## 经济代写|CS880 Game Theory

**Statistics-lab™**可以为您提供wisc.edu CS880 Game Theory博弈论课程的代写代考和辅导服务！

## 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）

- The stage game is shown in Table 1 .

\begin{tabular}{c|c|c|}

\hline & $\mathrm{H}$ & $\mathrm{L}$ \

\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 $\delta \rightarrow 1$.

(a) To find a subgame perfect equilibrium (SPE) that approaches a payoff of $(4,2)$ as $\delta \rightarrow 1$, we need to find a strategy for each player that is optimal at each stage of the game, given that the game will continue indefinitely with some probability $\delta \in [0,1)$. One possible SPE is as follows:

- In the first stage, Player 1 plays H and Player 2 plays L. This yields a payoff of $(3,1)$ for Player 1 and $(0,0)$ for Player 2.
- In all subsequent stages, both players play the following strategy:
- If the previous outcome was (H,L), play (H,L) again.
- If the previous outcome was (L,H), play (L,H) again.
- If the previous outcome was (H,H) or (L,L), play (L,H) with probability $p$ and (H,L) with probability $1-p$, where $p$ is the smallest value that satisfies the condition $\delta \geq \frac{1-p}{1+p}$.

The equilibrium payoff under this strategy is $(4,2)$ when $\delta \rightarrow 1$, because the players play (L,H) with probability 1 as $\delta$ approaches 1. Note that this is not the only SPE, and there may be other equilibria that also approach $(4,2)$ as $\delta \rightarrow 1$.

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

In a repeated prisoner’s dilemma game, each player has multiple strategies that they can use. One common strategy is called “tit-for-tat,” where a player cooperates in the first period and then in subsequent periods does whatever the other player did in the previous period.

If we repeat the game for two periods, each player has four possible strategies:

- Cooperate in both periods
- Defect in both periods
- Cooperate in the first period and then defect in the second period
- Defect in the first period and then cooperate in the second period

It’s important to note that the number of possible strategies increases with each additional period in a repeated game, making it more difficult to predict the outcome of the game.

## 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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