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# 经济代写| Four different best-response functions 微观经济学代写

## 经济代写

4.4. Four different best-response functions. Depending on mixing or not mixing the strategy set and/or the set of states of the world, we modify definition II.13:

DEFINITION II.22 (BEST-RESPONSE FUNCTIONS). Given $\Delta=(S, W, u)$, we distinguish four best-response functions:
\begin{aligned} s^{R, W} &: W \rightarrow 2^{S}, \text { given by } s^{R, W}(w):=\arg \max {s \in S} u(s, w) \ \sigma^{R, W} &: W \rightarrow 2^{\Sigma}, \text { given by } \sigma^{R, W}(w):=\arg \max {\sigma \in \Sigma} u(\sigma, w) \ s^{R, \Omega} &: \Omega \rightarrow 2^{S}, \text { given by } s^{R, \Omega}(\omega):=\arg \max {s \in S} u(s, \omega) \text {, and } \ \sigma^{R, \Omega} &: \Omega \rightarrow 2^{\Sigma}, \text { given by } \sigma^{R, \Omega}(\omega):=\arg \max {\sigma \in \Sigma} u(\sigma, \omega) \end{aligned}
If there is no danger of confusion, we stick to the simpler $s^{R}$ or $\sigma^{R}$ instead of $s^{R, W}$ etc.
EXERCISE II.11. Complete the sentence: $\sigma \in \sigma^{R, W}(w)$ implies $\sigma(s)=0$ for all
In line with lemma II.1, we obtain the following results:
THEOREM II.2. Let $\Delta=(S, W, u)$ be a decision situation in strategic form. We have

• $\sigma \in \Sigma$ and $\sum_{s \in s^{R, \Omega}(\omega)} \sigma(s)=1$ imply $\sigma \in \sigma^{R, \Omega}(\omega)$ and
• $\sigma \in \sigma^{R, \Omega}(\omega)$ implies $s \in s^{R, \Omega}(\omega)$ for all $s \in S$ with $\sigma(s)>0$.
These implications continue to hold for $W$ and $w$ rather than $\Omega$ and $\omega$.
Best-response functions $\sigma^{R, \Omega}$ can be depicted graphically. Consider, for example, the decision matrix
23
1. R.ATIONALIZABILITY
FIGURE TI.5. The best-response function
Let $\omega:=w\left(w \omega_{1}\right)$ be the probability of w w. We have b $_{1} \in s^{R, \Omega}$ in case of
$$\omega \cdot 4+(1-\omega) \cdot 1 \geq \omega \cdot 1+1+2$$
i.e., if $\omega \geq \frac{1}{4}$ holds. Remember that the best-response function is $\sigma$, $R$. $\Omega \rightarrow 2^{\Sigma}$. For $\omega \neq \frac{1}{4}$, there is exactly one best strategy, $\sigma=0$ (standing for $\left.\sigma=(0,1)=s_{2}\right)$ or $\sigma=1\left(\right.$ standing for $\left.\sigma=(1,0)=s_{1}\right)$, while $\omega=\frac{1}{4}$ implies that every pure strategy and hence every mixed strategy is best. We obtain
$$\sigma^{R, \Omega}(\omega)= \begin{cases}1, & \omega>\frac{1}{4} \ {[0,1],} & \omega=\frac{1}{4} \ 0, & \omega<\frac{1}{4}\end{cases}$$
and the graph given in figure II.5.
EXERCISE II.12. Sketch the best-response function $\sigma^{R, \Omega}$ for

4.4.四种不同的最佳响应函数。根据混合或不混合策略集和/或世界状态集，我们修改定义 II.13：

$$\开始{对齐} s^{R, W} &: W \rightarrow 2^{S}, \text { 由 } s^{R, W}(w):=\arg \max {s \in S} u(s , w) \ \sigma^{R, W} &: W \rightarrow 2^{\Sigma}, \text { 由 } \sigma^{R, W}(w):=\arg \max {\sigma \in \西格玛} u(\sigma, w) \ s^{R, \Omega} &: \Omega \rightarrow 2^{S}, \text { 由 } s^{R, \Omega}(\omega):=\arg \max {s \in S } u(s, \omega) \text {, 和 } \ \sigma^{R, \Omega} &: \Omega \rightarrow 2^{\Sigma}, \text { 由 } \sigma^{R, \Omega}(\omega):=\arg \max {\ sigma \in \Sigma} u(\sigma, \omega) \end{对齐}$$

• $\sigma \in \Sigma$ 和 $\sum_{s \in s^{R, \Omega}(\omega)} \sigma(s)=1$ 暗示 $\sigma \in \sigma^{R, \Omega}(\omega)$ 和