14

We do not know, given two perfect players White and Black, whether the game would necessarily end in a draw, or necessarily end in a win (for either Black or White).

However, can we prove that if there is a winning strategy, then it is for White? In other words, can we prove that Black must either lose or draw?

3
  • 1
    No, we cannot prove black must lose or draw. Maybe with quantum computers we'll be able to...
    – Tony Ennis
    May 4 '13 at 11:13
  • 1
    As an aside, a famous British grandmaster once jokingly said that in the initial position both sides are in mutual zugzwang. Hence, White is the first to damange his position so therefore chess is solved in Black's favor :)
    – Andrew Ng
    Jun 10 '13 at 21:16
  • I think you should specify that you are referring to "strategy" in a game-theoretical sense rather than a ¡"chess" sense
    – David
    Sep 5 '19 at 13:15
9

If there is such a proof, no one has found it, and I doubt very much that such a proof exists (it is hard to imagine a mathematically provable "guaranteed-draw" strategy as White). One would certainly expect White to have an advantage if anybody did, but there are some downsides to going first as well (you have to disclose information before your opponent), so it is theoretically possible that the downsides outweigh the upsides. That said, the probability of that being the case seems to be infinitesimal.

9
  • 3
    Thanks. Do you have a source?
    – Randomblue
    May 3 '13 at 22:41
  • 1
    I have no source for the assertion that no one has found such a proof other than that it is very unlikely that I wouldn't have heard of it (in addition to the reasons that such a proof is extremely unlikely to exist whether I had heard of it or not).
    – dfan
    May 4 '13 at 13:23
  • 4
    Yes, certainly we know that one of the players has a winning/drawing strategy, but we don't know if that player is White or Black. The question was "Can we prove that if there is a winning strategy, then it is for White?", and we do not have that proof.
    – dfan
    May 7 '13 at 11:35
  • 4
    White discloses the information of what move he made. Black can make his move based on that information. There are plenty of perfect-information games where the second player wins with best play for this reason. The most trivial example is a game of Rock Paper Scissors where the players disclose their choice in turn instead of simultaneously.
    – dfan
    Jun 19 '13 at 12:25
  • 1
    It's interesting to note, though, that you don't need to know what that strategy is in order to know it exists. For instance, if players make two moves per turn rather than one, the proof is quite easy. Assume White loses the game. Play 1.Nf3 and 2.Ng1. Now it's Black who loses, as he is "White at all effects" now, which is absurd. So, we've proven White has at least a guaranteed draw without having a clue on how to actually achieve it
    – David
    Sep 5 '19 at 13:16
5

It can theoretically be proven, but not with current technology.

If you take a brute force approach, there is some difficulty due to the number of positions.

In analysis of the Shannon Number, it is suggested that the game tree complexity is at least 10^123 for games of max length 80 moves. Let's assume that it is 10^123 for the purposes of this discussion.

10 ^ 81 = Estimated number of atoms in the universe

10 ^ 12 = Operations per second of a terahertz processor core (your processor probably runs at about 1/300th of this speed.)

10 ^ 7 = Rounded-off seconds per year

10 ^ 12 = 1 trillion years

Let's also assume that our processors can evaluate a chess position in only 1 processor cycle.

So, let's make every atom in the universe operate as a terahertz processor core for 1 trillion years.

Can we evaluate each position for 80-max length games?

No.

10^81 x 10^12 x 10^7 x 10^12 = 10^112

We fall short to the tune of being only 0.0000000001% complete with the calculation.

With advanced pruning (throwing out bad lines and their descendants), better technology, and some crafty programming... maybe we'll see 40-max games solved in our lifetime! We can also prune out positions that we've seen before (we can arrive there via transposition), but keep in mind it will take at least a CPU cycle to determine that we've evaluated the position before!

However, this should help you see why it's so far out of reach at the moment.

References

2
  • The question is not asking whether chess can effectively be solved, but whether we can prove whether the (inaccessible to us) result would have some particular characteristic (Black does not have a winning strategy).
    – dfan
    Jun 25 '13 at 14:08
  • 1
    This does answer the question in the context of brute force. The simplest method for proving a winning strategy is to analyze every position. I provide context on why this is not possible given current technology. Jun 25 '13 at 16:02
4

In theory, chess can be "solved", since it is a "finite" game with "perfect information". More precisely, there exists a strategy such that one player has a guaranteed win, or both players have a guaranteed draw given perfect play. Here's a technical article on the basic (well, basic for those familiar with economics/mathematics) concepts of Game Theory for those interested in the specifics. Essentially, every game which has "perfect information", i.e. each player can see all the pieces, and is aware of all the legal moves of said pieces at all points during the game (a counterexample of a perfect information game would be a card game, where you aren't able to see your opponent's hand), ** a finite number of players and a finite number of legal moves**, i.e. the game doesn't go on indefinitely, then it has a guaranteed winning or drawing strategy for one of the players.

In practice, we have neither the technology nor the intelligence (ok, maybe if all the best chess minds of today collaborated on finding the strategy, we may have sufficient intelligence required. MAYBE.) and time to do it manually.

To answer your question: Yes, there exists a winning (or drawing strategy). No, we don't know whether it is for white or for black.

Yes, chess is doomed to get solved someday. But we won't have the technology (in my opinion the only means of doing so) for it for many, many decades (hopefully even centuries) to come.

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  • 3
    The first part was implicit to my question.
    – Randomblue
    May 7 '13 at 9:58
  • I've read that article. It seems to me Backwards Induction (Zermelo’s Theorem) seems almost intuitive when worded as "The chess game must always end, thus given enough foresight, either player 1 or player 2 must have a forcing strategy."
    – ldog
    May 7 '13 at 10:38
  • Though of course it gives absolutely no insight into the game itself! If you imagine a novice player playing versus the best chess engine in the world, the novice player will always win or draw provided he has unlimited undo moves.
    – ldog
    May 7 '13 at 10:40
  • Just a comment on "chess is doomed to get solved someday" - this is of course true if Moore's law (basically, exponential growth of computing power) holds indefinitely. At the current rate, this would lead to chess being feasible to solve about 250 years from now. Not even the wildest extrapolations (discounting singularity theories) have this law holding that long (e.g. Intel expects the law to flatten out before 2020, due to quantum tunneling). I also have to wonder what kind of post-human civilisation would have that kind of processing power, only to turn it towards solving chess :)
    – Daniel B
    Jun 10 '13 at 7:59
  • 1
    No. Even with those minds working together, we wouldn't
    – David
    Sep 2 '19 at 13:32
0

In my opinion, I think the winning strategy is within the player's mind. Because your next move will depend on your opponent's move.

1
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    – Glorfindel
    Sep 27 '18 at 17:02
0

It is very unlikely that black could have a forced win since any line shown as winning for black could be played as white a tempo up. For example, if 1.e4, c5 is a forced win for black then white could play 1.c4 heading for the same line reversed.

-2

White has a slight advantage because it goes first. We're talking about 2% more wins at the grandmaster level. This slight advantage starts to level out as the game progresses. Taken to the extreme, in a perfectly played game, they're probably going to draw.

-5

White would have the advantage of opening the game, but I would doubt there is ever a winning strategy as you suggested.

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  • 4
    Please reread the question.
    – Randomblue
    May 3 '13 at 22:22
  • @Randomblue You are using the word "strategy" in a chess forum, so people assume you are using it with its chess meaning, rather than its game-theory meaning
    – David
    Sep 2 '19 at 13:35

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