I Who would win a perfect game of chess?

[SSequence]

And although chess players often speak of the "best move" in any given position, it's not clear that this concept is really well-defined. If one player is in a position to win, then as long as they maintain the possibility of winning, who's to say one move is better than another?

Which move wins most quickly, you say? What if many different moves can all result in the same quickest win (regardless of the other player's moves)?

I agree. Yes, if one player is in a "guaranteed winning position" [with perfect play from that point, regardless of the other player], then as long as the player keeps its guarantee of winning as such it isn't necessary at all there has to be a single best move from each game state.

An analogy would be having different ways to obtain the same high-score in a (single-player) score based game [game with ending, not endless].

P.S.
I haven't thought about this topic in detail for quite some time. Though I have wrote about it in length before. But naturally, most of it I expect it to be well-understood (and somewhat obvious) ... though there might be a few novel points. I do like the "hate to lost" point that I made since it isn't immediately obvious unless one thinks about it a little bit.

I am hesitant to link to my own post, but there seems to be fair amount of interest in this topic (more so than I expected). Here is the link. Maybe it would be useful or interesting for some, though there isn't anything specific to chess in my post.

I didn't mention in the post directly (due to length considerations), but for chess [or other multi-player games] a reasonable idealized view would be to see it from the perspective of a single player such as white/black (as a non-deterministic single-player game). But there can be some issues in such over-simplification [for example, consider a game where the actions of other player(s) could lead to endless play], which would need to be looked/described in more detail. The possibilities do seem to increase quite a bit with addition of "other" players. That's why I didn't add it in the post.

In the post, I assumed [for simplicity and focus on illustrating the basic-point] that there is a path to win/lose states from each game-state, but clearly that needs to be changed if there are draw possibilities [also increasing the "classifications" of game-states]. Actually, I tried to make the assumptions specific enough that there would be no (somewhat-natural) $L$ state either.

 

[PeroK]

I agree. Yes, if one player is in a "guaranteed winning position" [with perfect play from that point, regardless of the other player], then as long as the player keeps its guarantee of winning as such it isn't necessary at all there has to be a single best move from each game state.

In any game/algorithm, a solution in fewest steps can always be preferred and described as "best". That's part of the definition of "best" in algorithmic computations.

 

[SSequence]

I don't disagree, but the point was that there doesn't "necessarily" need to be a single best move from some game-state.

 

[PeroK]

I don't disagree, but the point was that there doesn't necessarily need to be a single best move from some game-state.

Let's take a simple example of King + Queen against King. This can be solved from any position and the minimum moves to checkmate calculated. There will typically be many different ways to deliver checkmate in this number of moves.

These solutions are still "better" than a solution where the pieces wander round the board for nearly 50 moves before delivering checkmate.

So, yes there may be more than one "best" move in a position. But this set of moves can still be preferred to others that are also "winning" moves.

 

[SSequence]

I haven't thought about the specific context (directly analogous to chess) that much [since it is somewhat different from context(s) which I thought about in more detail], but I would tend to agree.

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Yes, if our criterion is based on "minimum" no. of moves to win for example (with the assumption of a guarantee to win), then we would typically expect to cut-out a lot of sub-par solutions (ways of playing).

In fact, it doesn't seem a priori impossible that there might actually be a unique path leading to win in a minimum number of moves. I suppose this would be analogous to a single (unique) path giving the highest possible score in a single-player non-endless game.

 

[sysprog]

Let's take a simple example of King + Queen against King. This can be solved from any position and the minimum moves to checkmate calculated. There will typically be many different ways to deliver checkmate in this number of moves.

King + Queen against King can always be won in fewer than 11 moves. I don't care if I use all 10 moves, or even maybe a few more, if I have enough time. If I see a clear path to a win I don't care to try to find a shorter one.

In any game/algorithm, a solution in fewest steps can always be preferred and described as "best". That's part of the definition of "best" in algorithmic computations.

As you are doubtless well aware, algorithmic steps at the level of decision of which move to make don't have a 1-to-1 correspondence to chess moves, so winning the game in the fewest moves is not always preferable to winning with the most easily arrived at certainty, even if the latter winning sequence has a greater number of moves; the player must also consider the time on his clock.

In an actual game if I can see a mate in 4 easily, but would have to think longer to find a more elegant mate in 3, the 'best' move, in my view, is the one leading to the more easily found win.

Also, sometimes in chess one move is exactly as good as another; e.g when you have a 1-move back-rank mate by promotion of a pawn and a rook is as good as a queen for the purpose.

 

[PeroK]

K+Q vs K can always be done in fewer than 11 moves. I don't care if I use all 10 moves, or even maybe a few more, if I have enough time. If I see a clear path to a win I don't care to try to find a shorter one.

As you are doubtless well aware, algorithmic steps at the level of decision of which move to make don't have a 1-to-1 correspondence to chess moves, so winning the game in the fewest moves is not always preferable to winning with the most easily arrived at certainty, even if the latter winning sequence has a greater number of moves; the player must also consider the time on his clock.

In an actual game if I can see a mate in 4 easily, but would have to think longer to find a more elegant mate in 3, the 'best' move, in my view, is the one leading to the more easily found win.

We're talking about solving chess, not practical over-the-board play!

 

[sysprog]

We're talking about solving chess, not practical over-the-board play!

I get that. I think, as most players and programmers do, that neither side has a winning advantage at the outset. I was replying regarding the notion of 'best' move -- I think that for the 'solving chess' question, any path that leads from an arrived-at position to a certainty of winning is as good as any other; it doesn't matter how many moves it takes to win; how you win doesn't matter -- only whether you win matters.

