经济代写|博弈论代写Game Theory代考|Equilibria

An equilibrium of $\Gamma=\Gamma\left(u_i \mid i \in N\right)$ is an equilibrium of the utility measure $U$ as in (41). The joint strategic choice $\mathbf{x} \in \mathfrak{X}$ is thus a gain equilibrium if no player has an utility incentive to switch to another strategy, i.e.,
$u_i(\mathbf{x}) \geq u_i(\mathbf{y})$ holds for all $i \in N$ and $\mathbf{y} \in \mathcal{F}i(\mathbf{x})$. Completely analogously, a cost equilibrium is defined via the reverse condition: $u_i(\mathbf{x}) \leq u_i(\mathbf{y})$ holds for all $i \in N$ and $\mathbf{y} \in \mathcal{F}_i(\mathbf{x})$. Aggregated utilities. There is another important view on equilibria. Given the state $\mathbf{x} \in \mathfrak{X}$, imagine that each player $i \in N$ considers an alternative $y_i$ to its current strategy $x_i$. The aggregated sum of the resulting utility values is $$G(\mathbf{x}, \mathbf{y})=\sum{i \in N} u\left(\mathbf{x}{-i}\left(y_i\right)\right) \quad\left(\mathbf{y}=\left(y_i \mid y_i \in X_i\right)\right) .$$ LEMMA 6.1. $\mathrm{x} \subset \mathfrak{X}$ is a gain cquilibrium of $\Gamma\left(u_i \mid i \subset N\right)$ if and only if $G(\mathbf{x}, \mathbf{y}) \leq G(\mathbf{x}, \mathbf{x}) \quad$ holds for all $\mathbf{y} \in \mathfrak{X}$. Proof. If $\mathbf{x}$ is a gain equilibrium and $\mathbf{y}=\left(y_i \mid i \in N\right) \in \mathfrak{X}$, we have $u_i(\mathbf{x}) \geq u\left(\mathbf{x}{-i}\left(y_i\right)\right) \quad$ for all $y_i \in X_i$,
which implies $G(\mathbf{x}, \mathbf{x}) \geq G(\mathbf{x}, \mathbf{y})$. Conversely, if $\mathbf{x}$ is not a gain equilibrium, there is an $i \in N$ and a $y \in X_i$ such that
$$0<u_i\left(\mathbf{x}{-i}(y)\right)-u_i(\mathbf{x})=G\left(\mathbf{x}, \mathbf{x}_i(y)\right)-G(\mathbf{x}, \mathbf{x}) .$$ which means that $\mathbf{y}=\mathbf{x}{-i}(y) \in \mathfrak{X}$ violates the inequality.
Lemma $6.1$ reduces the quest for an equilibrium to the quest for a $\mathbf{x} \in \mathfrak{X}$ that maximizes the associated component function $g^{\mathbf{x}}: \mathfrak{X} \rightarrow \mathbb{R}$ of the aggregated utility measure $G$ with the values
$$g^{\mathbf{x}}(\mathbf{y})=G(\mathbf{x}, \mathbf{y})$$

经济代写|博弈论代写Game Theory代考|Randomization of matrix games

An $n$-person game $\Gamma=\left(u_i \mid i \in N\right)$ is said to be a (generalized) matrix game if all individual strategy sets $X_i$ are finite.

For a motivation of the terminology, assume $N={1, \ldots, n}$ and think of the sets $X_i$ as index sets for the coordinates of a multidimensional matrix $U$. A particular index vector
$$\mathbf{x}=\left(x_1, \ldots, x_i, \ldots, x_n\right) \in X_1 \times \cdots \times X_i \times \cdots \times X_n(=\mathfrak{X})$$
thus specifies a position in $U$ with the $n$-dimensional coordinate entry
$$U_{\mathbf{x}}=\left(u_1(\mathbf{x}), \ldots, u_i(\mathbf{x}), \ldots, u_n(\mathbf{x})\right) \in \mathbb{R}^n .$$
Let us now change the rules of the matrix game $\Gamma$ in the following way:
(R) Each player $i \in N$ chooses a probability distribution $p^{(i)}$ on $X_i$ and selects the element $x \in X_i$ with probability $p_x^{(i)}$.

