战略思考：博弈论介绍 Strategic Thinking: An introduction to Game Theory ECON2141
Proof. I first prove part a. By the definition of Nash equilibrium we have $U_{2}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \geq U_{2}\left(\alpha_{1}^{}, \alpha_{2}\right)$ for every mixed strategy $\alpha_{2}$ of player 2 or, since $U_{2}=-U_{1}$, $$ U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \leq U_{1}\left(\alpha_{1}^{}, \alpha_{2}\right) \text { for every mixed strategy } \alpha_{2} \text { of player } 2 . $$ Hence $$ U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right)=\min {\alpha{2}} U_{1}\left(\alpha_{1}^{}, \alpha_{2}\right) . $$ Now, the function on the right hand side of this equality is evaluated at the specific strategy $\alpha_{1}^{}$, so that its value is not more than the maximum as we vary $\alpha_{1}$, namely $\max {a{1}} \min {a{2}} U_{1}\left(\alpha_{1}, \alpha_{2}\right)$. Thus we conclude that $$ U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \leq \max {\alpha{1}} \min {\alpha{2}} U_{1}\left(\alpha_{1}, \alpha_{2}\right) . $$
$U_{1}\left(\alpha_{1}^{}, \alpha_{2}\right) \geq U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right)$ for every mixed strategy $\alpha_{2}$ of player 2 , 342 or $U_{2}\left(\alpha_{1}^{}, \alpha_{2}\right) \leq U_{2}\left(\alpha_{1}^{}, \alpha_{2}^{}\right)$ for every mixed strategy $\alpha_{2}$ of player 2 . Similarly, $$ U_{2}\left(\alpha_{1}, \alpha_{2}^{}\right) \geq U_{2}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \text { for every mixed strategy } \alpha_{1} \text { of player } 1 \text {, } $$ or $$ U_{1}\left(\alpha_{1}, \alpha_{2}^{}\right) \leq U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \text { for every mixed strategy } \alpha_{1} \text { of player } 1 \text {, } $$ so that $\left(\alpha_{1}^{}, \alpha_{2}^{}\right)$ is a Nash equilibrium of the game.
For a function $f$ that is twice continuously differentiable, $f$ has increasing differences if and only if $$ \begin{gathered} t^{\prime} \geq t \Rightarrow \frac{\partial f}{\partial x}\left(x, t^{\prime}\right) \geq \frac{\partial f}{\partial x}(x, t), \ \Leftrightarrow \frac{\partial^{2} f}{\partial x \partial t}(x, t) \geq 0, \forall x \in \mathcal{X}, \forall t \in \mathcal{T} . \end{gathered} $$ Given the definition of a function with increasing differences, we can formally define a supermodular game as follows: for all $i$,
$$ I_{p}(f)=\sum_{e \in \mathcal{P}} I_{e}\left(f_{e}\right) . $$ Note that this path latency is not a function of the corresponding path flow because it depends on the total flow on each of its edges. Consequently, the total latency $C(f)$ of a flow $f$ is defined as follows: $$ C(f)=\sum_{p \in \mathcal{P}} I_{p}\left(f_{e}\right) f_{p} $$ it can be easily shown that $C(f)$ depends solely on the edge flow, and is given by $$ C(f)=\sum_{e \in \mathcal{E}} I_{e}\left(f_{e}\right) f_{e} . $$
在约翰-冯-诺伊曼（John von Neumann）于1928年发表《论战略游戏理论》（On the Theory of Games of Strategy）一文之前，博弈论并没有作为一个独特的领域存在。冯-诺伊曼的原始证明采用了布劳威尔关于连续映射到紧凑凸集的定点定理，这成为博弈论和数理经济学的标准方法。在他的论文之后，他在1944年与Oskar Morgenstern合著了《游戏和经济行为理论》一书。这本书的第二版提供了一个效用的公理理论，它将丹尼尔-伯努利的旧效用理论（货币）作为一个独立的学科进行了重塑。冯-诺伊曼在博弈论方面的工作在1944年的这本书中达到了顶峰。这项基础性工作包含了为两人零和博弈寻找相互一致的解决方案的方法。随后的工作主要集中在合作博弈论上，它分析了个人群体的最优策略，假定他们之间可以执行关于适当策略的协议。
博弈论作业代写Game Theory代考
Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928. Von Neumann’s original proof used Brouwer’s fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli’s old theory of utility (of money) as an independent discipline. Von Neumann’s work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
博弈论课后作业代写
As an illustrative example of correlated equilibrium, we consider a variant of the game of Chicken, as shown in Table $3.10$ (this is the same game as in Example $3.3$ but with modified payoff values). The strategies $S T$ and $S$ refer, respectively, to staying straight and swerving. Using the payoffs in Table $3.10$ and the inequalities in (3.37), we conclude that a probability distribution at the correlated equilibrium must satisfy the following: $$ \begin{aligned} &(0-1) p_{11}+(5-4) p_{12} \geq 0 \ &(1-0) p_{21}+(4-5) p_{22} \geq 0 \ &(0-1) p_{11}+(5-4) p_{21} \geq 0 \ &(1-0) p_{12}+(4-5) p_{22} \geq 0 \end{aligned} $$ $$ \begin{gathered} \sum_{i, j \in{1,2}} p_{i j}=1 \ 0 \leq p_{i j} \leq 1, \forall i, j \in{1,2} \end{gathered} $$
