Chess is a Draw

[Uri Blass]

mr LK wrote: "evidence is so overwhelming that chess is a draw"

yep, in a current Iccf tourn at master level where i participate all games are
becoming a draw (2/3 finished and all draws0, and thats with participants rating 'only'
between 2300 and 2400 (but with comp use allowed). And not even all book moves were
boring/solid, i encountered some interesting opening moves in a few games, in one game
clearly a suboptimal line by Black in the French but then even with an advantage of 0.3 I now
can't convert it to a win (in a bishop/pawn endgame. In the current ICCF word championship
the late world champ GM Dronov passed away but for the rest all games are draws.
And yes for correspondence chess it -still- would be interesting to see how
some rules could be modified to reduce the draw problem (other option would
be similar methods as used in TCEC, e.g unbalanced openings or some lousy
openings with both sides to play the same opening (the latter not present
in the current tournament system.

Whereas a general deduction is unlikely, one of the most convincing evidence
imo is coming from the chinese database (confirming what i found earlier);
rules of chess are complicated but nevertheless graph theory may later
give some deeper insights in the game if some academic researchers would
continue to work on that (i recall some work done years ago). There probably
are other games or systems more interesting for the application of graph theory.
https://www.perplexity.ai/search/how-ha ... dBrvVUyQ#0

and this is what chatgpt4 has to say :

### Integrating Network Theory with Deductive Arguments to Explore Chess as a Draw
Using network theory along with deductive reasoning can create a robust framework to analyze whether chess might theoretically end in a draw. Here’s how these two methodologies can be conjoined:
#### 1. **Premises Based on Network Theory**
- **Nodes and Edges**: Considering each chess position as a node and each legal move as an edge, the game of chess can be visualized as a vast network or graph.
- **Finite Network**: Despite the complexity, the chess network is finite, as there are a limited number of possible positions and moves.
- **Recurrent Positions**: Some positions recur, indicating cycles in the network.
#### 2. **Deductive Reasoning from Network Analysis**
- **Starting Assumption**: If chess is a perfect information game with finite possibilities, then theoretically, an optimal play defined path should exist (as per Zermelo's Theorem).
- **Cyclic Paths Analysis**: Deductively, if you can prove that all starting positions either lead to cycles that include previously acknowledged drawn positions or directly result in such positions, then the game can be deduced to be a draw.
#### 3. **Exploring Connectivity for Drawn Networks**
- **Drawn Configuration Identification**: Begin with known drawn positions and states (e.g., insufficient material, perpetual checks). These are like "terminal nodes" in terms of game theory which do not lead to wins.
- **Backward Induction**: Start from these terminal drawn positions and explore backward through the game’s network to see if all paths inevitably lead to one of these positions.
#### 4. **Game Reductions and Simplification**
- **Reductive Deductions**: By applying reductive strategies (simplifying the game to known drawn positions and their reachability), one maps out paths from typical game starting points to these end-states.
- **Choke Points in Network**: Identify critical nodes (positions) that frequently transition to or sustain drawn games. Overwhelming connectivity to drawn nodes could underpin a deductive proof.
#### 5. **Further Deductive Inferences**
- **If there exists at least one non-drawn ascent path**: The entire premise can be challenged. Hence, one needs to deductively prove that every path has its end in a draw under optimal play.
- **Complete game survey**: Deductively, one might argue that given a comprehensive enough analysis or simulation, all play variations can be reduced to outcomes that predominantly link to draw nodes under optimal strategies.
#### 6. **Practical and Theoretical Limits**
- **Ideal Assumptions**: This approach assumes optimal play, which is more theoretical and not often practical.
- **Computational Constraints**: Even with strong deductive arguments and network mapping, the real-world computation and validation of these deductive pathways over the vast chess network pose significant challenges.
### Conclusion
Proving chess is a draw using network theory combined with deductive reasoning presents an intellectually stimulating but formidable challenge. Theoretically, if every game path can be linked to a cycle involving network nodes representing known draw positions, and no new winning strategies emerge from the analysis, the game could be deductively argued as drawn under optimal play. This, however, remains an overarching theoretical exercise, constrained by practical computational capabilities and the infinitude of real-game variables and strategies.

