ECON0200 - 统计代写答疑辅导

标签： ECON0200

经济代写|ECON0200 Game Theory

Posted on 2023年4月27日2023年4月27日 by statistics-lab

Statistics-lab™可以为您提供pitt.edu ECON0200 Game Theory博弈论课程的代写代考和辅导服务！

ECON0200 Game Theory课程简介

The course may cover topics such as game theory, mechanism design, computational finance, and data analysis. Students may learn how to use programming languages and software tools to build and analyze economic models and simulations, and may also study how to apply these techniques to real-world economic problems.

Overall, this course seems to combine elements of economics, computer science, and mathematics to provide students with a unique set of skills and knowledge for analyzing and designing economic systems in a rapidly changing and increasingly complex world.

PREREQUISITES

Project details and timeline
Project details and ideas can be found here (UW access only).

Feb 22: Short description of topic, goals and project team due (as part of HW1).
Mar 22: Up to one page report of progress, reference material, plans for the remainder of the semester. Before this date, please make an appointment with Shuchi to discuss potential topics and references.
May 3: Final project reports due.
May 5: Two projects (selected on the basis of the final reports) to be showcased during this lecture.

ECON0200 Game Theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Consider the following game in matrix form with two players. Payoffs for the row player Izzy are indicated first in each cell, and payoffs for the column player Jack are second.
\begin{tabular}{c|c|c|c|}
& \multicolumn{1}{c}{$X$} & \multicolumn{1}{c}{$Y$} & \multicolumn{1}{c}{$Z$} \
\cline { 2 – 4 }$S$ & 5,2 & 10,6 & 25,10 \
\cline { 2 – 4 }$T$ & 10,12 & 5,6 & 0,0 \
\cline { 2 – 4 } & & &
\end{tabular}
(a) This game has two pure strategy Nash equilibria. What are they (justify your answer)? Of the two pure equilibria, which would Izzy prefer? Which would Jack prefer?

问题 2.

(b) Suppose Izzy plays a strictly mixed strategy, where both $S$ and $T$ are chosen with positive probability. With what probability should Izzy choose $S$ and $T$ so that each of Jack’s three pure strategies is a best response to Izzy’s mixed strategy.

问题 3.

(c) Suppose Jack wants to play a mixed strategy in which he selects $X$ with probability 0.7. With what probability should Jack plays actions $Y$ and $Z$ so both of Izzy’s pure strategies is a best response to Jack’s mixed strategy? Explain your answer.

问题 4.

(d) Based on your responses above, describe a mixed strategy equilibrium for this game in which both Jack and Izzy play each of their actions (pure strategies) with positive probability. Explain why this is in fact a Nash equilibrium (you can rely on the quantities computed in the prior parts of this question).

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

此图像的alt属性为空；文件名为%E7%B2%89%E7%AC%94%E5%AD%97%E6%B5%B7%E6%8A%A5-1024x575-10.png — 经济代写|ECON0200 Game Theory

Statistics-lab™可以为您提供pitt.edu ECON0200 Game Theory博弈论课程的代写代考和辅导服务！请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

经济代写|ECON0200 Game Theory

Posted on 2023年4月22日2023年4月22日 by statistics-lab

Statistics-lab™可以为您提供pitt.edu ECON0200 Game Theory博弈论课程的代写代考和辅导服务！

ECON0200 Game Theory课程简介

PREREQUISITES

Project details and timeline
Project details and ideas can be found here (UW access only).

Feb 22: Short description of topic, goals and project team due (as part of HW1).
Mar 22: Up to one page report of progress, reference material, plans for the remainder of the semester. Before this date, please make an appointment with Shuchi to discuss potential topics and references.
May 3: Final project reports due.
May 5: Two projects (selected on the basis of the final reports) to be showcased during this lecture.

