Problem 5

\subsection*{Problem 5}
\begin{enumerate}
    \item \ \\
        \begin{center}
        \begin{tikzpicture}[scale=2.5,font=\footnotesize]
            \tikzstyle{level 1}=[level distance=15mm,sibling distance=22.5mm]
            \tikzstyle{level 2}=[level distance=15mm,sibling distance=6mm]
            \tikzset
                {
                    solid node/.style={circle,draw,inner sep=3,fill=black},
                    hollow node/.style={circle,draw,inner sep=3}
                }
            \node(0)[hollow node]{Tim Cook}
                child{node(1)[hollow node]{Bill Gates}
                    child{node [hollow node] {(5,5)} edge from parent node[left]{N}}
                    child{node [hollow node] {(15,9)} edge from parent node[left,xshift=3]{M}}
                    child{node [hollow node] {(12,7)} edge from parent node[right]{A}}
                edge from parent node[left,xshift=-15]{N}
                }
                child{node(1)[hollow node]{Bill Gates}
                    child{node [hollow node] {(9,15)} edge from parent node[left]{N}}
                    child{node [hollow node] {(0,0)} edge from parent node[left,xshift=3]{M}}
                    child{node [hollow node] {(8,7)} edge from parent node[right]{A}}
                edge from parent node[left,,xshift=3]{M}
                }
                child{node(1)[hollow node]{Bill Gates}
                    child{node [hollow node] {(7,12)} edge from parent node[left]{N}}
                    child{node [hollow node] {(7,8)} edge from parent node[left,xshift=3]{M}}
                    child{node [hollow node] {(10,10)} edge from parent node[right]{A}}
                edge from parent node[right,xshift=15]{A}
                }
        \end{tikzpicture}
        \end{center}
    \item \
        \begin{center}
            $A_T = \{A,M,N\} = 3$ \\
            $S_T = \{A,M,N\} = A_T$ \\
        \end{center}
    \item \
        \begin{center}
            $A_B = \{A,M,N\} = 3$ \\
            $S_B = \{(A,A) , (A,M) , (A,N) , (M,A) , (M,M) , (M,N) , (N,A) , (N,M) , (N,N)\}$ \\
        \end{center}
    \item By using backward induction: \\
    (a) If Tim Cook chooses N, Bill Gates selects M, so Apple's profit will be 15 \\
    (b) If Tim Cook chooses M, Bill Gates selects N, so Apple's profit will be 9 \\
    (c) If Tim Cook chooses A, Bill Gates selects N, so Apple's profit will be 7 \\
    Thus, to maximize Apple's profit, Tim Cook chooses N in the first stage. Therefore: \\
        \begin{center}
            $S_T^* = \{N\}$ and $S_B^* = \{(N,M)\}$
        \end{center}
    Where,
        \begin{center}
            $\pi_T^* =15$ and $\pi_B^* =9$
        \end{center}
    if we set $\pi_T^*$ as Tim's outcome at SPE and $\pi_B^*$ as Bill's outcome at SPE.
    \item The subgame perfect Nash equilibrium is where Tim Cook and his team at Apple do not advertise for their product but Bill Gates and his team at Windows should advertise for their product mildly. This case has occurred because Tim Cook and his team at Apple released their product (I-Phone Stellar) sooner than Bill Gates and Windows' product (Positron Vega). So, as long as there is no competitor in the market, there is no need to put effort for advertising. But conditions for Bill Gates and his team at Windows are not the same. They want to enter to a market which is pre-occupied, so they have to put effort on advertising, but since their product is very similar to Apple's product, a mild advertising could be enough for them.
    \item There are two Nash equilibria strategies for this game,
        \begin{center}
            $NE_1 = \{ M , (M,N) \}$ \\
            $NE_2 = \{ N , (N,M) \}$
        \end{center}
    By comparing this two with the findings in part 4, we see that one of these Nash equilibria strategies is the subgame perfect Nash equilibrium, but the other one is not. It is because of the nature of the game which is sequential and as it can be seen, the other Nash equilibria is exactly opposite and it could be the subgame perfect Nash equilibrium for the condition which Bill Gates chooses his advertising strategy first.
\end{enumerate}