## 经济代写|CS880 Game Theory

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## CS880 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.

## CS880 Game Theory HELP（EXAM HELP， ONLINE TUTOR）

- 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.

To find the worst Nash equilibrium (pure action) for each player in the game $G$, we need to find the equilibrium where each player gets their lowest possible payoff.

Player 1’s worst Nash equilibrium is when they play strategy $B$ and player 2 plays strategy $a$. In this equilibrium, player 1 gets a payoff of $-1$, which is their lowest possible payoff, and player 2 gets a payoff of $5$, which is their highest possible payoff among the Nash equilibria. Therefore, the payoffs for player 1 and player 2 in this equilibrium are $(-1,5)$.

Player 2’s worst Nash equilibrium is when they play strategy $c$ and player 1 plays strategy $A$. In this equilibrium, player 2 gets a payoff of $2$, which is their lowest possible payoff, and player 1 gets a payoff of $2$, which is their highest possible payoff among the Nash equilibria. Therefore, the payoffs for player 1 and player 2 in this equilibrium are $(4,2)$.

Note that there is no Nash equilibrium where both players get their lowest possible payoff simultaneously, as player 1’s lowest possible payoff is $-1$ and player 2’s lowest possible payoff is $2$.

- (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) To find a subgame perfect equilibrium (SPE) of $G^2$ where $(A,a)$ is played in the first period, we need to analyze the game in each period separately and then look for the SPE that results from those strategies.

In the first period, if player 1 plays $A$, player 2 has an incentive to deviate to $b$ to get a higher payoff of $5$ instead of $4$. Therefore, $(A,a)$ is not an SPE in the first period.

In the second period, if the game is played as a one-shot game, then the Nash equilibrium is $(C,c)$. However, since the game is repeated, player 2 has an incentive to deviate to $b$ in the second period to punish player 1 for playing $A$ in the first period.

Therefore, there is no SPE of $G^2$ where $(A,a)$ is played in the first period.

(ii) To find a Nash equilibrium of $G^2$ where $(A,a)$ is played in the first period, we need to look for a Nash equilibrium in each period that is consistent with the strategy profile played in the other period.

In the first period, if player 1 plays $A$, player 2 has an incentive to play $b$ to get a higher payoff of $5$ instead of $4$. Therefore, $(A,a)$ is not a Nash equilibrium in the first period.

In the second period, if the game is played as a one-shot game, then the Nash equilibrium is $(C,c)$. However, if player 1 expects player 2 to play $c$ in the second period, then player 1 has an incentive to deviate from $A$ to $B$ in the first period to get a higher payoff of $5$ instead of $4$. Therefore, $(A,a)$ is not a Nash equilibrium in the second period either.

Therefore, there is no Nash equilibrium of $G^2$ where $(A,a)$ is played in the first period.

## 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.

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