Best responses against computer strategies

In class today we spoke more about strategies which are methods for picking actions from action sets. Specifically we spoke about best responses: what is the best strategy when faced with a given strategy.

A recording of that class is available here.

I started class by discussing what the original expectations you had might have been about Game Theory and if anyone had any queries. No one did which is good news as I assume it means I’ve made the expectations of the class clear. If that’s not quite right and you still have any questions please get in touch!

I also reminded you that the deadline to form your groups was coming up, if you need a hand to get in touch with your assigned group please let me know.

After this we played matching pennies against strategies played by a computer. A large group of you seemed to grasp early that there was a “right” way to play against the first two strategies.

We took at look at calculating the best response \(\sigma_c*\) against the strategy \(\sigma_r=(x, 1-x)\).

We saw that depending on the value of \(x\) we had 3 possible best responses:

• If \(x=.8\) then \(\sigma_c*=(1, 0)\): we should always play the first: column.
• If \(x=.1\) then \(\sigma_c*=(0, 1)\): we should always play the second column.
• If \(x=1/3\) then \(\sigma_c^*\) can be any valid strategy: it does not matter what we do.

All of this was lead by the idea that the expectation for a given \(\sigma_c\) is a linear function. The function can either:

• Be increasing in \(y\): then it is optimised for \(y=1\).
• Be decreasing in \(y\): then it is optimised for \(y=0\).
• Be a flat line that does not depend on \(y\).

I pointed out that \(\sigma^rB\) is a row vector that is essentially the game from the column players point of view for a given value of \(\sigma_r\)*.

This leads to the concepts described in this section of the notes: nashpy.readthedocs.io/en/stable/text-book/best-responses.html#generic-best-responses-in-2-by-2-games

Which is that we could write down a generic form of \(\sigma_c^*\):

\[ \sigma_c^*= \begin{cases} (0, 1),&\text{ if } x > 1/3\\
(1, 0),&\text{ if } x < 1/3\\
\text{indifferent},&\text{ if } x = 1/3 \end{cases} \]

We spent some time then doing the same thing for \(\sigma_r^*\):

\[ \sigma_r^* = \begin{cases} (1, 0),&\text{ if } y > 1/2\\
(0, 1),&\text{ if } y < 1/2\\
\text{indifferent},&\text{ if } y = 1/2 \end{cases} \]

(where \(y\) is the probability of being in the first column).

I then passed briefly over the remaining sections of the notes:

• A general condition for a best response (in games that are bigger than 2 by 2) which is the only way the above process works

One thing I did not do in class was:

• Using Nashpy to check the best response condition for pairs of strategies