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As, from what I remember, the article says considering all possible games is meaningless. So what people usually do is get statistics for the average branching factor and the length of the game. With a program like chessV you can do this. Take grand chess for example. Run 1000 games in decent conditions. Say 4mins +3 secs time. Read the data. You probably need help here but I think it can be set up. Same with Omega chess or Shako (one of your favorites I remeber) for a few more known games.
It seems to me though that you aim for the more theoretical rather than applied part of math. There might be relevant conclusions to be drawn from things like number of captures and so on as long as games are chesslike enough.
For some games like go or havannah the game lenght is quite obvios based just on board size.
When the time I'll maybe try to deduce some such heuristics on games, I'll try on Apothecary as the final article on them is pending on my to do list along with chessV2 expeiments which I hope to be able to do soon!...
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Today I was looking at the wiki re: the mathemematical concept of Game Complexity, which led to the following sub-link on Game-tree Complexity, that gave me food for thought about CVs that might one day compete in terms of popularity with such well known games as Chess (which arguably is slowly being played out at elite level, in terms of opening theory):
https://en.wikipedia.org/wiki/Game_complexity#Game-tree_complexity
If the mathematically inclined on this CVP website are interested, perhaps there might be a way to evaluate the Game-tree Complexity of various CVs on CVP, and even slowly compile a list of those CVs with very large Game-tree Complexity number estimated for them; thus such CVs, if they ever become popular, might have a longer lifespan than e.g. Chess, and part of popularizing such CVs could be pointing out their great Game-tree Complexity number (as estimated).
Just as an example, here's a link to a 10x10 CV of my own invention:
https://www.chessvariants.com/rules/sac-chess
I'm not mathematically inclined, but I'd guess that the branching factor for Sac Chess (based on the Chess and Sac Chess armies operating at maximum centralized power for each piece in each game's setup) would be 2 to 3 times higher than for Chess on average, and I'd also guess that the average game of Sac Chess (especially for well-played games) would be about 2 to 3 times more than for Chess in terms of number of ply (a piece is traded in Chess about every 10 ply, I've read long ago, so a game of Chess [or possibly Sac Chess?!] would thus be over on average with a total of 32-2x(70/10)=18 units left on the board), and thus Sac Chess' Game-tree Complexity may be ((35 [=avg. # moves in chess] x2.5) to the power of 70 [=avg. # ply in a game of chess] x2.5, if I understand the Game tree Complexity estimate formula right) which may well easily exceed that of e.g. the game of Go - see the chart given further below in the sub-link on Game-tree Complexity that I first gave. Note that in played Sac Chess games I've looked so far, it seems a pair of pieces are traded about every 8 ply (rather than every 10 ply, as in Chess) on average, but this is based only on a small observation.