Under rule (R), the players are really playing the related $n$-person game $\bar{\Gamma}=\left(\bar{u}i \mid i \in N\right)$ with resource sets $P_i$ and utility functions $\bar{u}_i$, where (1) $P_i$ is the set of all probability distributions on $X_i$. (2) $\bar{u}_i(p)$ is the expected value of $u_i$ relative to the joint probability distribution $p=\left(p^{(i)} \mid i \in N\right)$ of the players. The $n$-person game $\bar{\Gamma}$ is the randomization of the matrix game $\Gamma$. The expected total utility value is $$\bar{u}_i\left(p^{(1)}, \ldots, p^{(n)}\right)=\sum{x_1 \in X_1} \cdots \sum_{x_n \in X_n} u_i\left(x_1, \ldots, x_n\right) p_{x_1}^{(1)} \cdots p_{x_n}^{(n)}$$

经济代写|博弈论代写博弈论代考|均衡

$\Gamma=\Gamma\left(u_i \mid i \in N\right)$的均衡是效用度量$U$的均衡，如(41)所示。因此，如果没有参与人有转换到另一种策略的效用动机，即
$u_i(\mathbf{x}) \geq u_i(\mathbf{y})$适用于所有$i \in N$和$\mathbf{y} \in \mathcal{F}i(\mathbf{x})$，那么联合战略选择$\mathbf{x} \in \mathfrak{X}$就是收益均衡。完全类似地，成本均衡是通过相反的条件定义的:$u_i(\mathbf{x}) \leq u_i(\mathbf{y})$适用于所有$i \in N$和$\mathbf{y} \in \mathcal{F}_i(\mathbf{x})$。聚合实用程序。关于均衡还有另一个重要观点。假设状态为$\mathbf{x} \in \mathfrak{X}$，假设每个玩家$i \in N$考虑的是当前策略$x_i$之外的替代策略$y_i$。得到的实用程序值的总和是$$G(\mathbf{x}, \mathbf{y})=\sum{i \in N} u\left(\mathbf{x}{-i}\left(y_i\right)\right) \quad\left(\mathbf{y}=\left(y_i \mid y_i \in X_i\right)\right) .$$ LEMMA 6.1。当且仅当$G(\mathbf{x}, \mathbf{y}) \leq G(\mathbf{x}, \mathbf{x}) \quad$对所有$\mathbf{y} \in \mathfrak{X}$都成立时，$\mathrm{x} \subset \mathfrak{X}$是$\Gamma\left(u_i \mid i \subset N\right)$的增益均衡。证据。如果$\mathbf{x}$是一个增益均衡，$\mathbf{y}=\left(y_i \mid i \in N\right) \in \mathfrak{X}$，我们有$u_i(\mathbf{x}) \geq u\left(\mathbf{x}{-i}\left(y_i\right)\right) \quad$对所有$y_i \in X_i$，
，这意味着$G(\mathbf{x}, \mathbf{x}) \geq G(\mathbf{x}, \mathbf{y})$。相反，如果$\mathbf{x}$不是一个增益均衡，则有一个$i \in N$和一个$y \in X_i$，使
$$0<u_i\left(\mathbf{x}{-i}(y)\right)-u_i(\mathbf{x})=G\left(\mathbf{x}, \mathbf{x}_i(y)\right)-G(\mathbf{x}, \mathbf{x}) .$$，这意味着$\mathbf{y}=\mathbf{x}{-i}(y) \in \mathfrak{X}$违反了不平等。引理$6.1$将对平衡的追求简化为对$\mathbf{x} \in \mathfrak{X}$的追求，该最大化了聚合效用度量$G$的相关成分函数$g^{\mathbf{x}}: \mathfrak{X} \rightarrow \mathbb{R}$，其值
$$g^{\mathbf{x}}(\mathbf{y})=G(\mathbf{x}, \mathbf{y})$$

经济代写|博弈论代写博弈论代考|矩阵博弈的随机化

$$\mathbf{x}=\left(x_1, \ldots, x_i, \ldots, x_n\right) \in X_1 \times \cdots \times X_i \times \cdots \times X_n(=\mathfrak{X})$$

$$U_{\mathbf{x}}=\left(u_1(\mathbf{x}), \ldots, u_i(\mathbf{x}), \ldots, u_n(\mathbf{x})\right) \in \mathbb{R}^n .$$现在让我们改变矩阵游戏的规则 $\Gamma$
(R)每个玩家 $i \in N$ 选择一个概率分布 $p^{(i)}$ 在 $X_i$ 并选择元素 $x \in X_i$ 有概率 $p_x^{(i)}$. 在规则(R)下，玩家实际上是在玩相关的$n$ -人游戏$\bar{\Gamma}=\left(\bar{u}i \mid i \in N\right)$，资源集$P_i$和效用函数$\bar{u}_i$，其中(1)$P_i$是$X_i$上所有概率分布的集合。(2) $\bar{u}_i(p)$是$u_i$相对于参与者的联合概率分布$p=\left(p^{(i)} \mid i \in N\right)$的期望值。$n$人游戏$\bar{\Gamma}$是矩阵游戏$\Gamma$的随机化。期望的总效用值是$$\bar{u}_i\left(p^{(1)}, \ldots, p^{(n)}\right)=\sum{x_1 \in X_1} \cdots \sum_{x_n \in X_n} u_i\left(x_1, \ldots, x_n\right) p_{x_1}^{(1)} \cdots p_{x_n}^{(n)}$$

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