This applies to most if not all games. In our paper "A googolplex of Go games" [1], we write
"Estimates on the number of ‘practical’ n × n games take the form b^l where b and l are estimates on the number of choices per turn (branching factor) and game length, respectively. A reasonable and minimally-arbitrary upper bound sets b = l = n^2, while for a lower bound, values of b = n and l = (2/3)n^2 seem both reasonable and not too arbitrary. This gives us bounds for the ill-defined number P19 of ‘practical’ 19x19 games of 10^306 < P19 < 10^924 Wikipedia’s page on Game complexity[5] combines a somewhat high estimate of b = 250 with an unreasonably low estime of l = 150 to arrive at a not unreasonable 10^360 games."
> Our final estimate was that it is plausible that there are on the order of 10^151 possible short games of chess.
I'm curious how many arbitrary length games are possible. Of course the length is limited to 17697 plies [3] due to Fide's 75-move rule. But constructing a huge class of games in which every one is probably legal remains a large challenge; much larger than in Go where move legality is much easier to determine.
The main result of our paper is on arbitrarily long Go games, of which we prove there are over 10^10^100.
[1] https://matthieuw.github.io/go-games-number/AGoogolplexOfGoG...
[2] https://en.wikipedia.org/wiki/Game_complexity#Complexities_o...
I remember from a lot of combinatorial problems (like cutting up space with hyper-planes or calculating VC dimension) that one sees what looks like exponential growth until you have a number of items equal to the effective dimension of the system and then things start to look polynomial.
BTW: I was going through some of your lambda calculus write-ups a while ago. Really great stuff that I very much enjoyed.
Or maybe the question should be what percent of games reach a position that has never before been seen?
There is additionally the 75-move rule where the the game is forced to be over without either player claiming the rule (the arbiter just ends the game), that rule would give an upper bound regardless of the players knowledge of the rules.
- After threefold repetition or 50 moves, either player may claim a draw.
- After fivefold repetition or 75 moves, the game is automatically drawn.
Most modern counts of the longest possible chess game, or the total number of possible chess games, are based on fivefold repetition and the 75-move rule.
Meanwhile, threefold repetition and the 50-move rule are still relevant in endgame tablebases, since they rule out certain forced mate sequences.