# Towards compositional game theory

I wrote this not just as a thesis, but (against all advice) as a resource for other people to learn about open games. In spite of some problems, it will probably remain my preferred reference on open games for years to come. It contains plenty of its own introduction, so I won’t introduce it again here.

There are two important pieces of errata:

• The SP-composition operator is broken and I no longer use it. I already knew this when I wrote the final version, so it’s emphasised plenty of times already, but it’s important enough that I mention it here again.
• There is a serious error in section 2.2.6, together with a portentous (and extremely embarrassing) footnote, where I assumed that all morphisms are comonoid homomorphisms. In fact that holds if and only if the underlying category is cartesian monoidal. This error was discovered by Josef Bolt. (A workaround is possible, but is work in progress and involves significantly more category theory.) On reflection, I wish I had specialised the entire thing to the category of sets.

I have resisted going down the rabbit-hole of keeping my thesis up-to-date, because correcting both of these would mean almost a complete rewrite with no benefit to me.

I learned the following things during and after writing my thesis:

• Writing a thesis takes a lot longer than it looks
• Investing two working days into learning about large-scale Latex programming was a good idea
• Naming Latex macros after their semantics rather than their syntax is a particularly good idea
• Although I still like the informal, rambling style I used, definitions, theorems and proofs should definitely be clearly demarcated
• I no longer want to write a book

## 5 thoughts on “Towards compositional game theory”

1. […] Towards compositional game theory, my PhD thesis, is by far the most complete exposition and also contains as much informal explanation as I could manage […]

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2. […] All I have to speak from here is personal anecdote. Open games form a symmetric monoidal category, which is viewed as a process theory and a formal underpinning of string diagrams. This view of monoidal categories has been used very effectively by Bob Coecke and his (many) collaborators, and I learned it from a steady stream of seminar speakers coming from Oxford to Queen Mary. In the months before the discovery of open games in late January 2015, I was thinking about the question “How to make game theory compositional?”, and searching for the right tool for the job. At the time I was mainly looking at tools from concurrency theory and game semantics such as event structures, and category theory had to justify itself as much as any other tool.  For several years I used category theory as a mere language, a point of view that I emphasised in my thesis. […]

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3. […] I had been working on open games for about 1.5 years and written my thesis about them, I moved to Oxford and took the chance to clean up the mathematical foundations. In […]

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4. +1 for semantical naming of macros. It’s one of my tenets, too.

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5. […] perfection has been an embarrassing thorn in my side since 2016 when I had to do major surgery on my PhD thesis because the category of “open games with subgame perfect equilibria” turned out to not […]

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