Something Hikaru Said

[rabbits23]

Robert: I only have a couple of comments to make. You mention the word perfection on several occasions apropos chess play and concede that it is difficult to pin down. My question is whether the the term 'perfect' can be justifiably applied to chess? The aim of the game is to get checkmate right? And, as a number of people have pointed out, to do this your opponent has to make mistakes. As far as I can see even the best chess programs make moves which an opponent can exploit. But does perfect play ever enter the equation and will it ever feature in the game? I mean what is it? Chess has so many variables that I doubt whether a human or a chess program could find a "so-called" perfect move for every possible position that arises during the course of a game. Now of course 1e4 is perfect, correct? And 1e4 e5 is perfect as well. Why? Because neither move leads to the downfall of their position. But after that it gets quite tricky. What if white plays 2f4? Is this an imperfect move and how can you tell?? Some people would say yes others no But but there is no way of knowing.There are too many factors involved I would argue. And this is why I love chess so much The uncertainty and unpredictability of it.

[Uri Blass]

None of this seems reasonable to me. The percentage of GM errors, for example. A GM makes far FEWER errors against a weak program than against a strong program, but not because he actually makes fewer errors, but because the opponent doesn't notice them and doesn't punish them.

I disagree here.
If the opponent does not play well it is easier not to make mistakes.

I clearly have games against humans when I did no significant mistakes based on computer analysis(no move reduce the evaluation by more than 0.2 pawn).

It is not because I am so strong but because it is easier not to make mistakes when the opponent does not play well.
If the opponent play well I expect myself to do more mistakes.

Uri

"no significant mistakes based on computer analysis" is meaningless when we are talking about PERFECT computer play. ANY mistake will be significant there. This extrapolation about what happens today is meaningless when we talk about a perfect chess opponent.

It is not meaningless because the computer is clearly stronger than me and find many mistakes.

If the computer find that I play more mistakes in games that I play against stronger opponents it means that it is easier to make mistakes against stronger opponents.

I do not have the 32 piece tablebases but my speculation is that there are many games when the winner did not do a mistake that change the theoretical result and also there are draws with no mistakes.
Of course the side who did no mistake could do mistakes in case of playing against a stronger opponent.

It is not easier to make mistakes against stronger opponents. It is simply more likely that they will understand your mistake and how they can exploit it. Your propensity for making mistakes is independent of the opponent, your brain cells don't suddenly change when you play a 2500 player vs a 1500 player.

As far as the "many games with no discernible mistakes" there are only "many games" because there are millions and millions of games that have been played. If your opponent spots the mistake and beats you only 1% of the time, that does NOT mean you played 99% of your games with no mistakes.

It is easier to make mistakes when the opponent help you to do mistakes.
If the opponent play weak moves I simply has good chances not to get positions when I do mistakes.

It is not that my brain change when I play against stronger player but
I simply need to solve harder problems that I cannot solve when
I do not need to solve hard problems against the weak player.

The opening moves I guess that I know to play perfectly because there are many drawing moves and after the opponent made a mistake it is easy not to make a mistake.

As an extreme example suppose that
I play against a player who make random moves.

I guess that part of the games are going to be 1.f3 e5 2.g4 Qh4 mate when I played perfectly because the opponent did not help me to do mistakes.

You can claim that maybe 1...e5 is a mistake(and I am not 100% sure because maybe 1...e5 draw when 1...d5 win) but my guess is that it is not a mistake.

Against non random player that is a very weak chess player something like the following can happen when again I cannot prove a mistake by white with today's software.

1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 Nf6 5.Nc6+ that win the queen can happen when after 5.Nc6+ I believe that it is easy to play perfectly(maybe I do not mate in the fastest way that is not important but I always play winning moves).

Your opponent can't help YOU "do mistakes". Not possible. He doesn't get to make moves for you. Both of you will still make mistakes at the same rate as always, but a stronger player will make fewer, and in particular fewer that the weaker player can grasp and punish. But he still makes mistakes just the same.

You are mixing playing skill with this mistake stuff. A player at a given level makes mistakes at a predictable rate. It just takes a stronger player to spot many of them. But just because they are not spotted, doesn't mean they are not there.

The stronger player make moves that lead to position when it is easier for the weaker player to make mistakes.

If you take a game that is solved like checker and 2 human non perfect players when one is significantly stronger but still usually lose against the perfect player then I believe that analysis can show that the winner often does not make mistakes based on computer analysis because the opponent made first a mistake and after the mistake it was an easy win.

I don't agree. We are not using the same definition of "mistake". A player of a given skill will make some predictable number of "outcome-altering" mistakes in a game. That's where his rating comes from in the first place. But his rating is not based only on the number of mistakes, it is based on the number of mistakes he makes that his opponent can take advantage of. The best GM on the planet might never lose a game, but that doesn't mean he doesn't make mistakes at his usual rate, it just means nobody can recognize and take advantage of them. But a perfect player will see every last mistake and win because of them.

