I feel that having Compositional game theory accepted for publication marks the end of the first chapter for me, and marks open games as a proper research topic. I want to record how open games came to be, told through the story of this paper, which took almost 3 years from writing to acceptance. (That’s roughly the same amount of time as the Higher Order Decisions/Games duo, although they definitely *felt* longer. Their story can wait for another blog post.)

# Blog

## The rising sea in applied mathematics

*The rising sea* refers to a particular approach to mathematical problem-solving, in which many small, apparently trivial steps are taken until the solution of a problem becomes itself trivial. It was poetically introduced by Alexander Grothendieck in his beautiful, auto-psychoanalytic *Récoltes et Samailles*, in which he imagines the mathematical problem as a landmass being swallowed as “the sea advances insensibly in silence”. This makes me think of Xerxes, all-powerful over humans, helpless against the power of the sea. Grothendieck views the mathematician and the problem as complimenting each other, the mathematician using the problem’s natural structure in its solution, rather than striking it with a foreign, invasive method.

###### (Sadly I’ve only read the small part of Récoltes et Samailles that is translated to English, and that is hard to find. The version which I originally read has disappeared from the internet, and I expect this link to become dead too, so now I’ve kept an offline version to be safe.)

## Slides on Breaking the Rules

I gave a talk last year at Duskofest in Oxford (a workshop celebrating the *n*th birthday of Dusko Pavlovic, where *n* is sufficiently large), based on my blog post Breaking the rules.

I have finally remembred to upload my slides here.

This is the only time I can remember enjoying creating slides.

## Towards compositional game theory

- PhD thesis, Queen Mary University of London
- Links: My preferred version, Official version

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.

## Compositional game theory reading list

The best starting point, for a reader who knows a little about both game theory and category theory, is the paper Compositional game theory.

Additional background and motivation is provided by the blog post A first look at open games and the preprint Compositionality and string diagrams for game theory.

By far the most complete exposition is my PhD thesis Towards compositional game theory. It is fully self-contained for readers who know category theory but not game theory.

If you don’t have background in category theory, my current recommendation is Seven sketches in compositionality by Brendan Fong and David Spivak.

## A first look at open games

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.

## Breaking the rules

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.

## Small article in Inspired Research

I have a small article in Inspired Research, the biannual magazine of the cs.ox department. Read it on page 16 here.

## Patch for ‘Coherence for lenses and open games’

- 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.

## A generalisation of Nash’s theorem with higher-order functionals

- In
*Proceedings of the Royal Society A*, 2013 - Links: published article, arXiv
- DOI:

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.

Continue reading “A generalisation of Nash’s theorem with higher-order functionals”