Thue-Morse (Fair Sharing) Sequence: A possible alternative to komi?

Not really related to komi but more to handicap: I’ve pondered a variant of handicap that allows people to place N (extra) stones in total, but they get to decide when. :slight_smile:

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A tough choice for an N-1 ranked person. Do I accept this stone which I have to make the best use of, or do I play at N-1 and get more rank if I win, and lose less if I lose…

… and how many stones is a free-placed stone worth? :slight_smile:

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Are AA two stones or two moves? :thinking:

Depending on interpretation, groups may now require three eyes for life.

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Oh! Cool idea… I think 2 moves meaning 2 eyes are still enough… But both would be interesting to try! :slight_smile:

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Currently, we use komi (bonus points for White) to try to resolve the inherent unfairness of the BWBWBWBWBWBW order of play, clearly favouring Black.

Correct me if I’m wrong, but from what AI numbers show and win rate of professional games, modern day komi should favour white more?

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Yes, 7.5 komi for area scoring slightly favours white. My comment (obviously poorly worded, sorry) was meaning that the move order alone was advantageous to black (which is why white needs komi in the first place) :slight_smile:

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Does getting to play twice allow you to fill-in two eyes (to capture a group with only two one-point eyes), or would each move need to be legal on its own? EDIT: I overlooked that this question was already brought up and addressed above.
There are a whole host of other issues to clarify and settle in such a variant.

A Thue-Morse sequence does not make the game fair without komi. One player would still have a theoretical advantage. Depending on how you handle passing (i.e., whether A can pass his first move, giving B the first single move, followed by two moves by A, and so on, essentially swapping positions with B), you could still make a strategy-stealing argument to say that player A has the advantage.

On another note, in an effort to quantify the value of individual moves (without so fundamentally changing the game), mathematician and Professor Elwyn Berlekamp devised a go variant called Environmental Go (aka Coupon Go). Basically, in this variant, there is a stack of coupons counting down from 20 to 0.5 (in half point increments), and on a player’s turn they can choose to either take the top coupon from the stack or play a stone. The values on any coupons taken is added to the player’s score at the end of the game. This allows the players to essentially bid for komi at the beginning of the game, and also forces the players to consider the value of sente/gote at each move thereafter.

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I said above:

Let me be more precise, since the word “fair” is somewhat vague and subjective. What I mean to say is that the Thue-Morse sequence does not necessarily remove the inherent advantage of one of the players.

This can be seen even in the simple “game” of splitting up a finite set of items (of various values), which is an example that has been used as motivation for the Thue-Morse sequence.

Consider if there are four items, worth $200, $125, $100, and $50. Clearly, player A (picking first and fourth) will get $250 of value, while player B only gets $225.

If the four items were instead worth $200, $150, $125, and $50, then player A would get $250 of value, while player B will get $275.

You might be able to argue that Thue-Morse was still a “fair” (or maybe even the “fairest”) method to divide the items in the above scenarios, however, clearly there can be an inherent advantage for one player over the other. In either case, the two people involved would argue about who should get to be player A or B, and perhaps the only “fair” way to further settle that would be a coin flip.

One final example: the four items are worth $200, $100, $75, and $25. Thue-Morse would give a clearly bad result, since whoever wins the coin flip to be player A gets to take $225 (vs $175 for player B), while obviously one could just split the fairly with $200 going to player A and $100 + $75 + $25 going to player B.

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I love this, as a game concept, not as a way to eliminate komi. Who wants to try (in a week, when holiday is over)?


As for fairness, that is not something that translates at all. Fairness here is used as a mathematical term, and thus does not correlate to the linguistic interpretation of what “fair” means.

In fact, since go is a perfect information game without draws, there must be a winner, so it can never be made “fair” without hiding some of the information or allowing draws to happen.

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I would like to try it, but I would need an easy way to know when to pass. Sadly the move-tree will be no help.

Edit: The move order is not as bad as I though

bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wbwb_bwbw_wb_bw_wbwb_bwbw_wbwb_bw_wb_bwbw_wb_bw_wbwb_bw_wb_bwbw

Edit2: python code to translate OGS move number to player action (play/pass)
def thueMorse(n: int) -> int: 
    return bin(n).count('1')%2

def move(n: int) -> None: 
    thueMorse_move = 0 
    for i in range(0, n): 
        if (i%2 == thueMorse(thueMorse_move)): 
            thueMorse_move += 1 
    if ((n-1)%2 != thueMorse(thueMorse_move-1)): 
        print('pass', end=' ')
    if n%2: 
        print('b') 
    else: 
        print('w') 
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Too bad that you can’t use PASS when planning conditional moves. That would make it fairly easy to play.

