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\centerline{\bf \Large Math 315 Homework 5}
\smallskip
\centerline{\bf Due Friday, March 10}
\bigskip
Solutions must be written in \LaTeX. You are encouraged to work with others on the assignment,
but you should write up your own solutions independently.
You should reference all of your sources, including your
collaborators.
\smallskip
\begin{enumerate}
\item Consider the extensive-form game given by the following game tree:
\begin{center}
\scalebox{.5}{\includegraphics{GameTree1.pdf}}
\end{center}
\begin{enumerate}
\item Use backward induction to determine each player's optimal strategy. What happens when each player uses their optimal strategy?
\item Identify all of Player 1's strategies and all of Player 2's strategies. Write this game in strategic form.
\item Find all of the pure strategy Nash Equilibria in the strategic form game.
\end{enumerate}
\bigskip
\item Consider the extensive-form game given by the following game tree. Player 0 is Nature, and the game has two information sets
as indicated in the game tree.
\begin{center}
\scalebox{.55}{\includegraphics{GameTree2.pdf}}
\end{center}
Identify all of Player 1's strategies and all of Player 2's strategies. Write this game in strategic form.
\bigskip
\item Consider the following strategic-form game:
\centerline{\hspace{2cm} Player 2} \vspace{-.5cm}
\begin{center}
$\begin{array}{c}
\\ \mbox{Player 1} \\ \end{array}\begin{array}{| c || c | c | c | c | c |}
\hline
& 1 & 2 & 3 & 4 & 5 \\ \hline
\hline
A & 2, -1 & 2, 0 & -5, -1 & 0, 0 & 1, -1 \\ \hline
B & 2, 2 & 3, 3 & -4, 3 & 1,3 & 1, -2 \\ \hline
C & 1, -1 & -2, 2 & -5, 4 & 0, 4 & -1, 3 \\ \hline
\end{array}$
\end{center}
\begin{enumerate}
\item Remove any rows or columns that are strictly dominated, and repeat until
no strictly dominated rows or columns remain. What is the resulting matrix after all strictly dominated strategies have been removed?
\item Find all pure strategy Nash equilibria in the game?
\end{enumerate}
\bigskip
\item Considered a simplified baseball game in which the pitcher chooses whether to throw a fastball or a curve ball, and
the batter chooses whether to anticipate a fastball or curve ball. Suppose that this is represented by the following strategic-form game:
\medskip
\centerline{\hspace{4cm}Batter}
\vspace{-.5cm}
\begin{center}
$\begin{array}{c} \mbox{Pitcher} \end{array}$$\begin{array}{| c || c | c |}
\hline
& \mbox{Anticipate Fastball} & \mbox{Anticipate Curve} \\ \hline \hline
\mbox{Throw Fastball} & 70, 30 & 80, 20 \\ \hline
\mbox{Throw Curve} & 85, 15 & 65, 35 \\ \hline
\end{array}$
\end{center}
\begin{enumerate}
\item Does this game have any pure strategy Nash equilibria?
\item Find the mixed strategy Nash equilibrium for the game.
\item What is batter's expected payoff if both players are using the mixed strategy Nash equilibrium? What is the
pitcher's expected payoff if both players are using the mixed strategy Nash equilibrium?
\end{enumerate}
\end{enumerate}
\end{document}