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Ever since the seminal work of Von Neumann and Morgestern Game Theory has grown into a formidable sector of pure and applied mathematics.

There are all sorts of games: perfect information, cooperative, topological games, combinatorial games, logical games, etc .

The list goes on (almost) forever.

But, I wonder (if the answer is well known, please forgive me, I am not an expert): is there a universal framework, grounded in either set theory or category theory (or even something entirely else) which subsumes all known games into a single corpus?

In other words, suppose someone asks you:

what is a game (as far as mathematics goes, of course) ?

Can one provide a single answer encompassing all of the above as special cases?

Obviously, such an answer should account for multi-players, concurrent or sequential moves, and many other things.

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    $\begingroup$ No, there is not. There is an attempt to give a framework for all imaginable extensive form games in the recent book The Theory of Extensive Form Games by Alós-Ferrer and Ritzberger (and the papers it is based on), but even there, gaps still exist. $\endgroup$
    – Michael Greinecker
    Commented Sep 17, 2016 at 20:04
  • $\begingroup$ Thanks Michael. But then, what is the obstruction? I find this state of affairs a bit disconcerting. It is as if we worked on many types of group theory, but we do not know what a group is... $\endgroup$
    – Mirco A. Mannucci
    Commented Sep 17, 2016 at 20:06
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    $\begingroup$ A very basic obstruction is that there is no obvious mathematical connection between cooperative and noncooperative game theory. For non-cooperative game theory, one can at least ask what is the most encompassing way to represent a strategic situation. Nothing like that is available for cooperative game theory. For the book in question, the obstruction is that it is less clear how one imposes the kind of measurability structures needed to get a satisfactory way of modeling randomization. $\endgroup$
    – Michael Greinecker
    Commented Sep 17, 2016 at 20:15
  • $\begingroup$ I think I would argue that either the group theory analogue is not good, or there is a good answer to the question. The point is that we know exactly what a group is, and if someone comes along and says "but I don't want to assume every element has an inverse" then we tell them go and invent monoids. If you can really properly define what you mean by a game then one can axiomatise this definition. For example a 2-player game just seems to me to be nothing more than a set, where player 1 chooses an element, and then player 2 chooses an element of the set P1 chose etc. That will do for mostgames $\endgroup$
    – znt
    Commented Sep 17, 2016 at 20:36
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    $\begingroup$ In what sense do you know what football is? I don't. I can point to a rule book, and I've played, but that tells me very little about what the abstract game is, and how you might state an optimal strategy. Is highly unsportsmanlike behavior youtube.com/watch?v=iHsg6Bb02L4 accepted as part of the game? Is part of the game deciding which players should be on the team, and which skills to train? $\endgroup$
    – Douglas Zare
    Commented Sep 18, 2016 at 0:40

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