Hi Andrew and welcome to OGS! I am afraid that the answer is “no.” This thread cannot be explained to anyone anymore :<
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My interpretation of smurphs comment, was
- in summary, it wasn’t very popular to use a thue morse sequence for the game moves, one reason being it became more complicated to figure out who’s turn it was next or in ten turns time and how that would affect life and death of groups, general tactics etc
- The discussion was kind of veering off into whether a Tak style opening was a good alternative to Komi, whether Komi and modified Komis were more suitable to evening out games between players of different strength than handicap stones, whether or not you can find two players who are the exact same strength on ogs, whether losing a game means the other player was skilful etc. So I interpret the comment as bringing the conversation back from mildly heated debates to whether the Thue-morse sequence for moves was a viable alternative to Komi, to which so far the answer seemed to be leaning toward no although some people wanted to try it out anyway, which is fair, and should be tested to see if the game is wildly different to go so as to be called a variant rather than an alternative to Komi in Go.
@mherrero415 Basically people have found that in games like Chess and Go there was an inherent advantage to going first because in some sense you get to dictate the opening, probably more so in chess than in go, but in Go you probably get the opportunity to play first in the key areas of the board when you’re the first player. So to even out the first move advantage white gets bonus points at the start of the game for going second.
The summary of the topic is can we replace this idea of giving white extra points for having to go second, to changing the move orders so it doesn’t go BWBWBWB… to something more fair like BWWBWBBW… which supposedly is fairer in the long run as mentioned by the OP with links to videos etc to find out more. The problem so far with this is that it could be harder to figure out whose turn it is next, given you’re in the middle of the game, and in fact if you want to know if you can capture or save your stones you need to know not just whose turn it is next, but whose turn it is for the next 5-10 moves. It wasn’t clear that this would be easy to know just looking at the sequence linked. That and supposed you play away from one part of the board and come back to it, let’s say 10 moves later, whether a sequence of moves you might have read out there still works/applies. Maybe black/white suddenly gets 2 moves in a row at a crucial point that breaks your sequence. So I think people said they would try it out, but I didn’t see any comments since whether it was easy/difficult/fun to play like this.
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That’s a good summary, but I just also want to note that some people (such as myself) made arguments of why Thue-Morse does not necessarily remove advantage disparity.
Let’s suppose you and me are in a room where there are three $100 bills on a table, along with a pile of $1 bills. If we had to take turns taking one bill at a time off the table, clearly the person going first would have an unfair advantage. However, I would be happy to let you go first, if we agree to use Thue-Morse alternation.
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Yeah I know, I thought your examples were fairly good. I think with respect to using it for turns in Go, the OP mentioned that the sequence got fairer the longer it was applied/followed for (at least in principle) although maybe there’s even examples where it takes a very long time to for it to approach even, mathematics has a lot of room for variety I suppose. But let’s say it could be plausible that a long game of ‘Go’ could be even with this sequence.
I know I missed a bunch of things people mentioned, I was just trying not to make a long post longer 
I also mentioned that you could adopt what they did in chess and play X amount of games alternating as black and white. I think in the World championship they did something like WBWBWBBWBWBW. Then they had rules for tie breaks like blitz games, since the main games had very long time controls.
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This is how you should use Thue-Morse to make things more fair.
Using the sequence on the moves in a game of go, even if it is a long game, does not necessarily have to make it more fair, as the values of moves are not fixed. Mathematics has a lot of room for variety, so applying a principle that works in one situation to a slightly different situation does not guarantee that the results will stay valid. Whereas Thue-Morse makes things fair in sharing games, it does not have to make things fair in other games.
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Using Thue-Morse you would still want to go first, since the person who goes first will always have either $100, $99 or $98 more than the other (the reason being that for any n, the moves 4n, 4n+1, 4n+2 and 4n+3 will have either person going exactly twice, both of them getting exactly 2 dollars in those four turns).
