Today was a fun class: thanks! We spoke about calculating utilities as well as best responses.

A recording of today is available here: cardiff.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=3819dbcc-aeac-487b-baaa-af9d00a567ce.

In class we finished covering how to calculate the expected utility for each player (the row and the column player) when we know what strategies they are playing (\(\sigma_r\) and \(\sigma_c\)). The notes on this are available at: nashpy.readthedocs.io/en/stable/text-book/strategies.html#calculation-of-expected-utilities.

I also showed how we could use Nashpy to directly compute expected utilities. The up to date notebook is available here.

After this discussion we moved on to looking at what is called a “best response”. We did this by playing the modified matching pennies game against the following 3 strategies:

- \(\sigma_r=(.2, .8)\)
- \(\sigma_r=(.9, .1)\)
- \(\sigma_r=(1/3, 2/3)\)

In the first two cases there was an intuitive way to play (and indeed a number of students got the maximum possible score) however in the final one it was not so clear.

Towards the end of the class we started looking at this a bit more formally however it was clear there was some confusion. I plan to revisit this on Friday.

The notebook I used to start looking at best responses is available here (at present this contains the code for the random sampling of the strategies – I will update it with more on Friday).

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