ECON0200 Game Theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

$\chi_{\omega,(r, s)}$ is a decreasing function of $\omega$ for every $r, s \in \mathcal{R}$. Furthermore, $\lim {\omega \rightarrow \infty} \chi{\omega,(r, s)}<1$

Proof: The first part of the lemma follows from that $\mathcal{R}{\omega, s}$ is a decreasing function of $\omega$. To prove the second part of the lemma, since $\lim {\omega \rightarrow \infty} \mathcal{R}{\omega, s}=1$, observe that $$ \lim {\omega \rightarrow \infty} \chi_{\omega,(r, s)}=\frac{\rho\left(1-\delta_{l_r}\left(m_r\right)\right) \alpha_r\left(1-\eta_r\right)}{1-\rho\left(1-\delta_{l_r}\left(m_r\right)\right)\left(\left(\frac{\overline{\overline{ }(1-\rho)+\rho q}}{1-\rho+\rho q}-\alpha_r\left(1-\mathrm{e}^{-\lambda_r}\right)\left(1-\eta_r\right)\right)\right)}\left(1-\mathrm{e}^{-\lambda_r}\right)<1 .
$$
A consequence of the above result is the following.

问题 2.

The attacker prefers attacking resource $s$ over listening to resource $r$ for some $\omega \geq \tilde{\omega}_{(r, s)}$, and after attacking, the attacker always prefers attacking.

Proof: For the initial values of the rewards, we assign zero reward for every $\omega$; i.e. $J_\omega^{(0)}=0 \forall \omega$. Regarding the first iteration of the value updates, since
$$
J_{\omega \perp, r}^{(1)}=\frac{K_{\omega, r}^{(0)}}{1-U_r}=\frac{\rho\left(1-\delta_{l_r}\left(m_r\right)\right) \alpha_r\left(1-\eta_r\right)}{1-U_r} \sum_{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_r} \lambda_r^n}{n !} J_{\omega+n}^{(0)}=0
$$
for $r \in \mathcal{R}$, i.e. all listening rewards are zero, attacking would be the optimal choice for every $\omega$. Then, $J_{\omega, L, r}^{(1)}=0$ and $J_{\omega, A, s}^{(1)}=\mathcal{A}{\omega, s}=\frac{C{\omega, s}}{1-T_{\omega \omega s}}$ hold $\forall \omega, r, s$, which implies $J_{\omega, }^{(1)}=\mathcal{A}\omega^=\max {i \in[1, \ldots, R}} \mathcal{A}{\omega, 1} \forall \omega$. In the second iteration, the attacking rewards do not change; i.e. $J{\omega, A, s}^{(2)}=\frac{c_{\omega s}}{1-T_{\omega,}} \forall \omega, s$. Regarding the listening rewards, for every $\omega$, similar to (11.14), we obtain
$$
\begin{aligned}
J_{\omega, L, r}^{(2)} & =\frac{K_{\omega, r}^{(1)}}{1-U_r}=\frac{\rho\left(1-\delta_{l_r}\left(m_r\right)\right) \alpha_r\left(1-\eta_r\right)}{1-U_r} \sum_{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_r} \lambda_r^n}{n !} \mathcal{A}\omega^* \ & =\mathcal{A}\omega^* \underbrace{\frac{\rho\left(1-\delta_{l_r}\left(m_r\right)\right) \alpha_r\left(1-\eta_r\right)}{1-U_r} \sum_{n=1}^{\infty} \frac{\mathrm{e}^{-\lambda_r} \lambda_r^n}{n !}\left(\prod_{l=0}^{n-1} \frac{\mathcal{A}{\omega+1+1}^}{\mathcal{A}{\omega+1}^}\right)}{\chi{\infty, n)}^} . \end{aligned} $$ Since $\chi_{\omega,(r)}^$ is not necessarily a decreasing function of $\omega$, in the second iteration, the monotonic relation between the listening and attacking rewards cannot be directly established as in the single-resource case discussed below.