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Sounds cool as a variant, I’d definitely give it a try.

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Here is another visualization of the move sequence (up to 1000), with double moves highlighted in red.

I think this move sequence would have a rather chaotic impact on the game. Not only will the double moves change life/death status, validity of tesuji/joseki/etc, the timing of any strategic move is important in relation to the chaotic sequence of double and single moves.

For example, if one were to ask “Is the L group dead?” under this variant, I think the answer might depend not only on who’s turn it is, but also precisely where in the move sequence you are.

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True, non-integer komi (to avoid draws) does not equalize the theoretical advantage (one color should always win under perfect play). In principle, there should be some integer komi where perfect play (under a reasonably specified ruleset to avoid no results and endless games) would result in a draw. However, in practice, an appropriately set non-integer komi does seem to at least narrow the practical advantage disparity vs not using komi at all (the winning chances for two equally skilled players seem to move closer toward 50-50).

I am not at all convinced that Thue-Morse would narrow the practical advantage disparity. Maybe it does or maybe it even makes the advantage disparity worse. It’s just a completely different game, still with players taking turns, but just sometimes they are allowed to play two stones and sometimes just once, in a big chaotic and asymmetric mess.

I expect that if people try to play this variant, they will initially find roughly balanced performance, simply for the fact that all learned go tactics, strategy, and theory would have to be thrown out and everyone will be on the same initial footing back at the drawing board. I think only basic reading (visualization) skill will be readily transferable, but one will have to actively resist learned patterns, intuition, shape, gut feelings, etc.

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I think the opposite is true… thue-morse literally exists to remove the advantage of going first. This is purely an order of play equaliser and says nothing about either player’s ability to recognise the best move, it simply corrects the advantage that one side would have of going first, meaning it doesn’t matter if you play first or second, because it evens out rather quickly.

The only reason Go has no draws is because we artifically set komi to be a non-integer value. Go in its natural state can (and indeed the mythical “perfect game” would) end in a draw. You can also just play with NZ rules who use integer komi and allow draws :wink:

Sounds fun, doesn’t it? :smiley: (because obviously what Go really needs is MORE complexity!) :rofl:

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Bots would love this variant because ladders (their only weakness) NEVER work :smiley:

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@Kosh v me (experiment 1)

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yes, i think this is the best way for deciding komi, even better than the current rule of 6.5

another suggestion i read somewhere else is the “i cut, you choose” approach. player A placed a black stone somewhere on the board, and then B can decide whether he wants black or white. this way, A will try his best to make the 1st move neutral (whatever it means).

all these would seems much simpler than the ABBABAAB .

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No it doesn’t. Take for example Tic-Tac-Toe, in which the Thue-Morse sequence will guarantee a win for the first player (play in the center).

Of course that game is too small to be taken serious, in this manner, but the same will hold for 4 in a row on a large board (where the first player wins at the 7th move). I’m pretty sure the first player will win 5 in a row as well (although I didn’t go through all combinations to check).

The reason the sequence is fair in other situations (let’s say dividing a pile of cookies by choosing them one by one following the Thue-Morse sequence), is that your move does not affect the value of the other possible moves (choosing a cookie does not change the desirability of the other cookies). I don’t see the same thing happening with moves in a game, since the moves drastically change the value of the other moves.

At the n-th move, write n in binary and count the number of 1’s. If it’s odd, then it is white’s turn, if it is even, then it is black’s turn. To keep track of which “real” move it is (as the move counter will not help), you could just write the move numbers in chat each turn, or you could count the stones on the board + prisoners.

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@andysif: This is known as the Pie Rule. I think it is even more fair than a coin toss.

@Kosh: I had never considered Komi bidding before. I think it is genius. My only edit would be to switch your suggestion of a coin toss with using the Pie Rule instead. If neither player can agree on who gets to suggest the first move, then I am 100% for deciding that dispute with a coin toss :stuck_out_tongue:.

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