I am trying for an exercise to use Fictitious Play to find a mixed Nash equilibrium for the Matching Pennies zero-sum game with matrix:
$$A = \begin{bmatrix}1 & -1\\-1 & 1\end{bmatrix}$$
According to Fictitious play, the row and column player update their strategies in step $t$ according to the following formula:
$$x^t = \frac{1}{t} \cdot \sum_{i=0}^t e_{i,t}^m$$ $$y^t = \frac{1}{t} \cdot \sum_{i=0}^t e_{j,t}^n$$
Where $e_{i,t}^m$ is a $m$-dimensional unit base vector (= pure strategy). In this case, A is quadratic so $m = n$.
Now in class we said that $x^0 = y^0 = 0^n$ (zero-vector).
According to me trying to solve the exercise, this initialisation doesn't make much sense because Fictitious Play is going to converge after 2 steps:
Choose $x^0 = y^0 = 0^n$ such that both players can select a random pure strategy in step $t=1$, e.g. $e_{i,1} = \begin{bmatrix}1\\0\end{bmatrix}$ and $e_{j,1} = \begin{bmatrix}0\\1\end{bmatrix}$ such that $x^1 = \begin{bmatrix}1\\0\end{bmatrix}$ and $y^1 = \begin{bmatrix}0\\1\end{bmatrix}$
Then if we calculate the best response for $e_{i,2}$:
$$x^TAy^1 \Rightarrow \begin{bmatrix}0\\1\end{bmatrix}$$
and $e_{j,2}$
$$(x^1)^TAy \Rightarrow \begin{bmatrix}1\\0\end{bmatrix}$$
So we end up with:
$$x^2 = \frac{1}{2}\left( \begin{bmatrix}0\\0\end{bmatrix} + \begin{bmatrix}1\\0\end{bmatrix} + \begin{bmatrix}0\\1\end{bmatrix} \right) = \begin{bmatrix}1/2\\1/2\end{bmatrix}$$
and
$$y^2 = \frac{1}{2}\left( \begin{bmatrix}0\\0\end{bmatrix} + \begin{bmatrix}0\\1\end{bmatrix} + \begin{bmatrix}1\\0\end{bmatrix} \right) = \begin{bmatrix}1/2\\1/2\end{bmatrix}$$
Which already is the Nash equilibrium.
Did I get something wrong about Fictitious Play?
$y^2$ is wrong in what you wrote above:
You started with (Heads, Tails), that is Heads for player 1 and Tails for player 2. Then, in the next round, player 1 wants to match Tails, and player 2 wants to mismatch Heads, so the next pure strategy profile is (Tails, Tails).