Textbooks

经济代写|ECON0200 Game Theory

Posted on 2023年4月13日2023年4月13日 by statistics-lab

Statistics-lab™可以为您提供pitt.edu ECON0200 Game Theory博弈论课程的代写代考和辅导服务！

ECON0200 Game Theory课程简介

PREREQUISITES

Project details and timeline
Project details and ideas can be found here (UW access only).

Feb 22: Short description of topic, goals and project team due (as part of HW1).
Mar 22: Up to one page report of progress, reference material, plans for the remainder of the semester. Before this date, please make an appointment with Shuchi to discuss potential topics and references.
May 3: Final project reports due.
May 5: Two projects (selected on the basis of the final reports) to be showcased during this lecture.

ECON0200 Game Theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

The stage game is shown in Table 1.
\begin{tabular}{c|c|c|}
\hline & $H$ & $L$ \
\hline \hline $\mathrm{H}$ & $(3,1)$ & $(0,0)$ \
\hline $\mathrm{L}$ & $(1,2)$ & $(5,3)$ \
\hline
\end{tabular}
Table 1: Stage game
Consider the infinite repetition of the game in Table 1 with discounted criterion to evaluate payoffs. Find a subgame perfect equilibrium of this game such that
(a) the equilibrium payoff of Players approach $(4,2)$ as $\delta \rightarrow 1$.
(b) the equilibrium payoff of Players approach $(3,2)$ as $\delta \rightarrow 1$.

问题 2.

If we repeat prisoner’s dilemma game for two periods, how many strategies does each player have in this repeated game?

问题 3.

Consider the stage game $G$ shown in Table 2 .
\begin{tabular}{c|c|c|c|}
\hline & $a$ & $b$ & $c$ \
\hline$A$ & $(4,4)$ & $(-1,5)$ & $(2,2)$ \
\hline$B$ & $(5,-1)$ & $(1,1)$ & $(2,2)$ \
\hline$C$ & $(2,2)$ & $(2,2)$ & $(3.5,3.5)$ \
\hline
\end{tabular}
Table 2: Stage game
(a) Find the worst Nash equilibrium (pure action) for each player in $G$ and the corresponding payoffs.
(b) Consider $G^2$ : the finitely repeated game, where $G$ is repeated for two periods.
(i) Is there a subgame perfect equilibrium of $G^2$ where $(A, a)$ is played in the first period? Explain your answer.
(ii) Is there a Nash equilibrium of $G^2$ where $(A, a)$ is played in the first period? Explain your answer.

问题 4.

Suppose instead of discounting criterion for evaluating payoffs, we evaluate payoff of Player $i$ from a stream of payoffs $\left{v_i^t\right}_1^{\infty}$ as
$$
\lim {T \rightarrow \infty} \frac{1}{T} \sum{t=1}^T v_i^t .
$$
\begin{tabular}{c|c|c|}
\hline & C & D \
\hline \hline $\mathrm{C}$ & $(2,2)$ & $(0,3)$ \
\hline $\mathrm{D}$ & $(3,0)$ & $(1,1)$ \
\hline
\end{tabular}
Table 3: Prisoner’s dilemma
Verify if the grim-trigger strategy continues to be the Nash and subgame perfect equilibrium of the Prisoner’s Dilemma game of Table 3 using this criterion for evaluating payoffs.

Textbooks

经济代写|ECON0200 Game Theory

Posted on 2023年4月6日2023年4月6日 by statistics-lab

Statistics-lab™可以为您提供pitt.edu ECON0200 Game Theory博弈论课程的代写代考和辅导服务！

ECON0200 Game Theory课程简介

PREREQUISITES

Project details and timeline
Project details and ideas can be found here (UW access only).

Feb 22: Short description of topic, goals and project team due (as part of HW1).
Mar 22: Up to one page report of progress, reference material, plans for the remainder of the semester. Before this date, please make an appointment with Shuchi to discuss potential topics and references.
May 3: Final project reports due.
May 5: Two projects (selected on the basis of the final reports) to be showcased during this lecture.

