[jdleesmiller]

(Author of the article here.) I agree that it's somewhat unsatisfying that the output of all this is an enormous table of numbers rather than some nice simple rules.

I think it would be interesting to try some of the methods for approximately solving MDPs on this game to see how complicated the machine learning model that approximates the value function (or state-action function) has to be to get reasonable accuracy. Maybe some simple rules would emerge...

[PennRobotics]

What has consistently worked for me is to mostly use 3 directions, forcing all of the large tiles to the bottom row, and crowding the space above the largest number(s) so going in the fourth direction keeps those large numbers along the bottom.

All the way to 1024 is autopilot by just keeping small numbers out of the bottom row. Then, the thinking begins---plan each move to keep the 32s, 64s, 128s on the same edge of the board until you make an inevitable mistake or have bad luck working in the remaining 2x4 or 2x3 section. This last section is where your analysis algorithm would really add value. Plus, using just three directions until it's time to condense large blocks should reduce your computation a lot.

Do you think regular use of all four directions is necessary to get more consistently to higher values like 16k or 32k? (Actually, is 32k possible?) I feel like once you have most of the blocks on board to get to 8k except for perhaps one 256, you can't really have a random 2 or 4 in the corner or it's Game Over in 10 moves or fewer. Moving in the fourth direction makes the 'corner small number' a guarantee when the board is filled like this.

Playing all large numbers on the bottom also means you can calculate an optimal play (using the smaller numbers) in small (2x3, 2x4, 3x3) regions as long as the result comes out at the same grid location consistently---near the smallest large number.

Then, optimization of the large numbers becomes a separate problem. Do you put everything sequential in the bottom row?

4096-2048-512-128

Or do you put high values at the ends?

4096-128-512-2048

Or low values at the ends? (This is not the answer. Getting a medium value from one end to the other---consistently---without messing everything else up is not probable and probably not possible.)

Or should the high values make a wedge along two sides?

512

4096-2048-128

In summary, I don't think trying to solve the full state space makes sense. This game feels like two separate isolatable optimization problems---small and large numbers---plus a little bit of glue when the isolation breaks down: starting a game, fixing a stuck situation, or collapsing a series of large numbers.

Have you thought about using your worst cases to find rules to make Evil 2048 even more evil? Or see how an MDP works against Evil 2048 to get the best outcome in worst case scenarios?

Oh-- Great article and analysis, by the way.