An AI for 2048 – Part 3 Biased Random Blind Strategies

March 23, 2014 in Games by hundalhh | Permalink

 

I will be continuing to post articles on AI for the game 2048.  The first two articles were:

 

PROGRAMMING

In this article, I will make some comments on the cyclic blind strategies and report on random blind strategies.  The cyclic results can be generated from the Mathematica source code that I posted earlier.  The Biased Random results below can be obtained by just modifying the strategy function in the Mathematica source code.

 

CYCLIC BLIND

In the Cyclic Blind Article, we saw that left-down-right-down had the highest performance of any length 4 or less cyclic strategy with an average end tile total (AETT) of about 680.  The other cyclic strategies fell into the following seven groupings:

 

  1. All four directions alternating vertical and horizontal (dlur, drul)  420 AETT.
  2. All four directions without alternating vertical and horizontal (durl, dulr, dlru, …)  270 AETT.
  3. Three directions with one direction repeated twice in a row where the directions that were not repeated were opposite (durr, dllu, dull, drru, …) 390 AETT
  4. Three directions with one direction repeated twice in a row where the directions that were not repeated were perpendicular (drrl, duur, ddlu, …) 330 AETT
  5. Three directions with one direction repeated, but not twice in a row, and the directions that were not repeated were perpendicular (dudr, duru, dldu …) 295 AETT
  6. Three directions with no repetition (dlu, dur, …)  360 AETT
  7. One Horizontal Direction + One vertical direction with or without repetition 60 AETT.

 

BLIND RANDOM BIASED STRATEGIES

The other blind strategy to consider is independent random biased selection of directions.  For example, someone might try rolling a six sided die and going left on 1, right on a 2 and down for rolls of 3, 4, 5, or 6.  I ran 1000 games each for every possible biased random strategy where the strategy was of the form

left with probability L/T, right with probability R/T,  up with probability U/T, down with probability D/T,

where T was the total of L, R, U, and D, and L, R, U and D between 0 and 4.

The results are tabulated below

AETT    MAX   
        TILE  L/T   R/T   U/T   D/T
11.138   16   1.    0.    0.    0.
55.416   128  0.57  0.    0.43  0.
56.33    128  0.67  0.    0.33  0.
57.17    64   0.5   0.    0.5   0.
57.732   128  0.6   0.    0.4   0.
70.93    16   0.6   0.4   0.    0.
71.246   32   0.57  0.43  0.    0.
71.294   32   0.5   0.5   0.    0.
71.752   32   0.67  0.33  0.    0.
234.42   256  0.4   0.4   0.1   0.1
238.182  256  0.38  0.38  0.12  0.12
241.686  256  0.44  0.33  0.11  0.11
245.49   256  0.44  0.44  0.11  0.
246.148  256  0.4   0.3   0.2   0.1
247.046  256  0.5   0.25  0.12  0.12
247.664  256  0.36  0.36  0.18  0.09
250.372  256  0.43  0.29  0.14  0.14
253.09   256  0.44  0.22  0.22  0.11
254.002  512  0.5   0.38  0.12  0.
255.178  256  0.33  0.33  0.22  0.11
255.242  256  0.33  0.33  0.17  0.17
255.752  256  0.33  0.33  0.25  0.08
257.468  512  0.43  0.43  0.14  0.
258.662  512  0.36  0.27  0.27  0.09
258.832  256  0.57  0.29  0.14  0.
258.9    256  0.36  0.27  0.18  0.18
260.052  256  0.3   0.3   0.2   0.2
261.676  256  0.38  0.25  0.25  0.12
261.902  256  0.31  0.31  0.23  0.15
262.414  256  0.27  0.27  0.27  0.18
263.702  256  0.27  0.27  0.27  0.2
264.034  256  0.31  0.23  0.23  0.23
264.17   512  0.57  0.14  0.14  0.14
264.2    256  0.29  0.29  0.29  0.14
264.226  256  0.31  0.23  0.31  0.15
264.232  256  0.29  0.29  0.21  0.21
264.33   256  0.5   0.17  0.17  0.17
265.328  256  0.33  0.22  0.22  0.22
265.512  256  0.4   0.2   0.3   0.1
265.724  256  0.4   0.2   0.2   0.2
265.918  512  0.33  0.25  0.25  0.17
265.986  256  0.3   0.3   0.3   0.1
266.61   256  0.25  0.25  0.25  0.25
266.796  256  0.29  0.21  0.29  0.21
266.812  256  0.31  0.31  0.31  0.08
267.096  256  0.33  0.17  0.33  0.17
267.5    512  0.33  0.25  0.33  0.08
267.804  256  0.36  0.18  0.27  0.18
267.99   256  0.3   0.2   0.3   0.2
268.414  512  0.43  0.14  0.29  0.14
268.634  256  0.44  0.11  0.33  0.11
271.202  256  0.38  0.12  0.38  0.12
271.646  256  0.33  0.22  0.33  0.11
271.676  512  0.5   0.33  0.17  0.
272.128  256  0.36  0.18  0.36  0.09
273.632  256  0.5   0.12  0.25  0.12
279.82   256  0.4   0.1   0.4   0.1
281.958  256  0.4   0.4   0.2   0.
291.702  512  0.44  0.33  0.22  0.
306.404  256  0.38  0.38  0.25  0.
306.508  512  0.5   0.25  0.25  0.
310.588  512  0.43  0.29  0.29  0.
310.596  512  0.36  0.36  0.27  0.
321.016  512  0.4   0.3   0.3   0.
333.788  512  0.33  0.33  0.33  0.
340.574  512  0.5   0.17  0.33  0.
341.238  512  0.57  0.14  0.29  0.
342.318  512  0.44  0.22  0.33  0.
347.778  512  0.36  0.27  0.36  0.
355.194  512  0.38  0.25  0.38  0.
366.72   512  0.5   0.12  0.38  0.
369.35   512  0.43  0.14  0.43  0.
371.63   512  0.4   0.2   0.4   0.
384.962  512  0.44  0.11  0.44  0.

There are a few things you might notice.

  • The  best strategy was 44% Left, 11% Right, and 44% down with a AETT of 385 which was much better than pure random (25% in each direction), but not nearly as good as the cyclic strategy left-down-right-down (AETT 686).
  • The AETT varies smoothly with small changes in the strategy.  This is not surprising.  You would think that 55% up, 45% down would have nearly the same result as 56% up and 44% down.  The Extreme value theorem then tells us that there must be a best possible random biased strategy.
  • The top 17 strategies in the table prohibit one direction with AETTs of 280 to 385.
  • If the strategy uses just two directions, then it does very poorly.

 

Well that’s it for blind strategies.  Over the next few posts, we will introduce evaluation functions, null moves, quiescent search, alpha-beta prunning, and expectimax search.