Even if you are not into chess, I think you will enjoy reading about Caruana’s amazing performance at Sinquefield.

(The cast comes off my hand today.  I will be able to type with both hands soon!)

Monte Carlo Tree Search (survey) is working rather well for Go.

So far we have only considered blind strategies and a simple greedy strategy based on maximizing our score on every move.  The results were interesting because the blind cyclic strategy left-down-right-down did surprisingly well and we saw that a biased random strategy which strongly favored two orthogonal moves like left and down with a small amount of one other move like right outperformed all of the other biased random strategies.

If we want to do better, we need something more sophisticated than a blind strategy or pure, short-term, score greed.

The first approach used by game AI programmers in the olden days was to simply program in the ideas that the programmer had about the game into the game AI. If the programmer liked to put an X in the middle of a tic-tac-toe board, then he hard-wired that rule into the computer code.  That rule alone would not be enough to play tic-tac-toe, so the programmer would then just add more and more decision rules based on his own intuition until there were enough rules to make a decision in every game situation. For tic-tac-toe, the rules might have been

• If you can win on your next move, then do it.
• If your opponent has two in a row, then block his win.
• Take the center square if you can.
• Take a corner square if it is available.
• Try to get two squares in a single row or column which is free of your opponent’s marks.
• Move randomly if no other rule applies.

Though this approach can be rather successful, it tends to be hard to modify because it is brittle.  It is brittle in two ways: a new rule can totally change the behavior of the entire system, and it is very hard to make a tiny change in the rules.  These kind of rule based systems matured and incorporated reasoning in the nineties with the advent of “Expert Systems“.

Long before the nineties, the first game AI programmers came up with another idea that was much less brittle.  If we could just give every position a numerical rating, preferably a real number rather than an integer, then we can make small changes by adding small numbers to the rating for any new minor feature that we identify.

In tic-tac-toe, our numerical rating system might be as simple as:

• Start with a rating of zero
• Add 100 to the rating if I have won,
• Subtract 100 if my opponent has won or will win on his next move,
• Add 5 if I own the central square,
• Add 2 points for each corner square that I own,
• Add 1 point for every row, column, or diagonal in which I have two squares and my opponent has no squares.

So, for example, the position below would be scored as

2 (for the upper left corner x) + 2 (for the other corner) + 1 (for the top row) = 5.

Now, in order to decide on a move, we could just look at every possible move, rate each of the resulting boards, and choose the board with the highest rating. If our tic-tac-toe position was the one below and we were the x player,

then we would have six possible moves.  Each move could be rated using our board rating system, and the highest rated board with x in the center and a rating of 8 would be chosen.

$\ $

This approach was first applied to computer game AI by the famous theoretician Claude Shannon in the paper “Programming a Computer for Playing Chess” (1949), but the idea without the computers actually predates Shannon by hundreds of years.  For instance, the chess theoretician Giambatista Lolli published “Osservazioni teorico-pratiche sopra il giuoco degli scacchi ossia il Giuoco degli Scacchi” in 1763.  He was the first to publish the idea of counting pawns as 1, bishops as 3, knights as 3, rooks as 5, and queens as 9 thereby giving a rating to every board position in chess (well, at least a rating for the pieces).  Shannon modified this rating by subtracting 0.5 for each “doubled” pawn, 0.5 for each “isolated” pawn, 0.5 for each “backward” pawn, and adding 0.1 for each possible move.  Thus, if the pieces were the same for both players, no captures were possible, and if the pawns were distributed over the board in a normal fashion, Shannon’s chess AI would choose the move which created the most new moves for itself and removed the most moves from his opponent.

Shannon significantly improved on the idea of an evaluation function by looking more than one move ahead.  In two players games, the main algorithm for looking a few moves ahead before applying the evaluation function is the alpha-beta search method which will be tested in a later post.

Let’s try to find and test a few evaluation functions for 2048.

Perhaps the simplest idea for evaluating a position is related to flying. According the great philosopher Douglas Adams,

“There is an art to flying, or rather a knack. Its knack lies in learning to throw yourself at the ground and miss. … Clearly, it is this second part, the missing, that presents the difficulties.”  (HG2G).

Suffice it to say that in 2048, the longer you delay the end of the game (hitting the ground), the higher your score (the longer your flight).  The game ends when all of the squares are filled with numbers, so if we just count every empty square as +1, then the board rating must be zero at the end of the game.

I ran 1000 simulated games with the Empty Tile evaluation function and got an average ending tile total of 390.  The strategy ended the games with a maximum tile of  512 tile 25 times, a 256 tile 388 times, 128 491 times, 64 92 times, and 32 4 times.