My comment: some interesting insights here in the above, whereby i do not agree with the 'infinitured of
real-game variables and strategies' After all chess is a finite game and draw eg. by 3 pos repetition is a draw,
in fact chatgpt4 in the conclusion seems to contradict itself regarding it stated earlier.
Whether this method would be easier then selective (alfa/beta type) number crunching in checkers
(or Othello) style of course remains a question.

PS as for mr Sanders (?) maybe i overreacted a little, he just asked for a scientific 'explanation' not a proof
(in fact i tried to make a last edit, but this then was refused, a time out, ah well no big deal).
There is a Lewis F Sanders btw, Fide rated 1840 or so. As for explanation(s) that chess is draw, 'scientific' or
not 'fsanders' may not have seen all earlier discussions, there was a thread 'is there a project to solve chess'
with a lot of yes/no arguments about what a real proof should be, and then this thread with also some
arguments from my experience. We have scientific explanations that the earth isn't flat, general relativity
to explain gravity And yep, we there also is a Chinese database showing that in chess White can
not find a fundamental opening advantage from the opening stage, which maybe isn't a
'scientific' explanation but nevertheless imo a significant empirical finding. If i fly in one direction
from London and then after many hours endup in London again, having viewed some curved horizons,
it still maybe doesn't qualify as a 'scientific' explanation that the earth isn't flat (unless many scientists
would do it and write a paper, i suppose). In a similar way inspecting the Chinese database means almost
by deductive reasoning that White can not find a forced win (with other engines e.g Torch
or whatever the tree again would be slightly different but with new analysis the drawish nature
(and thus imo the draw result) would again be confirmed. And for those not believing this there's
my 10 k$ challenge; if i would be rich i would offer a million, no big deal.
So good luck again, and now instead of number crunching you can also try to use network
theory to prove in chess there is a forced win for White (thus having solved the game).
tip: don't try 1.g4 for such a purpose

1)nobody believe chess is a win for white so I see no reason people try to prove it is a win for white.
2)Even for things that people believe they may not believe that they can prove.

I believe nobody is going to prove in the next 10 years even that chess is at least a draw for white that is clearly easier task than proving that chess is a draw.

[jefk]

UB wrote:
1 "nobody believe chess is a win for white"
well years ago a former world champion in correspondence chess and computer expert
dr Hans Berliner believed it should be a win with 1.d4 ('The System).
Even nowadays besides you, mr Kaufman, myself, (at least) the GMs Adorjan, Rowson, etc, and a few
others there still are many i guess (incl mentioned mr sanders maybe) who suspect that it might
be a win for White. Not that i care, but they are wrong, and in some ways we are now
seeing a paradigm shift, eg. as result of high level correspondence chess.

2 "nobody is going to prove in the next 10 years even that chess is at least a draw for white
(that is clearly easier task than proving that chess is a draw).""

Well in game theory there's the concept of dominant (or weakly dominant) strategies.
And as White happens to be the first mover in chess, White has probably has a weakly dominant
strategy and thus the better chances (example after e4 f6 d4 Black shouldnt play ...g5).
Ergo this implies that chess is 'at least a draw for White''
QED

disclaimer i'm not an ivory tower nor ivy league mathematician
(yeah i know but i also know the earth isn't flat btw)

As for the number crunching i wouldn't exclude that international draughts will be
'weakly' solved (like checkers) within approx six years or so (although the endgame
phase in this game still can be pretty sharp) And subsequently chess within approx ten
years or so. Network theory is a bit more difficult (and would be in interesting
challenge for an AGI) but then also imo has a higher spinoff, although some
hardware computing experts might disagree.

