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\author{Sebastian van Strien (Imperial College)}
\title{M?PA48  Games and Dynamics: \\
\quad \\
%Alternative title: Evolutionary learning theory \\ 
Alternative title: \\
\quad  \\ 
 Dynamical learning models (in strategic environments) \\ 
\quad  \\ 
Three exercises for all the project\\}
\date{\today} 


\begin{document} 

{\bf  Exercises for all the project}

 
 \begin{enumerate}
% \item Show that if $(p,q)$ is a NE then the matrix $p_{ij}=p_iq_j$ is in the CE set.   
% \lsp{4cm} 
\item Consider  $A=\left( \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{array}\right)$
and the corresponding replicator system 
\begin{equation} \dot x_i = x_i ((Ax)_i - x\cdot A x). \tag{*} \end{equation} 
Determine the NE and ESS points of this system. Draw the indifference lines, and 
best response regions in the appropriate triangle $\Delta$. Determine the 
singularity points of (*), and determine the eigenvalues at these singularities, and thus whether they 
are saddle points, attracting or repelling.
Sketch the phase portrait of (*).   You are allowed to use theorems in the lecture notes. 
%
%Take $P=x_1x_2x_3$ and let $\hat x=(1/3) {\bf 1}$. 
%Then $\dot P / P = (\hat x - x) \cdot A x$. Write $x=\hat x + h$. Note that $A\hat x = {\bf 1}$ and $h\cdot {\bf 1}=0$. 
%So  $(\hat x - x) \cdot A x =\left(  \begin{array}{c} h_1 \\ h_2 \\ h_3 \end{array} \right) \cdot  \left(  \begin{array}{c} 2h_1+h_3 \\ h_1+2h_2 \\ h_2+2h_3 \end{array} \right) = 2(h_1^2+h_2^2+h_3^2) + (h_1h_3+ h_2h_1 + h_3h_2)$
%
%Write $x=\hat x + h$. Then $\dot h_i  = (1/3+h_i) ( (A (\hat x + h))_i - (\hat x + h ) A (\hat x +h ) = (1/3) 
%( ( A h)_i  - \frac{1}{3} {\bf 1} Ah)  + h.o.t. = (1/3) ( (Ah)_i - (h_1+h_2+h_3))$.
%So the linearisation matrix is 
%$$\left( \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 
% \end{array} \right)$$

 \item  Consider the 
 $$\mbox{ battle of the sexes game:} \left(\begin{array}{cc} (2,1) & (0,0) \\ (0,0) & (2,7) 
\end{array}\right)$$
(where we use the 2nd notation).
Compute the NE, CCE and CE sets. 
\item 
We say that two $m \times n$ bimatrix games $(A,B)$ and $(\tilde A,\tilde B)$ 
are \emph{(linearly) equivalent}, $(A,B) \sim (\tilde A,\tilde B)$, if 
there exist $c,d>0$ and  $c_1,\ldots,c_n , d_1, \ldots,d_m \in \R$ so that 
\[
{\tilde a}_{ij} = c \cdot a_{ij} + c_j \quad \text{ and } \quad {\tilde b}_{ij} = d \cdot b_{ij} + d_i
\qquad \text{for } i = 1,\ldots,m \text{ and } j = 1,\ldots,n.
\]
Here we use the 2nd notation for bimatrix games. 
Show that if $(A,B)$ and $(\tilde A,\tilde B)$ are equivalent then
\begin{enumerate}
\item  the replicator dynamics associated to   $(A,B)$ and to $(\tilde A,\tilde B)$ are the same (or do you need 
$c=d$ for this?);
\item  $\BR_A=\BR_{\tilde A}$, 
$\BR_B=\BR_{\tilde B}$ and the best response dynamics associated to $(A,B)$ and to $(\tilde A,\tilde B)$ are the same. 
\item the NE, CE and CCE sets associated to $(A,B)$ and to $(\tilde A,\tilde B)$ are the same. 
\item the Hart and Mas-Colell regret matching algorithms for these games give the same dynamics. 
\item  explain why the reinforcement learning algorithms from Section 6.1 of the lecture notes are not necessarily the same
for the games $(A,B)$ and to $(\tilde A,\tilde B)$.  
\end{enumerate}



\end{enumerate}

\end{document} 