Sorry guys, but I won the 3x3 to 1024 on the first attempt, so the next 99 of you will only see losses. ;)
> We can also see it being ‘lazy’ — even when it has two high value tiles lined up to merge, it will continue merging lower value tiles. Particularly within the tight constraints of the 3x3 board, it makes sense that it will take the opportunity to increase the sum of its tiles at no risk of (immediately) losing — if it gets stuck merging smaller tiles, it can always merge the larger ones, which opens up the board.
One flaw in the strategy I noticed was failure to prioritize getting higher values together when board space is scarce. That is, it seems to employ the same strategy regardless of board free space. (Maybe they addressed this, admittedly didn't read all.) Or maybe that is better than my strategy.
Boardspace creates uncertainty - on a wide open board there are more places the new tiles can appear. It seems possible that controlling the amount (or, more precisely, the position) of boardspace helps reduce the odds of new tiles appearing in unhelpful places, so prematurely combining tiles could be an antistrategy.
Strictly speaking, all we can say is that the optimal strategy used in my post is no worse than ballenf's. It's possible that ballenf's strategy is also optimal --- i.e. that they are equally good. :)
Yes! Optimal strategies for playing a game too complicated to reason out completely in your head, or which has some random element often aren't unique.
Cepheus http://poker.srv.ualberta.ca/ is a Heads Up Limit Texas Hold 'Em strategy (two players, no betting strategy needed beyond whether to fold / call / raise) that is proved to be approximately optimal. There absolutely must be other strategies that are similar, and in particular circumstances one would dominate another, it's just that since they're all optimal if they played random hands over time they'd all break even.
> We can also see it being ‘lazy’ — even when it has two high value tiles lined up to merge, it will continue merging lower value tiles. Particularly within the tight constraints of the 3x3 board, it makes sense that it will take the opportunity to increase the sum of its tiles at no risk of (immediately) losing — if it gets stuck merging smaller tiles, it can always merge the larger ones, which opens up the board.
One flaw in the strategy I noticed was failure to prioritize getting higher values together when board space is scarce. That is, it seems to employ the same strategy regardless of board free space. (Maybe they addressed this, admittedly didn't read all.) Or maybe that is better than my strategy.