[Edit: Ah, I just discovered there are three $100 bills, where I thought it was a single one. Of course you want to go second for exactly the same reason as above]
There are a lot of special cases that will have Thue-Morse not resulting in an equal share of loot when dividing items. Your example is one of them, but there are plenty of others. It is the fairest way to divide things with fixed value, but it is not necessarily going to become an equal share if you keep going long enough.
According to this source, the sequence becomes fair, if the list of items decreases in value by a given fixed percentage (the second item is worth 90% of the first, the third item 90% of the second, the fourth 90% of the third, etc.), and this percentage is quite high. If the percentage is low, Thue-Morse will not be optimal. The closer the percentage gets to 100%, the fairer Thue-Morse will be.
I’ve also tested a few distributions of values using Thue-Morse, simple alternation (ABAB) and repeating ABBA.
- If the values are distributed by a uniform random distribution, Thue-Morse and ABBA are both very fair (almost 0.3% of the value of the average item), while alternation scores quite badly (50%)
- If the values are distributed with a (approximation of) normal distribution, then both Thue-Morse and ABBA seem about equally fair (about 1% of the value of one item), whereas alternation seems to be around 45%.
- If the values of the n-th item is a polynomial function, Thue-Morse is exactly 0 at turns that are some multiple of some power of 2 (depending on the polynomial). I find this one the most fascinating. Alternation scores horribly, ABBA scores slightly less horribly, but still terrible.
- Exponentially increasing values don’t work for any of the sequences, although Thue-Morse is fairer than ABBA. The problem here is probably that the largest value is so much larger than the other ones, that having the first turn is an incredible advantage.
- Exponentially decreasing values do work for Thue-Morse, although it needs to decrease relatively slowly. This is what the article above shows.
- The harmonic progression 1/2, 1/3, 1/4, 1/5, etc. drops too fast to become fair, although once again Thue-Morse is fairest (39%), followed by ABBA (43%) and worst, as usual, alternation (70%). (here the percentages are not of the average value, but of the highest value) [Edit: I think I might have made an error here, but no time to check it now]
One strategy that should be fair is to let the person choose who at the moment of his turn has the least value. This way of dividing the turns will always be fairest, and can be shown to roughly equal to the Thue-Morse sequence in several cases (such as the exponentially decreasing one with a small factor, or the uniform distribution). It would make the 100 dollar and 1 dollar bill example fair, for example.
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Any mathematical claim about the fairness properties of Thue-Morse must be qualified with technical assumptions about the underlying game, e.g., how value would be distributed across the sequence.
Further, discussing any fairness properties, it is important to distinguish between the following:
- Fairer: reduces the advantage disparity.
- Fairest: reduces the advantage disparity as much as possible.
- Perfectly Fair: reduces the advantage disparity to zero.
Remarks:
- Each condition is a stronger statement that the previous.
- Fairer and fairest do not necessarily imply that the disparity has been reduced to zero.
- Fairer and fairest are relative statements. Fairer requires comparison to at least one other alternative (e.g., simple alternation). A reasonable claim of “fairest” should compare to a reasonably large and comprehensive set of alternatives.
- As a quirk of how we use the language, “perfectly fair” is often just shortened to “fair”, which makes the phrase appear linguistically weaker than “fairer” or “fairest”, while it is actually the strongest claim.
One might be able to argue that Thue-Morse is fairer or even fairest (among some group of alternatives), but that is not enough to show that it is perfectly fair.
One can easily construct examples where Thue-Morse is not perfectly fair, even if it can be shown to be fairest among several alternatives. For example, dividing an odd number of equally valued items.
5, 4, 3, 2, 2
The perfectly fair (and hence fairest) turn sequence is ABABB. Thue-Morse and the greedy scheme that you described produce sub-optimal results.
Finding a fair way to divide a finite set of items is also known as a partition problem, which is NP-complete. Although this problem has efficient practical algorithms (due to the existence of a pseudo-poly-time dynamic programming solutions), it’s NP-hardness suggests that simple greedy algorithms (or a fixed sequence like Thue-Morse) are unlikely to be optimal in general.