Well, that didn’t seem like anything spectacular.  Let’s try to group like tiles—if there are two 128 tiles, let’s try to put them side-by-side.  Of course, the idea of putting like tiles together is obvious.  Slightly less obvious is the idea of trying to put nearly equal tiles together.  Over at stackoverflow.com, the user ovolve, the author of the first 2048 AI, suggested the criteria of “smoothness” in his great post about how his AI worked.  There are some interesting ways to measure smoothness with Fourier transforms, but just to keep things simple, let’s measure smoothness by totaling the differences between side-by-side tiles.  For example, the smoothness of the board below would be

2+2+4 (first row) + 30+32 (second row) + 14+ 48 + 64  (third row) + 12 + 16 (fourth) + 2+0+2 (first col) + 30+16 + 0 (second) + 4 + 64 + 64   + 128 + 128

which sums to 662.

Once again I ran a 1000 games using smoothness as an evaluation function. The average ending tile total was 720, a new high.  The strategy had a maximum ending tile total of  1024 3 times, 512 263 times, 256 642 times, 128 87 times, and 64 5 times.  This looks very promising.

Let’s try ovolve’s last criteria — monotonicity.  The idea of monotonicity is that we prefer that each row or column is sorted.  On the board above, the top row

4 > 2 < 4 > 0 is not sorted.  The second 2 < 32 > 0 = 0 and forth 4 < 16 > 0 = 0 rows are sorted only if the spaces are ignored. The third row 2 < 16 < 64 < 128 is fully sorted.  Let’s measure the degree of monotonicity by counting how many times we switch from a less than sign to a greater than sign or vice-versa.  For the first row we get 2 changes because the greater than sign 4 > 2 changed to a less than sign 2 < 4 and then changed again in 4 > 0. We also get two changes for 2 < 32 > 0, no sign changes in the third row (perfectly monotonic), and two changes in the last row.

I ran 100 games using the monotonicity evaluation function.  The average ending tile total was 423.  The maximum tile was 512 16 times, 256 325 times, 128 505 times, 64 143 times, and 32 11 times.

I have summarized the results of these three simple evaluation functions in the table below.

Well, clearly Smoothness was the best.

There are three great advantages to numerical evaluation functions over if-then based rules.  One, you can easily combine completely different evaluation functions together using things like weighted averages.  Two, you can plot the values of evaluation functions as the game proceeds, or even plot the probability of winning against one or two evaluation functions.  Three, in 2048, there is the uncertainty where the next random tile will fall.  With numerical evaluation functions, we can average together the values of all the possible boards resulting from the random tile to get the “expected value”  (or average value) of the evaluation function after the random tile drop.

Well, that’s it for my introduction to evaluation functions.  Next time we will focus on more sophisticated evaluation functions and the powerful tree search algorithms that look more than one move ahead.

Have a great day.  – Hein

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

• An AI for 2048 — Part 1 – which described the game 2048 and reported the results of pure random play and a very simple greedy strategy that just tried to maximize it’s score on every move.
• An AI for 2048 — Part 2 Cyclic Blind Strategies – which reported on strategies that just cyclically repeated the same key stokes.

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.

A guy named Phil asked me to post my code, so here it is.  It’s in Mathematica, so I imagine that most people will have trouble reading it.  I will probably rewrite the code in a more common, faster language later.

(* Mathematica code for simulating 2048
   - by Hein Hundal (Public Domain)

   Visit  http://gabrielecirulli.github.io/2048/

   for more detials.
*)

(* collapse[v] takes a list of values and returns a 
   collapsed list of values where two consecutive equal 
   values are summed into one value. *)

collapse[v_List] := PadRight[
   collapseAux[Cases[v, _Integer]]  , 4, "."];

collapseAux[{}] = {};
collapseAux[{x_}] = {x};
collapseAux[v_List] := If[ v[[1]] == v[[2]],
   Prepend[ collapseAux[Drop[v, 2]], v[[1]]*2],
   Prepend[ collapseAux[Drop[v, 1]], v[[1]]]];

vGlobalMoves = Characters["lrud"];
mGlobalEmptyBoard = Table[".", {4}, {4}];

move[mBoard_, sMove_String] := Switch[sMove,
   "l", collapse /@ mBoard,
   "r", Reverse /@ collapse /@ Reverse /@ mBoard,
   "u", Transpose[ collapse /@ Transpose[ mBoard ]],
   "d", Reverse[ Transpose[ 
        collapse /@ Transpose[ Reverse[ mBoard]]]],
   _, Throw[{"move::illeagal move", sMove}]];

(* game1Turn[ mStart_List, randFunc_, moveStrat_]
      Performs one turn of the game.
      - mStart is a 4 x4 game matrix where every elemet 
        is either a number 2, 4, 8, ... or the string ".".
      - randFunc is any function that take a positive 
        integer n as input and outputs a positive integer 
        between 1 and n.
      - moveStrat is any function that takes a game board as 
        an input and gives as an output one of the four 
        characters u, d, l, r.
      - The output of game1Turn is a new board state.  *)

game1Turn[ mStart_List, randFunc_, moveStrat_] :=
  Module[{sMove, mBoard, mEmpty, iSpot, iVal},
   sMove = moveStrat[mStart];
   mBoard = move[mStart, sMove];

