of reaching \(I\)): \(\mathbb{P}(I|b)=\sum_{x\in I}\mathbb{P}(x|b)\). Def 10.49 (Bayes' rule for an assessment): \((p,b)\) satisfies Bayes' rule iff \(\forall I\in\mathcal{I}\) with \(\mathbb{P}(I|b)>0\), \(p(x)=\dfrac{\mathbb{P}(x|b)}{\mathbb{P}(I|b)}\) (\(\forall x\in I\)) (10.24).

注记 10.24–10.25 / Remarks 10.24–10.25 10.24：给定 \(b\)，满足贝叶斯法则的评估 \((p,b)\) 由 (10.24) 唯一钉定。10.25：\(b\) 完全混合时贝叶斯法则约束每个节点/信息集；但 \(b\) 非完全混合时，可能有 0 概率被达的信息集，贝叶斯法则对其无约束，可任意赋信念——这正是 Example 10.18 之问题源。10.24: given \(b\), the assessment \((p,b)\) satisfying Bayes' rule is uniquely pinned by (10.24). 10.25: if \(b\) is completely mixed, Bayes' rule constrains every node/info set; but if not, some info set may have 0 probability of being reached, where Bayes' rule imposes no restriction and beliefs can be assigned arbitrarily — the source of the issue in Example 10.18.

Definitions 10.50–10.52（convergence, consistency, sequential equilibrium） Def 10.50（逐点收敛）：\(b^n\to b\) 当且仅当 \(\forall i,I,a\)，\(b^n_i(a,I)\to b_i(a,I)\)；\(p^n\to p\) 当且仅当 \(\forall x\in D\)，\(p^n(x)\to p(x)\)。Def 10.51（一致性）：\((p,b)\) 一致当且仅当 \(\exists\{(p^n,b^n)\}\) 使 \(p^n\to p\)、\(b^n\to b\)、\(\forall n\) \(b^n\) 完全混合且 \((p^n,b^n)\) 满足贝叶斯法则。Def 10.52（序贯均衡）：\((p,b)\) 序贯均衡当且仅当 \((p,b)\) 一致且序贯理性。Def 10.50 (Point-wise convergence): \(b^n\to b\) iff \(\forall i,I,a\), \(b^n_i(a,I)\to b_i(a,I)\); \(p^n\to p\) iff \(\forall x\in D\), \(p^n(x)\to p(x)\). Def 10.51 (Consistency): \((p,b)\) is consistent iff \(\exists\{(p^n,b^n)\}\) with \(p^n\to p\), \(b^n\to b\), every \(b^n\) completely mixed, and \((p^n,b^n)\) satisfying Bayes' rule. Def 10.52 (Sequential equilibrium): \((p,b)\) is a sequential equilibrium iff it is consistent and sequentially rational.

三组关系 / Three relations & Theorems 10.11–10.12 1. 序贯理性＋贝叶斯法则 ⇏ 向前看能力（逆向归纳剖面）；有时 ⇒ SPNE（Ex 10.17），有时 ⇏ SPNE（Ex 10.18）。2. 序贯理性＋贝叶斯法则＋完全混合 ⇒ 向前看能力 ⇒ SPNE。3. 序贯理性＋一致性 ⇒ 序贯均衡 ⇒ 向前看能力 ⇒ SPNE。
Thm 10.11：若 \((p,b)\) 是 \(\Gamma\) 的序贯均衡，则 \(b\) 是 \(\Gamma\) 的 SPNE；若 \(\Gamma\) 完美信息且 \(b\) 只含纯策略，则 \(b\) 是逆向归纳剖面。Thm 10.12：有完美回忆的有限博弈 \(\Gamma\) 至少有一个序贯均衡。1. Sequentially rational + Bayes' rule ⇏ look-ahead capability (backward induction profile); sometimes ⇒ SPNE (Ex 10.17), sometimes ⇏ SPNE (Ex 10.18). 2. Sequentially rational + Bayes' rule + completely mixing ⇒ look-ahead ⇒ SPNE. 3. Sequentially rational + consistency ⇒ sequential equilibrium ⇒ look-ahead ⇒ SPNE.
Thm 10.11: if \((p,b)\) is a sequential equilibrium of \(\Gamma\), then \(b\) is a SPNE; if \(\Gamma\) has perfect information and \(b\) contains only pure strategies, \(b\) is a backward induction profile. Thm 10.12: a finite game \(\Gamma\) with perfect recall has at least one sequential equilibrium.