## 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）

1. Recall the MLA game as described on Slide 4 of Lecture 9 with its equilibria described on slides 6 and 11. This game involves three members of a legislative assembly (three MLAs) deciding how to vote on a pay raise.

Suppose now that each of the MLAs votes in sequence, using some pre-determined order (e.g., alphabetical). Let’s call the MLAs 1, 2 and 3: MLA 1 votes first; MLA 2 votes next knowing exactly how MLA 1 voted; and finally MLA 3 votes knowing the two previous votes.
(a) Draw the extensive form game tree for this sequential voting game. How will each MLA vote assuming that a subgame perfect equilibrium is played? What is the payoff for each of the MLAs? Explain.
(b) Describe one profile of Nash equilibrium strategies for this extensive form game that is not subgame perfect.
(c) Now suppose that the MLAs have seen a recent poll, which could potentially influence the payoff of MLA 3. Specifically, the new poll indicates a level of fiscal conservatism in the district of MLA 3 that makes it uncertain how voters will react if she votes for a pay raise. As such MLA 3 ‘s payoffs are now random variables. She will realize her original payoffs with probability 0.4 ; however, with probability 0.6 she will receive the following payoffs:

• MLA 3 ‘s revised payoffs:
• if the raise passes, and MLA 3 votes for: -1
• if the raise passes, and MLA 3 votes against: 2
• if the raise fails, and MLA 3 votes for: -3
• if the raise fails, and MLA 3 votes against: 0
In other words, there is a $60 \%$ chance that her payoff when voting for a raise will be reduced by 2 . All three MLAs are aware of this poll and its implications.
Notice now that the payoffs for MLA 3 are random events. What is the matrix form of the game in which all three MLAs vote simultaneously, using the expected payoffs of MLA 3 ? Draw the extensive form game tree for this new game assuming they vote sequentially (in the same order as in part (a)). How will each MLA vote assuming that a subgame perfect equilibrium is played? What is the payoff for each of the MLAs? Explain.

(a) The extensive form game tree for this sequential voting game is as follows:

Assuming a subgame perfect equilibrium is played, each MLA will vote as follows:

• MLA 1 will vote for the pay raise, as this gives her the highest payoff of 3, regardless of what the other MLAs do.
• MLA 2 will vote against the pay raise, as this gives her the highest payoff of 2, given that MLA 1 voted for it.
• MLA 3 will vote against the pay raise, as this gives her the highest expected payoff of -0.6, given that MLA 1 voted for it and MLA 2 voted against it.

The payoffs for each MLA are:

• MLA 1: 3
• MLA 2: 2
• MLA 3: -0.6

(b) One profile of Nash equilibrium strategies that is not subgame perfect is for all MLAs to vote against the pay raise. This is a Nash equilibrium because no MLA can improve her payoff by changing her vote, given the votes of the other two MLAs. However, this strategy is not subgame perfect, because MLA 1 could improve her payoff by deviating and voting for the pay raise.

(c) The matrix form of the game in which all three MLAs vote simultaneously, using the expected payoffs of MLA 3, is:

The extensive form game tree for this new game assuming they vote sequentially (in the same order as in part (a)) is:

Assuming a subgame perfect equilibrium is played, each MLA will vote as follows:
• MLA 1 will vote for the pay raise, as this gives her the highest payoff of 2.4, regardless of what the other MLAs do.
• MLA 2 will vote against the pay raise, as this gives her the highest payoff of 1.6, given that MLA 1 voted for it.
• MLA 3 will vote against the pay raise, as this gives her the highest expected payoff of -0.

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}{} & \multicolumn{1}{c}{$Y$} & \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?
(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.
(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.
(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).
(e) If we swap two of Izzy’s payoffs in this matrix – in other words, if we replace one of his payoffs $r$ in the matrix with another of his payoffs $t$ from the matrix, and replace $t$ with $r$ we can make one of his strategies dominant. What swap should we make, which strategy becomes dominant, and why is it now dominant?

(a) The pure strategy Nash equilibria are $(S,Y)$ and $(T,Z)$ because neither player has an incentive to deviate unilaterally. In the first equilibrium, Izzy prefers $S$ and Jack prefers $Y$. In the second equilibrium, Izzy prefers $T$ and Jack prefers $Z$.

(b) In order for Jack to be indifferent between playing $Y$ and $Z$, the expected payoff of playing $Y$ should be equal to the expected payoff of playing $Z$. Thus, we must have $0.5 \times 5 + 0.5 \times 10 = 0.5 \times 25$, which simplifies to $7.5 = 12.5p + 0p$, where $p$ is the probability of Izzy playing $S$. Solving for $p$, we get $p = 0.4$. Therefore, Izzy should play $S$ with probability $0.4$ and $T$ with probability $0.6$.

(c) In order for Izzy to be indifferent between playing $S$ and $T$, the expected payoff of playing $S$ should be equal to the expected payoff of playing $T$. Thus, we must have $0.7 \times 5 + y = 0.7 \times 10 + z$, where $y$ is the expected payoff of Jack playing $Y$ and $z$ is the expected payoff of Jack playing $Z$. Since Jack wants to play $X$ with probability $0.7$, he wants to make Izzy indifferent between playing $S$ and $T$. Thus, we must also have $0.3 \times 5 + y = 0.3 \times 10 + z$. Solving these two equations simultaneously, we get $y = 7$ and $z = 7.5$. Therefore, Jack should play $Y$ with probability $0.5$ and $Z$ with probability $0.5$.

(d) One mixed strategy equilibrium is for Izzy to play $S$ with probability $0.4$ and $T$ with probability $0.6$, and for Jack to play $Y$ with probability $0.5$ and $Z$ with probability $0.5$. This is a Nash equilibrium because neither player has an incentive to deviate unilaterally. If Izzy deviates by changing the probabilities of $S$ and $T$, then Jack’s best response is to continue playing $Y$ and $Z$ with equal probabilities. Similarly, if Jack deviates by changing the probabilities of $Y$ and $Z$, then Izzy’s best response is to continue playing $S$ and $T$ with the same probabilities as before.

(e) If we swap the payoff of $S$ and $T$, then $T$ becomes dominant. Specifically, if we replace $5$ with $10$ and replace $10$ with $5$, then the payoff matrix becomes:

\begin{tabular}{c|c|c|c|} \multicolumn{1}{c}{} & \multicolumn{1}{c}{$Y$} & \multicolumn{1}{c}{$Y$} & \multicolumn{1}{c}{$Z$} \ \cline { 2 – 4 }$S$ & 10,12 & 5,6 & 0,0 \ \cline { 2 – 4 }$T$ & 25,10 & 10,6 & 5,2 \ \cline { 2 – 4 } & & & \end{tabular}

Now, if Izzy plays

## 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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