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.
- A first look at open games, a blog post serving as a zero-background introduction
- Compositionality and string diagrams for game theory, a paper that is forever-unfinished and unsubmitted, but was written for an audience of economists and goes into more detail while avoiding mathematical technicalities
- 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
Even I think open games are hard to understand, and I invented them.
(Perhaps this is just me though. Grothendieck wrote “The very idea of scheme is of infantile simplicity — so simple, so humble, that no one before me thought of stooping so low.” [Grothendieck, Récoltes et Samailles, translated by Colin McLarty] So simple, in fact, that it took me years before I understood the definition of a scheme.)
Here I will give the best starting-out intuition I can give for open games, based on a few years of giving research talks consisting of three-quarters introduction. I’ll make no attempt to explain how they work — for that, section 2 of my thesis is still the best thing.
As might be expected, the rules of the game are an important concept in game theory. But the way that game theory treats its all-important rules is very un-subtle: it is firmly built into the epistemic foundations that the rules are common knowledge, which makes it extremely difficult to talk about breaking the rules. If any player breaks the rules, or even if any player suspects another player of breaking the rules (up to any level of epistemic reasoning), you are simply outside the scope of your model. Of course the possibility of breaking any individual rule, and the consequences for doing so, can be manually built into your game, but then it is unclear whether it can reasonably be called a ‘rule’ any more.
I have a small article in Inspired Research, the biannual magazine of the cs.ox department. Read it on page 16 here.
- Coherence for lenses and open games, arXiv v2
I have been working on referee reports from version 1 (which was rejected for good reasons). I have uploaded an intermediate state of corrections, for the benefit of the reviewers of my extended abstract for STRING’17. This put me in the awkward position of uploading a paper that I know probably still contains some errors, although it’s less wrong than the previous version.
This is my first paper, written in the first few months of my Ph.D. and published quickly. Four and a half years later it is still technically my best paper by the usual (wrong) metrics. Obviously now I wouldn’t dare to do something as outrageous as submitting a paper to such a highly-ranked journal.