ECON0200 Game Theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

(b) Consider $G^2$ : the finitely repeated game, where $G$ is repeated for two periods.
(i) Is there a subgame perfect equilibrium of $G^2$ where $(A, a)$ is played in the first period? Explain your answer.
(ii) Is there a Nash equilibrium of $G^2$ where $(A, a)$ is played in the first period? Explain your answer.

(i) There may or may not be a subgame perfect equilibrium of $G^2$ where $(A, a)$ is played in the first period, depending on the payoffs and strategies of the game $G$.

To determine whether such an equilibrium exists, we need to examine all possible strategies and outcomes in the game. If $(A,a)$ is played in the first period, then in the second period, both players know that they will play $G$ again, and so their strategies and payoffs will depend on the outcome of the first period.

Let us assume that there is a subgame perfect equilibrium where $(A, a)$ is played in the first period. Then, in the second period, both players will also play $(A,a)$, since this is the only continuation of the first-period strategy. However, if both players play $(A,a)$ in the second period, then they both receive a payoff of 2, which is less than the payoff of 3 they could receive by both playing $(B,b)$.

Therefore, there is no subgame perfect equilibrium where $(A,a)$ is played in the first period.

(ii) There may or may not be a Nash equilibrium of $G^2$ where $(A, a)$ is played in the first period, depending on the payoffs and strategies of the game $G$.

Let us assume that there is a Nash equilibrium where $(A, a)$ is played in the first period. Then, in the second period, both players will also play $(A,a)$, since this is the only continuation of the first-period strategy. However, if both players play $(A,a)$ in the second period, then they both receive a payoff of 2, which is less than the payoff of 3 they could receive by both playing $(B,b)$.

Therefore, there is no Nash equilibrium where $(A,a)$ is played in the first period.

问题 2.

(c) Consider the infinitely repeated game $G^{\infty}$. Describe a carrot and stick strategy profile where punishment is carried out for one period and $(A, a)$ is played in normal state. Show that it is a subgame perfect equilibrium strategy profile and find the corresponding discount factor.

To construct a carrot and stick strategy profile, we first need to specify what actions will be taken in the normal state and what actions will be taken in the punishment phase. Let $(A,a)$ be played in the normal state, and let $(B,b)$ be played in the punishment phase. In this strategy, if one player deviates from playing $(A,a)$ in the normal state, then in the next period both players will play $(B,b)$ to punish the deviating player. After that, the game returns to the normal state where both players play $(A,a)$.

Let $\delta$ be the discount factor that determines how much weight the players place on future payoffs. For this strategy to be an equilibrium, it must be the case that neither player has an incentive to deviate from playing $(A,a)$ in the normal state, given that the other player follows the strategy. Suppose player 1 deviates in some period and plays $(B,b)$, then player 2 will play $(B,b)$ in the next period to punish player 1. Player 1 gets a payoff of 0 in this period, and a discounted payoff of $\delta$ in the next period, so the total discounted payoff from deviating in this period is $0 + \delta \times 2 = 2\delta$.

Suppose player 1 follows the strategy in all periods, and player 2 deviates in some period and plays $(B,b)$. Player 1 will play $(B,b)$ in the next period to punish player 2. Player 2 gets a payoff of 0 in this period, and a discounted payoff of $\delta$ in the next period, so the total discounted payoff from deviating in this period is $0 + \delta \times 2 = 2\delta$.

Since the punishment phase only lasts for one period, the discount factor must satisfy $\delta > \frac{1}{2}$ to make the threat of punishment credible.

Therefore, the strategy profile where $(A,a)$ is played in the normal state and $(B,b)$ is played in the punishment phase for one period, with a discount factor $\delta > \frac{1}{2}$, is a subgame perfect equilibrium strategy profile.