 

[PeroK]

I get that. I think, as most players and programmers do, that neither side has a winning advantage at the outset. I was replying regarding the notion of 'best' move -- I think that for the 'solving chess' question, any path that leads from an arrived-at position to a certainty of winning is as good as any other; it doesn't matter how many moves it takes to win; how you win doesn't matter -- only whether you win matters.

That's your definition of "best": any move that retains a winning position. An alternative definition of "best" is any move that wins by the least number of moves.

Your definition is rather odd in that if there is a checkmate in one move, that is not the best move. It's just as good - by your definition - not to deliver checkmate. It only becomes the best move when you near the 50-move draw rule. If you remove that rule then with "best play" (your definition) you might never win. You could play indefinitely with K + Q against K, claiming every move is "best" but never delivering checkmate. Only the 50-move rule provides you with the impetus to finish the game!

In any case, describing not delivering mate when you have the chance as "best play" is stretching a point.

 

[sysprog]

That's your definition of "best": any move that retains a winning position. An alternative definition of "best" is any move that wins by the least number of moves.

Your definition is rather odd in that if there is a checkmate in one move, that is not the best move. It's just as good - by your definition - not to deliver checkmate. It only becomes the best move when you near the 50-move draw rule. If you remove that rule then with "best play" (your definition) you might never win. You could play indefinitely with K + Q against K, claiming every move is "best" but never delivering checkmate. Only the 50-move rule provides you with the impetus to finish the game!

In any case, describing not delivering mate when you have the chance as "best play" is stretching a point.

I don't claim that sub-optimal moves can ever be legitimately called 'best' -- I think that 'not incorrect' would be a better name for any move that preserves a certainty of winning in finite time, including the optimal move; for purposes of the 'solving chess' problem optimal moves are not 'best' and are not definitively better than sub-optimal moves that are equally certain to produce a win.

For a working solution to 'solving chess', every winning line is as good as the optimal one, so there isn't, in my view, a 'best' move in every situation -- when there is only 1 move that leads to victory, I would call that move 'correct' rather than 'best'; however, I mostly don't strongly disagree with your point. It's true that chess problem solutions require that the fewest moves be made in accomplishing the goal.

This is just humor, but when I was kid, sometimes when someone would refuse to resign a very obviously lost position, just so he could inflict on me the drudgery and tedium of proving my position was a winning one, I would deliberately delay checkmate; I would proceed to confine his king to 2 squares, then capture every one of his pieces and pawns, then promote all of my remaining pawns, then place my pieces in some tidy arrangement, and only then deliver checkmate -- punishment for trying punish me instead of resigning and playing again or not.

 

[jbriggs444]

You could play indefinitely with K + Q against K, claiming every move is "best" but never delivering checkmate.

An interesting point. A strategy which is clearly not best in which each individual move is "best". However, any loop prevention heuristic could succeed at avoiding this. The "fastest mate is preferred" heuristic is just one possibility. Though a very convenient one.

 

[SSequence]

The notion of an "optimal move" on a given game-state should still be quite well-defined though [optimal in the sense that it retains an already winning position].

A lot of terminology would remain constant within games that have finite number of states. Very briefly, I think something similar to this would be enough (number of other ways would be possible I guess). We might define a strategy as a 1-1 correspondence between game-states and action/move of the player at that game-state. So, for a two-player deterministic games, if a given state is a winning state for some player, then we can define any "action/move" as "optimal" if it is part of "some" winning strategy [starting from the given/chosen state].

So, a strategy that is simply based on selecting "optimal moves" from each game-state may not be an optimal strategy. That would merely be a necessary condition.

Making the notions in above paragraph precise shouldn't be difficult. It isn't much different from post#182 [except that the context of a single-player non-deterministic has some differences from two-player deterministic (different classifications of states), which one needs to account for properly].

 

[PAllen]

There is a real problem defining best play in the context of chess assuming the starting position is drawn. Minimizing moves to mate objectively chooses one or more best moves in a winning position. But if it is drawn, there is no such simple criterion for favoring one or a few moves out of all those that preserve the draw. For a minimal definition of perfect play, you don’t need to distinguish, but to achieve wins against an imperfect player, randomly choosing a move that preserves the draw will be very poor at achieving a win. The truly optimal approach to play against an imperfect player would be to have a model of that player’s “error profile”, and would be different against different players.

 

[SSequence]

There is a real problem defining best play in the context of chess assuming the starting position is drawn.

Agreed.

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From a practical perspective, assuming that start-state for some board-game (not chess) isn't a draw, I think there would also be some difference between storage of optimal strategy and execution of it under real time-constraints? It doesn't seem implausible that the former might be possible practically but the latter might not be? It seems worth mentioning as an aside explicitly, but this is an entirely different topic though.

 

[user30]

While chess hasn't been solved yet, other games have. For example, I know that in in some games, like connect four, if both players play perfectly, the player who goes first will always win. On the other hand, some games, like tic tac toe, a perfect game will result in a draw; in fact, I recently found out that this is true for checkers as well. What I'm wondering though, is if it's possible to predict which scenario a perfect game of chess would lead to even without having fully solved it yet, and if it is possible, what the answer is.

I'm a chessplayer and the general consensus is that the game is a theoretical draw. The game is fairly sensitive to what we call "tempos" (how many moves have I moved "forward" compared to my opponent) but it's likely not enough to force a win.

While it is true that chess is incredibly complex, a lot of lines are garbage. A super sophisticated future super computer might very well solve it with perfect pruning (excluding irrelevant lines).

But not in our lifetime.

 

[Halc]

Perfect chess is likely a draw, but if there is a win, it will be white.
Other games like go and reversi probably do not end in a draw, and especially in the reversi case, it is unclear if it is an advantage to go first.

I haven't read the entire thread, so all this has probably been brought up.