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Just a side note: The argumentation here is always based on picking games, where each player can choose a value out of a fixed (but not necessarily limited) heap of values. In a go game, the value of a move depends on the previous made moves.
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Yes, that’s true. Analyzing how Thue-Morse would affect go would be astronomically more complex.
However, the purpose of using these simple picking games are to demonstrate the properties and limitations of Thue-Morse. If Thue-Morse does not even make a simple picking game perfectly fair (or even the fairest), then it weakens any intuition or arguments that claim that Thue-Morse might make go perfectly fair.
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Well the results are in…
As a game, Thue-Morse-Go works. We didn’t make any mistakes with the ‘forced passes’ and I dare say anyone playing it regularly would soon become very familiar with when those passes occur.
Was it “Fair” with zero komi? I don’t know. It felt fairly even. There were mistakes on both sides and I think it’s fair to say that Thue-Morse-Go increases the opportunities for significant mistakes on both sides.
As Black, my opening strategy was to control the centre. I hoped that the increased complexities of Thue-Morse-Go Life&Death would favour me that way but I found the increased complexities of staying connected may well have balanced that out.
All in all, an interesting exercise. I wonder if players more familiar with assessing ko threats would also adapt to Thue-Morse-Go more readily as both involve consecutive moves by one side.
Thanks to @BHydden for the interesting and unusual game.
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Will OGS recognize draws in 0 komi games?
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Moderators can manually rule a game a tie, but currently draws are weird and usually display as B+0
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Draws are considered wins by B+0 unless a moderator ties the game during (!) the scoring phase, in which case the tie is properly displayed and handled correctly by the ranking system.
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Here is something interesting with your visualization.
Looks like these blue dots are in a non-repeating sequence.
Well, it’s kind of the point that this sequence does not have repeating patterns. For every sequence X that occurs inside the sequence, you can be sure that the sequence XXX does not occur, so whenever something repeats twice, it won’t do it a third time.
The sequence is fractal-like: if you take every second element of the sequence, you get the original sequence back. For example, it starts as
B W W B W B B W W B B W B W W B W B B W B W W B B W W B W B B W
In fact, this is another way to generate the sequence: start with just the sequence BW, and then to get the next sequence, you replace each B with BW, and W with WB simultaneously. Doing this infinitely often gives the Thue-Morse sequence.
Another name for the sequence also mentions the name Euwe, refering to the chess grandmaster Max Euwe, who independently thought up this sequence as a way to make infinite chess games possible. There used to be a chess rule, that it was not allowed to make a sequence of moves repeat more than 3 times (or else the game was declared a draw).
Take the following position, black to move:
There are two short sequences of moves that are different: Black can put either queen to e5, White blocks the check with the horse that can move, Black moves the queen back to the original position, White moves the horse to the original position. Since black has a choice of which queen to move, he can let 0 correspond to moving the queen at a5, and 1 to moving the queen at e1.
An infinite game can then be played by making the move sequences in order of the Thue-Morse(-Euwe) sequence: 0110100110010110…
Since never in this sequence of moves there is a situation where the exact same sequence of moves is played three times in a row, it does not violate the (old) rules of chess, and thus it would be an infinite game. Nowadays there is a new rule, which states that if there have been 50 moves without capture or pawns moving, then the game is a draw, hence it is no longer possible to play infinite games.
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Yes that is one of the features, be it deemed benefit or detractor; the series is infinitely non repeating.
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Thanks, I think I wrote that last time being half jet-lagged, but you’re totally correct 
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Okay, now I’m wanting to play. Anyone up for a fast correspondence game of Thue Morse Go?
My preferred time controls would be fisher, 2d + 6h/move, max 2 days, no pause on weekends, though I’m willing to go a bit faster or slower. Remember that the passes to enforce Thue-Morse move order will result in extra time.
Also, 0 komi; agree to ask a mod to adjudicate as a draw if that is the result.
I’m not familiar enough with different rulesets to know for sure, but I think Japanese Rules would work best for this?
Feel free to send me a challege (username: Samraku).
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