What i usually talk about regarding chess, although it might become a bit boring i admit,
is an 'ultraweak solution' (that it's a draw) but then there's no firmly established
definition of an ultraweak solution (except an unclear wikipedia article)

[towforce]

A related question: is it possible to prove that chess cannot be solved without generating a high proportion of the game tree?

I know that a lot of people think that chess cannot be solved without generating most of the game tree (or at least a big enough proportion of it to make it impossible), but I think that they're wrong.

Here's a mini-drama to make the point:

Player 1 (beginner): I cannot prove that one can mate with king and rook against king. There are just too many different moves to try.

Player 2 (intermediate): It is possible to prove that: when you know the method, you can firstly prove that the king can be forced to the edge of the board, and secondly you can prove that the king can be mated when it's trapped there. However, the position on my board is a bit more complicated than that, and I cannot prove that I can get a mate here.

Player 3 (strong): I know the method to mate from that position, and therefore I can prove that it's possible. However, on my board, the position is more complicated than that, and I don't know if I can prove whether I can get a mate from here.

The point of the mini-drama is that the more you know about the game, the more complex the positions which you can prove the outcome of. It's obviously possible for a machine to know more about chess than a human (though this isn't quite true yet), and it might be possible for a machine to know enough to be able to prove the outcome from the starting position.

[Uri Blass]

Well in game theory there's the concept of dominant (or weakly dominant) strategies.
And as White happens to be the first mover in chess, White has probably has a weakly dominant
strategy and thus the better chances (example after e4 f6 d4 Black shouldnt play ...g5).
Ergo this implies that chess is 'at least a draw for White''
QED

It is not a proof because there are symmetric positions when the side to move is losing because of zugzwang.

Foir example white kg1 pawns h6 f6 black Kg8 pawns h3 f3.

[jefk]

towforce wrote:
"the more you know about the game, the more complex the positions which you can prove the outcome of."

sure, this is a sort of meta-knowledge, leading in some endgame situations like you mentioned to knowledge about methods and strategies how to play in such positions. But anyway it's the result of experience acquired by by having played (and/or observed) many games. Finding the right move -without calculation in a tree- is often the result of pattern cognition, a process which is mostly done in the temporal lobe of the brain (whereby calculation probably takes mostly place in the frontal lobes); compare it with face recognition, it's a subconscious process and as we know from Nakamura by some humans can be done with amazing speed; with in addition coupling it to the right playing plan (choosing an appropriate move), just like we often can couple a face with a name (if we know it, ie it's a known face).

Alfazero played millions of -complete games against itself as result of which the neural network parameters were established; similar nowadays with the Nnue engines. Upon further inspection i presume that from Alfazero knowledge it should be possible to establish that White cannot win with sufficient strong play by Black. Anyway using a system as Alfazero would imo make more sense certainly as first step than mindless numbercrunching like it was done for checkers (not that i condemn that finding, but for chess and certainly later for Go it requires such enormous computing powers that it makes more sense to look at more efficient methods. But because games as chess are played move per move, there always - some ways- also are 'trees' involved.

As for my knowledge and experience of the game, although i referred many times to the Chinese database to illustrate the 'drawish' nature of the game, i have played tenthousands of computer-computer games (as well as analyzed many possibly hundreds of GM games), which is important of course to know how the middle game will progress after the opening. By inspecting games were won by White (with proper book play for Black), i found that such games always were only won because of one or more mistakes (or one or more severe suboptimal moves for Black. In other words, i could always correct the moves for Black in such a way that the new outcome of the game would be a draw. Again, not a solid mathematical 'proof' but like mr Kaufman wrote, the (combined) evidence is overwhelming. Such findings may not qualify as a 'scientific explanation' for skeptics or math purists, but for me are sufficient to know that the outcome of a perfect play is a draw, with similar certainty as knowing the earth is not flat (the discussion about how many 9's we can add to the nr 99.999999... certainty in this respect imo is irrelevant. Sure, i know that the game of chess can be regarded as a mathematical game, and thus in principle it (also) could be 'proven' mathematically that the game is a draw with 100.000.. pct certainty; yet this imo would not give more certainty to the knowledge which I achieved already by now that it's a draw.)

