The Anti-Pattern Game

Here is a small sport I though of when taking a course on modal logic. It's a bit too tedious to be played by people, but I had great fun finding a profitable stategy for it - and there are still easy questions about this sport I can't answer.

The foundations

The anti-sample game is a two-player sport. It is performed utilizing black and white pebbles (like Go). These pebbles are placed in sequence on a line. On their flip the participant has a simple choice: to proceed the sequence with a white or a black pebble.

A participant looses if they put down a pebble such that the sequence contains a pattern. A pattern is a sequence of pebbles repeated 3 times in a row.

A participant wins if the opposite participant looses.

Example

Here's a simlpe sport gained by player 1:

Player two misplaced because after their closing turn, the sequence ended with the pattern: ● ○ ○ ● repeated trice.

Player two could have chosen to play ○ instead on the ultimate flip, but that will even be a loosing move since then ○ would have been repeated trice. So if player two may have gained, it must have changed its course a number of moves ago.

A winning strategy - Ready Player 1

The obvious query about any such two-participant game is weather any participant has a successful technique. Often this may be determined by a finiteness situation, however here we might no less than think about that the sport might go on to generate an infinitely long sequence.

To unravel this query I wrote a brief Haskell program which does a brute power search to find a profitable strategy. Actually, the Haskell program wasn’t that short - as a result of I wished to use modal logic to formulate the profitable technique condition and needed to make applicable definitions.

Because it seems, there is a successful technique for participant 1, and they will win the sport inn less than twenty-two strikes. The Haskell program finds the solution represented as a finite tree, which decides the moves of participant 1 given any doable move of player 2.

Generalisations

Regardless that the game is solved, there are extra questions we could pose if we make small adjustments to the game. A few of these questions I don't yet know the reply to.

Cooperating gamers

This question is simple to formulate: If the palyers have been cooperating, may the sport go on for ever? Put extra exactly:

Is there an infinite binary sequence the place no contigous subsequence is of the type σσσ for some sequence σ of non-zero length?

First off, it is clear that this game can proceed for a long time. Here's a recreation with a thousand performed moves:

At first sight this may appear disorderly, however there's numerous repetition right here:

- The sequence of the primary three pebbles ● ● ○ covers most of the sport: Image

Image

The truth that ● ● ○ is so outstanding in this instance, suggest a method to generate an infinite recreation sequence:

This makes stream an inductively outlined stream of pebbles, which makes up an infinitely lengthy game. An attention-grabbing query is then: Can participant 1 force the game to go on idefinitely (assuming participant 2 is not going to unfastened if avoidable)?

As an aside: This stream, whereas devoid of direct repetition is very compressible. It is also an instance of information which responds effectively to being compressed several times. Listed here are the first 1 million moves of the stream compressed twice. The file is a mere 512 bytes, and unpacks to a 26kb file, which once more unpacks to 3Mb. That’s a compression factor of virtually 6000!1

More colours

What if the sport had more colours? What if there have been three colours of pebbles, not just two?

More repetition allowed

The selection that three repetitions form a sample is arbitrary. What if we allowed three repetitions, however prohibited 4? Clearly the games would be longer, but would it make a necessary difference to wether a player has a successful technique?

More gamers

What if there are more gamers? This might influence the dynamics, and that i do not know of a profitable technique for the three player variant. Since the sport might go on for ever, it is not obvious that there is a winning technique for any participant when he controls only a 3rd of the moves.

There can be a question of who winns in a three participant game. Below is a document of one, where participant 1 places the loosing pebble. But did participant 2 or player three win?

Different profitable situations would give different incentives. Listed here are two possible winning situations:

Cooperative victory: If a player looses the opposite two win.Exclusive victory: If participant n looses, then player n + 1 (mod 3) wins.

In the cooperative variant it is obvious that any two players might cooperate to win - mainly as a result of they out-number the third. But in a neurtal setting, who might cooperate? Player 1’s first transfer has no impact, but participant 2 might play the same colour as Player 1, with a purpose to open up for a cooperation with player 1. Then participant three has a pressured move, and when Player 1 is ready to make his move once more, the board appears to be like as follows:

If participant 1 plays ● now, he and participant 2 can pressure player 3 to unfastened:

If player 1 as an alternative plays ○ the enjoying area is more open, and participant three will get to chose his move:

But regardless that participant 2 and participant 3 can cooperate to pressure player 1’s strikes on this setup, participant 3 would free the sequence:

Thus, for the cooperative variant, the question is: Can both participant 1 or participant 2 select to cooperate with participant three as an alternative? The answer remains elusive for the time being.

Within the unique variant the player 2 does not want to help player 1 win, so

just isn't acceptable to player 2. However, it now unclear if any participant has a profitable strategy, or if given good play from each player the game would go on idefinitely.

The factor will get higher because the stream progresses: the primary a hundred million strikes might be compressed to a small 7.2kb file - unpacking it fully you may discover that your text editor will wrestle whereas opening the resulting file.↩︎

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