fair share go

phillip1882 · #1

i believe this would make for a fair game without komi.
use the fair share sequence,
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 etc. for move order.

CDavis7M · #2

It would be a different game entirely if the players could play twice in a row.

gennan · #3

Groups may need more than 2 eyes when players can play more than 1 stone at a time, or are multiple moves on a turn not considered simultaneous?

phillip1882 · #4

Quote:

or are multiple moves on a turn not considered simultaneous?

yes, each move has to be go legal. your not playing two move in a row, your taking two turns.

Matti · #5

phillip1882 wrote:

i believe this would make for a fair game without komi.
use the fair share sequence,
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 etc. for move order.

How do you suggest to keep track whose move is it?

tundra · #6

Matti wrote:

phillip1882 wrote:

i believe this would make for a fair game without komi.
use the fair share sequence,
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 etc. for move order.

How do you suggest to keep track whose move is it?

If nothing else, one could print out the sequence on paper beforehand. (It is always the same, and is not difficult to generate.) Then, each time a move is made, cross out the current number. The next number tells us whose move it is. A bit low-tech, but it should work.

Btw, for those who may not have seen this sequence before, it has actually been known and studied for a while. Mathematicians sometimes refer to it as the Thue–Morse sequence. It has been independently rediscovered several times.

(To take a personal example, I once knew someone who discovered it for himself when he was in Grade 6, i.e., 11 or 12 years old. He was, like the OP, trying to find a way to make turn-based games more fair. I was rather impressed, though we never actually tried it out.)

(Also, the wikipedia page mentions another application: allocating items between two people as part of, say, a divorce settlement. For example, the wife might get first choice from the pile, then the husband gets second and third choices, then the wife gets fourth choice, and so on, just as in the Thue-Morse sequence. But I do not know if this has ever been tried in real life.)

As for whether this is necessary to make go into a fair game, I am doubtful. (Indeed, it is not even clear if Thue-Morse would make it fair. There might still be an advantage for one of the players.) We already have ways to balance things between players, namely komi and, if needed, handicap stones.

Bill Spight · #7

phillip1882 wrote:

i believe this would make for a fair game without komi.
use the fair share sequence,
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 etc. for move order.

tundra wrote:

Btw, for those who may not have seen this sequence before, it has actually been known and studied for a while. Mathematicians sometimes refer to it as the Thue–Morse sequence. It has been independently rediscovered several times.

(To take a personal example, I once knew someone who discovered it for himself when he was in Grade 6, i.e., 11 or 12 years old. He was, like the OP, trying to find a way to make turn-based games more fair. I was rather impressed, though we never actually tried it out.)

(Also, the wikipedia page mentions another application: allocating items between two people as part of, say, a divorce settlement. For example, the wife might get first choice from the pile, then the husband gets second and third choices, then the wife gets fourth choice, and so on, just as in the Thue-Morse sequence. But I do not know if this has ever been tried in real life.)

As for whether this is necessary to make go into a fair game, I am doubtful. (Indeed, it is not even clear if Thue-Morse would make it fair. There might still be an advantage for one of the players.) We already have ways to balance things between players, namely komi and, if needed, handicap stones.

(Emphasis mine.)

It may apply to Nim, but I doubt if it applies to go. To quote Wikipedia: https://en.wikipedia.org/wiki/Thue–Morse_sequence#Equitable_sequencing

Wikipedia wrote:

In their book on the problem of fair division, Steven Brams and Alan Taylor invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called balanced alternation, or taking turns taking turns taking turns . . . , as a way to circumvent the favoritism inherent when one party chooses before the other.

(Emphasis mine.)

For instance.

Black wins 3x3

Click Here To Show Diagram Code: [go]$$ Black wins 3x3 $$ ------- $$ | . C . | $$ | C 1 C | $$ | . C . | $$ -------[/go]

The marked points may seem preferable to the unmarked points for both players, but isn't that so only for Black? They help Black to form an eye. Even if White takes two of them on his next turn, Black can form an eye with the other two, and win the game.

