Suppose we compare that whole function with Mr. Neumman’s function, and compare how good are the probable moves you’d make versus him making. On most chess positions, Mr. Neumann’s move would probably be better. [...] That’s the detailed complicated actually-true underlying reality that explains why the Elo system works to make excellent predictions about who beats who at chess.
This explanation is bogus. (Obviously, the conclusion that Elo scores are practically meaningful is correct, but that’s not an excuse.)
Mr. Humman could locally-validly reply that Tessa is begging the question by assuming that there’s a fact of the matter as to one move being “better” than another in a position. Whether a move is “good” depends on what the opponent does. Why can’t there be a rock-paper-scissors–like structure, where in some position, 12. …Ne4 is good against positional players and bad against tactical players?
Earlier, Tessa does appeal to player comparisons being “mostly transitive most of the time”—but only as something that “didn’t have to be true in real life”, which seems to contradict the claim that some moves in a position are better on the objective merits of the position, rather than merely with respect to the tendencies of some given population of players.
The actual detailed complicated actually-true underlying reality is that by virtue of being a finite zero-sum game, chess fulfills the conditions of the minimax theorem, which implies that there exists an inexploitable strategy. You can have rock-paper-scissors–like cycles among particular strategies, but the minimax strategy does no worse than any of them.
The implications for real-world non-perfect play are subtler. As a start, Czarnecki et al. 2020 (of Deepmind) suggest that “Real World Games Look Like Spinning Tops”: there’s a transitive “skill” dimension along which higher-skilled strategies beat lower-skilled ones, but at any given skill level, there’s a non-transitive rock-paper-scissors–like plethora of strategies, which explains how players of equal skill can nevertheless have distinctive styles. The size of the non-transitive dimension thins out as skill increases (away from the “base” of the top—see the figures in the paper).
This picture seems to suggest that rather than being total nonsense, the problem with Humman’s worldview is in his attribution of it to the “top tier”. Non-transitivity is real and significant in human life—but gradually less so as we approach the limit of optimality.
This picture seems to suggest that rather than being total nonsense, the problem with Humman’s worldview is in his attribution of it to the “top tier”. Non-transitivity is real and significant in human life—but gradually less so as we approach the limit of optimality.
i was with you until this, but i must pick a nit. Humman’s error is far earlier: he is not near that limit.
even in the “spinning top” model, we would expect that non-transitivity rarely matters: between two players picked at random, skill differences will dominate. you’ll need strong selection effects (players selected from a particular Elo bracket, e.g.) before play-style can matter.
perhaps paradoxically, i would expect more non-transitive matchups at the very highest level, as that is where these selection effects are strongest. indeed, there are different “best” players for classical, blitz, 960, … . at the local club, Neumman will win all of these, and the visiting IM will never lose a game, no matter the time control, or opening position.
I agree Tessa’s explanation isn’t especially good, though it’s maybe more “incomplete” than “bogus”.
I don’t think the minimax theorem comes anywhere close to implying the existence of some sort of true optimal strategy, though, which I think becomes clear if you consider two types of chess bots. Bot A plays the same move as (something like) LeelaPieceOdds—unless that move would be moving from a game theoretically won position to a draw or loss, or from a draw to a loss, in which case it, say, randomly selects from all the moves that don’t do that, or ideally picks a move that humans are inclined to blunder against (maybe LeelaPieceOdds’s second choice, or something.)
On the other hand, Bot B immediately resigns whenever the position is a game theoretic loss, and immediately offers a draw whenever the position is a game theoretic draw. If its opponent rejects the draw offer, Bot B prefers to stay in states with the fewest opportunities for its opponent to blunder.
While both are “inexploitable”, Bot A beats humans every time, and Bot B draws them every time. (Unless chess is a game theoretic win for (WLOG) white, in which case Bot B wins as white but immediately resigns as black.) If you made Bot B a chess.com account, it very well might literally never break 1000 Elo.
So in pathological cases the non-transitivity can get pretty bad. The tops result sounds really neat but I haven’t read it yet and don’t know exactly how Bot B would fit into it—obviously “<1000 Elo” isn’t really going to be a description of Bot B that captures how it fits into such a structure very well.
To make the pair of strategies slightly more concrete, we could say that Bot A always picks LeelaPieceOdds’s top choice, out of moves that don’t hurt the game-theoretic status, and Bot B always picks its bottom choice out of those moves.
It would be fun to watch Bot B play a novice human, followed by a grandmaster. In both cases it might look a lot like a bored genius playing a newbie—hanging some pieces to even the odds, then playing competently with what’s left.
Why can’t there be a rock-paper-scissors–like structure, where in some position, 12. …Ne4 is good against positional players and bad against tactical players?
I would say that in that situation, the move is bad, but being a positional player after your opponent makes that move is also bad.
hmm it didn’t strike me when reading but that paragraph in OP is kinda bad as you say.