   (* only add a new piece if the board changed *)
   If[ mBoard =!= mStart,
      mEmpty = Position[mBoard, "."];
      iSpot = randFunc[Length[mEmpty]];

      (* the new board tile will either be a 4 or a 2 *)
      iVal = If[ randFunc[10] == 1, 4, 2];
      mBoard = ReplacePart[mBoard, mEmpty[[iSpot]] -> iVal]
    ];
   mBoard];

(*  gameManyTurns  - executes iDo turns of the game  *)
gameManyTurns[mStart_List, randFunc_, moveStrat_, iDo_Integer] := 
   NestList[game1Turn[#, randFunc, moveStrat] &, mStart, iDo];

(******************* Display Results of Multiple Runs **********)

periodTo0[m_List]  :=  m /. "." -> 0;
maxTile[m_List]    := Max[Flatten[periodTo0[m]]]
totalTiles[m_List] := Total[Flatten[periodTo0[ m ]]];

rand1[i_Integer] := 1 + RandomInteger[i - 1];

(* rand2[m]   replaces a random entry on the board m with a 2 *)
rand2[m_List] := ReplacePart[ m, 
   (RandomInteger[3, {2}] + 1) -> 2];

runSeveralGames[ randFunc_, moveStrat_, iDo_Integer]  := 
   Module[{},
   Table[
    (* run a single game for 100 turns *)
    ten1 = gameManyTurns[rand2[mGlobalEmptyBoard], 
               randFunc, moveStrat, 100];

    (* keep going until there is not change for 50 moves *)
    While[ ten1[[-50]] =!= ten1[[-1]]  && Length[ten1] < 10000,
       ten1 =  Join[ten1, gameManyTurns[ten1[[-1]], 
                    randFunc, moveStrat, 100]]
    ];

    ten2 = TakeWhile[ ten1, # =!= ten1[[-1]] &];

    (* output a list {# turns of the game, tile Total,
    maximum tile} for each game *)
    {Length[ten2], totalTiles[Last[ten1]], maxTile[ten1[[-1]]]},
    {iDo}]];

stats[mRes_List] := Module[{mRN = N@mRes},
   {Mean[mRN], StandardDeviation[mRN], Max[mRes[[All, 3]]],
    Tally[mRes[[All, 3]]] // Sort}];

(******** Blind Cyclic Strategy ****************************)

(* createCyclicStrategy - creates a cyclic strategy function 
   from the string s.  If s = "uddl", then the strategy function 
   will repeat the sequence move up, move down, move down, and 
   move left indefinitely. *)

createCyclicStrategy[sMoves_String] := Module[
   {exHeld, iCount = 1},
   exHeld = Hold[
     Function[ m, chars[[ Mod[iCount++, iStringLength] + 1]]]];
   ReleaseHold[
    exHeld /. {chars -> Characters[sMoves] ,
      iStringLength -> StringLength[sMoves]}]];

testOneStrategy[] := Module[{},
   stratDRDL = createCyclicStrategy["drdl"];
   mRes = runSeveralGames[rand1, stratDRDL, 100];
   stats[mRes]];

In my last post, I described how to play 2048 and reported the results of some simple simulations.  Namely, I reported the results of two very simple strategies:

• Random play will achieve an average ending tile total (AETT) of 252 and usually the highest tile will be a 64 tile or a 128 tile (about an 85% chance of ending with one of those).
• A pure greedy strategy where the player just tries to make the largest new tile possible results in an average ending tile total (AETT) of 399 and usually the highest tile will be a 128 tile or a 256 tile (about an 87% chance of ending with one of those).

Blind Strategies

Perhaps surprisingly, there are “blind” strategies that will do even better than the pure greedy strategy.  Blind strategies are strategies that ignore the current state of the board.  I tried two types of simple blind strategies “biased random” and “repeating sequences”.  I will summaries the results of the biased random trials in my next post.

The simplest repeating strategy would be just hitting the same key every time—down, down, down, ….   Of course that does not do well.  Nor would down, up, down, up, ….  So the simplest repeating strategy worth trying would be down, right, down, right or some other combination of horizontal and vertical moves.  I ran 100 games for every possible repeated sequence of two to four moves and found that the best of these sequences was down, right, down, left  or a version of that sequence in disguise.  The results are captured in the table below.

Notice that restricting your moves to only two out of the four possibilities does poorly.  The Down, Left, Down, Left… strategy had an average tile total of only 44.38 at the end of the game and that strategy never created a tile above 64.

The best strategies were down-right-down-left or that same strategy in disguise (dlul, dldr, and drur are really the same strategy).  Notice that this strategy does better than pure greedy which had an AETT of 399.