Var 1

Click Here To Show Diagram Code: [go]$$ Var 1 $$ ------- $$ | . 2 . | $$ | 3 1 2 | $$ | . 3 . | $$ -------[/go]

Var 2

Click Here To Show Diagram Code: [go]$$ Var 2 $$ ------- $$ | . 2 . | $$ | 3 1 3 | $$ | . 2 . | $$ -------[/go]

tundra · #8

Hello Bill:

I agree with the general view, that Thue-Morse may not always lead to a fair game. But I am not sure if your example follows the Thue-Morse sequence.

The sequence is:
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 ...
Or, to make the application to Go more explicit, for players Black and White:
B W W B W B B W W B B W B W W B ...

In particular, the first five moves are:
B W W B W, with the fifth move being White's.
However, your example seems to follow:
B W W B B, with the fifth move being Black's.

Looking forward to your update (unless I am mistaken, in which case, mea culpa) ;-)

Bill Spight · #9

tundra wrote:

Hello Bill:

I agree with the general view, that Thue-Morse may not always lead to a fair game. But I am not sure if your example follows the Thue-Morse sequence.

The sequence is:
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 ...
Or, to make the application to Go more explicit, for players Black and White:
B W W B W B B W W B B W B W W B ...

In particular, the first five moves are:
B W W B W, with the fifth move being White's.
However, your example seems to follow:
B W W B B, with the fifth move being Black's.

Looking forward to your update (unless I am mistaken, in which case, mea culpa) ;-)

It's symmetrical. The advantage is to the first player, whoever that may be. If the empty points correspond to the items on the pile, then after the first play, their preferences are different for each player. That is different from Nim, where no item in any Nim pile is preferable to any other item in that pile. OC, the piles need not be equivalent.

Here is another example from a modified game of tictactoe in CGT. As we know, tictactoe is a tie with correct play. However, as the first player gets the last play, it is, under CGT a first player win. So let's change the rules so that each player gets four moves. This new tictactoe game is a second player win. Using go stones to play tictactoe, suppose that the first player (Black, by convention) starts in the corner.

White wins new tictactoe

Click Here To Show Diagram Code: [go]$$ White wins new tictactoe $$ ------- $$ | . . 1 | $$ | . 2 . | $$ | . . . | $$ -------[/go]

Now

is the only play to win, because White can now prevent Black from making three in a row, and White can win by getting the last move. But if Black makes the second play, any point will allow Black to make three in a row.

phillip1882 · **#10**

for tic tak toe
agian the move order is x o o x o x x o

so lets day x make the strongest first move, the center.

Code:

   |   |
---|---|---
   | x |
---|---|---
   |   | 

now o has two moves.
lets say he plays two diagonals

Code:

 o |   |
---|---|---
   | x |
---|---|---
   |   | o 

now its x's turn. let say he plays a diagonal as well.

Code:

 o |   |
---|---|---
   | x |
---|---|---
 x |   | o 

now white get one turn.
obviously he has to block with the other diagonal.

Code:

 o |   | o
---|---|---
   | x |
---|---|---
 x |   | o

now x gets two turns.
and he has two ways of making 3 in a row. so lets try something else for the o player.
lets try two diagonals on the same line.

Code:

 o |   |
---|---|---
   | x |
---|---|---
 o |   |  

now its x's turn so he's forced to block.

Code:

 o |   |
---|---|---
 x | x |
---|---|---
 o |   |  

now it's o's turn. again forces to block

Code:

 o |   |
---|---|---
 x | x | 0
---|---|---
 o |   |  

now x get two turns, and he can easily get 3 in a row.

it seems black can win no matter what white does.

i don't know about whether that would be true for go as well though.

Bill Spight · **#11**

OK, my mistake.

One possible Thue-Morse sequence for go.

Variation 1

Click Here To Show Diagram Code: [go]$$ Variation 1 $$ ------- $$ | . 2 7 | $$ | 4 1 2 | $$ | . 3 5 | $$ -------[/go]

(

omitted )

White has two choices for :w8:

. Whichever one he plays, Black captures all the White stones next.

Variation 2

Click Here To Show Diagram Code: [go]$$ Variation 2 $$ ------- $$ | 4 2 7 | $$ | . 1 2 | $$ | . 3 5 | $$ -------[/go]

(

omitted )

Ditto.

