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All I ever found about this were natural language rules à la Asimov's three laws of robotics. The questions are straightforward questions:

1) Is there a well-posed mathematical definition of game on a graph? If so, what is it?

2) Is there a well defined category of games on graphs? If so, what is it?

But i'd love to hear some comments about issues involved in this question.

After some time working on a game on a graph, such as the timber game, or maker-breaker games, one can make $\textit{ad hoc}$ definitions, which usually involve defining a state space and player $1$ and player $2$ allowed moves by defining to which states each can proceed in the state space, from a given state. Can this be formalized into categorical language in a way that encompasses all known games on graphs?

I'm looking for something along the lines of:

given an enriched graph category $\mathcal{Graph}_S$ (where $S$ is some extra structure on the graphs such as labeling, weights, etc...), the associated $S$-game category is the category $\mathcal{Game}_S$ whose objects are ....

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  • $\begingroup$ Did you try to google? I have got lots of interesting papers on this topic. If yes, why are you not satisfied? $\endgroup$
    – András Bátkai
    Commented Sep 1, 2019 at 15:52
  • $\begingroup$ I googled definition of game on graph and category of games on graphs and found nothing mathematically well defined or well-posed. can you provide the names of these papers? $\endgroup$
    – IpsumPanEst
    Commented Sep 1, 2019 at 15:57
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    $\begingroup$ Why should there be such a thing, and what would you hope to learn about it? You can simulate any finite game as a game on a graph, just draw the strategy tree... And if you want to rule out this kind of game, I think you will end up with a natural language definition which is moreover not nice. $\endgroup$
    – user36212
    Commented Sep 1, 2019 at 19:15

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