PS i mentioned GM Adorjan ("Black is ok" but he passed away, ok let's replace him a by e.g. Carlsen, players like him -and Nakamura- have an enormous amount of metaknowledge about the game (and Carlsen knows it's a draw, he once stated for example that one side can only win if the other side makes a mistake). Nb in this forum also 'jouni' here in this forum agreed, and more will follow i guess (unless interest will fade away because after a while it simply becomes a fact, well at least for most of us, just like the earth isn't flat).

[towforce]

Alfazero played millions of -complete games against itself as result of which the neural network parameters were established; similar nowadays with the Nnue engines. Upon further inspection i presume that from Alfazero knowledge it should be possible to establish that White cannot win with sufficient strong play by Black.

Thank you for a good, thoughtful post! Unfortunately, I'm not quite in agreement, because I think that the knowledge and the method of training for NNs for chess makes them unsuitable for proving the outcomes of positions. I certainly think it's possible though - but we'd have to undertake the process in a different way. Meanwhile, we're learning invaluable lessons about how to prepare machines to do intelligent work.

My solution for solving chess: (1) find a deep pattern that enables accurate evaluation of most chess positions at ply 1 (2) reverse engineer that pattern to uncover what it's telling us about chess that we have missed

Obviously I have no proof, but my intuition is telling me strongly that we have missed some relatively simple but very important things about chess. I think that what we do know about chess is strongly biased towards the types of patterns that human brains prefer. When Deeper Blue beat Kasparov in 1997, several GMs (including Daniel King in his book about the match) said that we have to accept that there's more than one way to play chess, alluding to the fact that chess based on generating a deep game tree on the fly produces a different style of play to strong humans.

[jefk]

ok tf, you may have a point.

earlier on, i mentioned the concept of 'balanced games' and also referred to
complexity theory. while i did some search not much is known (yet?) in this
specific topic (*).
What is known from the initial position, are the rules of chess,
how pawns move, knights move, and so on. While pawns cannot move
backwards, other pieces can, so there is such a large degree of freedom
in the game (this relates to axioms of choice in game theory) that
it maybe should be possible to classifiy the game as some sort of
'balanced game'' which is in equilibrium with perfect play from
the start (and thus neither side can force a win). This again, also
is (scientific) 'intuition' (which is allowed in science, well at least
physics btw, keeping in mind that lateron verification is needed
to confirm such thoughts). But thereby i'm also thinking about
systems in biology, which there often are equilibrium situations,
often resulted from evolution and biologists unlike mathematicians
may find such line of reasoning acceptable.
Although for the time being, i don't think we will resolve this any
time soon at least not within the framework of pure math.

(*) also went in the topic of Nash equilibrium but if a game has
a Nash equilibrium this doesn't mean that it must be a draw

[towforce]

A quick look at how humans and ANNs learn:

Humans:

1. Identify a position they don't understand

2. Study that position until they understand it deeply


ANNs

1. Data file with billions of positions (many orders of magnitude more than a human will see in their lifetime)

2. Each data point consists of a position and a value

3. The ANN encodes an approximation of that position/value pair data

The ANN method of training is unsuitable for proving the outcome of positions, whereas the human, who has done the in-depth work, will know how to prove the outcome for positions for which they have sufficient knowledge.

What I am saying is that it would be possible to train ANNs to have the ability to do proof of outcome, but it would require a completely different training method. I don't know how easy it would be to do this, or how far it could be taken, though. There may be an exponential factor that makes it difficult to take all the way to the chess opening position.

[Jouni]

Yes draw: TCEC S26 - Rapid Chess Bonus: 45/45 draws so far.

[Jouni]

100 games and 100 draws. It's over now folks! SF and Lc0 plays perfect chess from opening position.