> the minimax theorem
a less complicated and more correct reference is Zermelo’s theorem, this is combinatorial game theory not economic (a further complication of reality is that the wiki article is kinda bad, it’s just induction). The theory also explains that in fact some chess moves are better than others in a mathematical sense, because some worsen the position more than others (e.g. Win->Draw vs Win->Win). Though it doesn’t match what Tessa says about vectors with lengths very well.
This explanation is bogus. (Obviously, the conclusion that Elo scores are practically meaningful is correct, but that’s not an excuse.)
Mr. Humman could locally-validly reply that Tessa is begging the question by assuming that there’s a fact of the matter as to one move being “better” than another in a position. Whether a move is “good” depends on what the opponent does. Why can’t there be a rock-paper-scissors–like structure, where in some position, 12. …Ne4 is good against positional players and bad against tactical players?
Earlier, Tessa does appeal to player comparisons being “mostly transitive most of the time”—but only as something that “didn’t have to be true in real life”, which seems to contradict the claim that some moves in a position are better on the objective merits of the position, rather than merely with respect to the tendencies of some given population of players.
The actual detailed complicated actually-true underlying reality is that by virtue of being a finite zero-sum game, chess fulfills the conditions of the minimax theorem, which implies that there exists an inexploitable strategy. You can have rock-paper-scissors–like cycles among particular strategies, but the minimax strategy does no worse than any of them.
The implications for real-world non-perfect play are subtler. As a start, Czarnecki et al. 2020 (of Deepmind) suggest that “Real World Games Look Like Spinning Tops”: there’s a transitive “skill” dimension along which higher-skilled strategies beat lower-skilled ones, but at any given skill level, there’s a non-transitive rock-paper-scissors–like plethora of strategies, which explains how players of equal skill can nevertheless have distinctive styles. The size of the non-transitive dimension thins out as skill increases (away from the “base” of the top—see the figures in the paper).
This picture seems to suggest that rather than being total nonsense, the problem with Humman’s worldview is in his attribution of it to the “top tier”. Non-transitivity is real and significant in human life—but gradually less so as we approach the limit of optimality.
i was with you until this, but i must pick a nit. Humman’s error is far earlier: he is not near that limit.
even in the “spinning top” model, we would expect that non-transitivity rarely matters: between two players picked at random, skill differences will dominate. you’ll need strong selection effects (players selected from a particular Elo bracket, e.g.) before play-style can matter.
perhaps paradoxically, i would expect more non-transitive matchups at the very highest level, as that is where these selection effects are strongest. indeed, there are different “best” players for classical, blitz, 960, … . at the local club, Neumman will win all of these, and the visiting IM will never lose a game, no matter the time control, or opening position.
I agree Tessa’s explanation isn’t especially good, though it’s maybe more “incomplete” than “bogus”.
I don’t think the minimax theorem comes anywhere close to implying the existence of some sort of true optimal strategy, though, which I think becomes clear if you consider two types of chess bots. Bot A plays the same move as (something like) LeelaPieceOdds—unless that move would be moving from a game theoretically won position to a draw or loss, or from a draw to a loss, in which case it, say, randomly selects from all the moves that don’t do that, or ideally picks a move that humans are inclined to blunder against (maybe LeelaPieceOdds’s second choice, or something.)
On the other hand, Bot B immediately resigns whenever the position is a game theoretic loss, and immediately offers a draw whenever the position is a game theoretic draw. If its opponent rejects the draw offer, Bot B prefers to stay in states with the fewest opportunities for its opponent to blunder.
While both are “inexploitable”, Bot A beats humans every time, and Bot B draws them every time. (Unless chess is a game theoretic win for (WLOG) white, in which case Bot B wins as white but immediately resigns as black.) If you made Bot B a chess.com account, it very well might literally never break 1000 Elo.
So in pathological cases the non-transitivity can get pretty bad. The tops result sounds really neat but I haven’t read it yet and don’t know exactly how Bot B would fit into it—obviously “<1000 Elo” isn’t really going to be a description of Bot B that captures how it fits into such a structure very well.
I like this observation!
To make the pair of strategies slightly more concrete, we could say that Bot A always picks LeelaPieceOdds’s top choice, out of moves that don’t hurt the game-theoretic status, and Bot B always picks its bottom choice out of those moves.
It would be fun to watch Bot B play a novice human, followed by a grandmaster. In both cases it might look a lot like a bored genius playing a newbie—hanging some pieces to even the odds, then playing competently with what’s left.
I would say that in that situation, the move is bad, but being a positional player after your opponent makes that move is also bad.
hmm it didn’t strike me when reading but that paragraph in OP is kinda bad as you say.
> the minimax theorem
a less complicated and more correct reference is Zermelo’s theorem, this is combinatorial game theory not economic (a further complication of reality is that the wiki article is kinda bad, it’s just induction). The theory also explains that in fact some chess moves are better than others in a mathematical sense, because some worsen the position more than others (e.g. Win->Draw vs Win->Win). Though it doesn’t match what Tessa says about vectors with lengths very well.