经济代写|博弈论代写Game Theory代考|ECON3050

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|Backward induction, Kuhn’s Theorem

Let $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ be an extensive form game with perfect information. Recall that $A(\varnothing)$ is the set of allowed initial actions for player $i=P(\varnothing)$. For each $b \in A(\varnothing)$, let $s^{G(b)}$ be some strategy profile for the subgame $G(b)$. Given some $a \in A(\varnothing)$, we denote by $s^a$ the strategy profile for $G$ in which player $i=P(\varnothing)$ chooses the initial action $a$, and for each action $b \in A(\varnothing)$ the subgame $G(b)$ is played according to $s^{G(b)}$. That is, $s_i^a(\varnothing)=a$ and for every player $j, b \in A(\varnothing)$ and $b h \in H \backslash Z, s_j^a(b h)=s_j^{G(b)}(h)$.
Lemma 2.11 (Backward Induction). Let $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ be a finite extensive form game with perfect information. Assume that for each $b \in A(\varnothing)$ the subgame $G(b)$ has a subgame perfect equilibrium $s^{G(b)}$. Let $i=P(\varnothing)$ and let a be the $>_i$-maximizer over $A(\varnothing)$ of $o_a\left(s^{G(a)}\right)$. Then $s^a$ is a subgame perfect equilibrium of $G$.

Proof. By the one deviation principle, we only need to check that $s^a$ does not have deviations that differ at a single history. So let $s$ differ from $s^a$ at a single history $h$.

If $h$ is the empty history then $s=s^{G(b)}$ for $b=s_i(\varnothing)$. In this case $o\left(s^a\right)>_i o(s)=o_b\left(s^{G(b)}\right)$, by the definition of $a$ as the maximizer of $o_a\left(s^{G(a)}\right)$.

Otherwise, $h$ is equal to $b h^{\prime}$ for some $b \in A(\varnothing)$ and $h^{\prime} \in H_b$, and $o(s)=o_b(s)$. But since $s^a$ is a subgame perfect equilibrium when restricted to $G(b)$ there are no profitable deviations, and the proof is complete.
Kuhn [22] proved the following theorem.
Theorem $2.12$ (Kuhn, 1953). Every finite extensive form game with perfect information has a subgame perfect equilibrium.

Given a game $G$ with allowed histories $H$, denote by $\ell(G)$ the maximal length of any history in $H$.

Proof of Theorem 2.12. We prove the claim by induction on $\ell(G)$. For $\ell(G)=0$ the claim is immediate, since the trivial strategy profile is an equilibrium, and there are no proper subgames. Assume we have proved the claim for all games $G$ with $\ell(G)<n$.

Let $\ell(G)=n$, and denote $i=P(\varnothing)$. For each $b \in A(\varnothing)$, let $s^{G(b)}$ be some subgame perfect equilibrium of $G(b)$. These exist by our inductive assumption, as $\ell(G(b))<n$.

Let $a^* \in A(\varnothing)$ be a $\leq_i$-maximizer of $o\left(s^{a^}\right)$. Then by the Backward Induction Lemma $s^{a^}$ is a subgame perfect equilibrium of $G$, and our proof is concluded.

经济代写|博弈论代写Game Theory代考|Classical examples

• Extensive form game with perfect information. Let $G=\left(N, A, H, P,\left{u_i\right}_{i \in N}\right)$ be an extensive form game with perfect information, where, instead of the usual outcomes and preferences, each player has a utility function $u_i: Z \rightarrow \mathbb{R}$ that assigns her a utility at each terminal node. Let $G^{\prime}$ be the strategic form game given by $G^{\prime}=\left(N^{\prime},\left{S_i\right}_{i \in N},\left{u_i\right}_{i \in \mathbb{N}}\right)$, where
$-N^{\prime}=N$.
• $S_i$ is the set of $G$-strategies of player $i$.
• For every $s \in S, u_i(s)$ is the utility player $i$ gets in $G$ at the terminal node at which the game arrive when players play the strategy profile $s$.

We have thus done nothing more than having written the same game in a different form. Note, however, that not every game in strategic form can be written as an extensive form game with perfect information.

Exercise 3.1. Show that $s \in S$ is a Nash equilibrium of $G$ iff it is a Nash equilibrium of $G^{\prime}$.

Note that a disadvantage of the strategic form is that there is no natural way to define subgames or subgame perfect equilibria.

• Matching pennies. In this game, and in the next few, there will be two players: a row player (R) and a column player (C). We will represent the game as a payoff matrix, showing for each strategy profile $s=\left(s_R, s_C\right)$ the payoffs $u_R(s), u_C(s)$ of the row player and the column player.

博弈论代考

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