As for new tictactoe:

loses

Click Here To Show Diagram Code: [go]$$ loses $$ ------- $$ | 2 . 1 | $$ | 3 . 4 | $$ | 2 . . | $$ -------[/go]

Now, even though Black gets two moves, she cannot make three in a row.

----

But the point is that the empty points on the board do not act like items in a Nim heap. For the Thue-Morse sequence to approach fairness, it appears that the original game must be impartial. Neither go nor tictactoe are impartial. Nim and Kayles are, as is the Brams-Taylor division game, and others. So there is no particular reason to think that the Thue-Morse sequence will approach fairness for go or tictactoe or five-in-a-row, or, I suppose, Othello.

In go, the opponent's best play is my best play is often correct, but not always.

Bill Spight · **#12**

Also, one thing that gives the first player an advantage in a game with alternating play is when the value of plays decrease over time. In general, that is so in go, but sente is the main exception, as sente raises the local temperature.

But early in the game actual sente are rare, so perhaps the Thue-Morse sequence would be good for early play.

First, let's apply it to the first four moves.

Regular go, parallel opening 1

Click Here To Show Diagram Code: [go]$$ Regular go, parallel opening 1 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 4 . . . . . , . . . . . 1 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 2 . . . . . , . . . . . 3 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Let both players start on the 4-4 points. White can enforce a parallel opening with :w2:

.

Regular go, parallel opening 2

Click Here To Show Diagram Code: [go]$$ Regular go, parallel opening 2 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 2 . . . . . , . . . . . 1 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 4 . . . . . , . . . . . 3 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

In this sequence it is Black who can enforce a parallel opening with :b3:

.

Thue-Morse diagonal opening

Click Here To Show Diagram Code: [go]$$ Thue-Morse diagonal opening $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 2 . . . . . , . . . . . 1 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 3 . . . . . , . . . . . 2 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

The Thue-Morse sequence allows White to enforce the diagonal opening. A diagonal opening probably does not favor White, but this fact mitigates any advantage Black might have in the first four moves. So far, so good.

How about extending the Thue-Morse sequence to the first eight moves?

Thue-Morse 8 move sequence, 1

Click Here To Show Diagram Code: [go]$$ Thue-Morse 8 move sequence, 1 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . 5 . . 4 . . | $$ | . . . 2 . . . . . , . . . . . 1 5 6 . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 3 . . . . . , . . . . . 2 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

In regular go, :w4:

is now considered good in the AI era. However, the two :b5:

moves make it bad, by transposition turning it into a premature invasion of a 4-4 6-3 corner. This can happen with a parallel opening, as well.

Thue-Morse 8 move sequence, 2

Click Here To Show Diagram Code: [go]$$ Thue-Morse 8 move sequence, 2 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 2 . . . . . , . . . . . 1 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . 4 . . | $$ | . . . . . . . . . . . . . . . 5 5 . . | $$ | . . . 3 . . . . . , . . . . . 2 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

is also OK in regular go, but the two :b5:

moves give White a broken shape. Neither :b5:

is good in regular go, which is why we don't see it, but the fact that it is sente makes it good in the Thue-Morse sequence.

So it seems to me that maybe the Thue-Morse sequence is not a bad idea for the first four moves. For instance,

Thue-Morse 4 move Go Seigen opening, 1

Click Here To Show Diagram Code: [go]$$ Thue-Morse 4 move Go Seigen opening, 1 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 2 . . . . . , . . . 2 . , 1 . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

In his later years Go Seigen advocated the two space high approach to :b1:

, with the idea of the 4-4 follow-up in the top left corner. The Thue-Morse sequence allows White to enforce that position.

Thue-Morse 4 move, Go Seigen opening, 2

Click Here To Show Diagram Code: [go]$$ Thue-Morse 4 move, Go Seigen opening, 2 $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . 2 . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . , 1 . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . 2 . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Go Seigen also advocated the mukai komoku, which the Thue-Morse sequence allows White to enforce.

All of these possibilities that the B-W-W-B sequence give to White seem to me to reduce Black's first move advantage and also to make for interesting openings.

But later in the game, letting one player make two moves in a row messes up the game.

fair share go

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