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.
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.
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.
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.