As of today, nobody has a clue about how many mistakes a GM makes, because you have to first recognize the mistake, and that's often impossible in chess. Until you find a perfect player.

There was a time where some deep blue moves were labeled as "mistakes" based on GM analysis. Deep analysis showed that they were not mistakes, the humans just couldn't follow what was going on. 7 piece endings are already problematic for human understanding. What happens when there are 32 piece files in this hypothetical discussion?

I wonder what happens if you use chinook to analyze games of humans in checkers.

Does chinook find that a human player who is not the perfect player does relatively less mistakes when he play against significantly weaker player
relative to the case of playing against significantly stronger player?

I believe that chinook is going to find that not only that a player does less mistakes against weaker players but also that in part of the games the stronger player played perfectly because the opponent did a mistake in the beginning and after it winning is easy.

If we find that I am right then it support my speculation about chess when I believe that I play relatively less mistakes(moves that change the outcome of the game) against weaker players when in part of the games I do not do mistakes.

[Uri Blass]

700-1500 Elo is certainly meaningless. Where does that come from? 20 years ago the assumption was that 2800 was the upper bound on Elo. That seems to have bitten the big banana. The only thing that bounds Elo is that the best player will be hard-pressed to get more than 800 above the second-best player. But then the second best can get to 800 below the 3rd.

Well, I will show how that is "meaningless". Some 2-3 years ago I computed the estimate of the rating of the perfect engine as limiting ELO of Houdini 3 to infinite number of nodes (time control). Duncan refers to that post of mine. Today I repeated with thousands of games this estimate with Komodo 9.3 to infinite time control (doubling in time). I guess you consider ELO gain per doubling as "close to 70 ELO points", which is totally misleading, and the diminishing returns are thought by you as "hard to measure".

My today's tests:
First doubling: 10s+0.1s vs 5s+0.05s -- 148 ELO points
Second doubling: 20s + 0.2s vs 10s+0.1s -- 128 ELO points
Third doubling: 40s+0.4s vs 20s+0.2s -- 110 ELO points

Other data:
Close to eighth doubling -- CEGT 40/20' -- 65 ELO points
Close to ninth doubling -- CCRL 40/40' -- 55 ELO points

Diminishing returns are very visible here. I fitted these results with the relevant curve. Here is the plot of ELO gain per doubling in time:


The red dots are not data, they are predictions, first for Larry's 45'+15'' on 24 cores level against humans in odds matches, for which Larry got as Komodo performing at 3250 FIDE ELO level, second red dot shows TCEC level of 150'+30'' on 24 faster cores. One sees that at those high levels the gain per doubling is below 40 ELO points.

The fit: the fitting curve chosen is relevant as I show. It is a/(1+b*(number of doublings)^c). {a,b,c} are parameters to fit, and {c} is the relevant exponent here. If {c} is between 0 and 1, then we do have diminishing returns, but the total ELO is unbounded for higher and higher number of doublings, therefore the rating of perfect engine is undetermined and high. If c>1, then the perfect engine has a definite limited rating which can be derived.

It turns out that the best fit is c=1.56, which is significantly larger than 1, and the ELO of the perfect engine can be computed and is not very high. I actually derive this, it is not assumed.

Do you have a more relevant fitting expression? I assume only that diminishing gains are going to 0 gain per doubling to infinite number of doublings, and this is related to the draw ratio going to 100% to infinite time control (doublings). Do you have a more plausible model which shows that the draw ratio has in fact lower limit than 100%?

Having the fit, I can compute the ELO of the perfect engine by summing up all gains from doublings starting from established by Larry FIDE ELO 3250 (close to 13th doubling in time). It is ~1300 ELO points above Larry's 24 core Komodo. Therefore, Komodo 9.3 shows a FIDE ELO of the perfect engines at about 3250+1300 = 4550 ELO points. CCRL rating would be 100-200 ELO points higher, as their rating is computer rating. And very close to 4800 CCRL I got 2-3 years ago with Houdini 3.

As you see, 700-1500 ELO points estimate for improvement over Komodo I gave previously is not meaningless at all, my model here is simple, robust, and consistent with earlier results (including one or two by Don Dailey). If you want to dismiss it, it's surely not by "we don't know" mantra, because it seems it is mostly you who "doesn't know".

Again, your "diminishing returns" applies to programs of today and earlier. Forget the "doubling" stuff, we are not talking about doubling. We are talking about an infinite improvement, all the way to perfect play. If you take programs from 1970, you would get a completely different perspective vis a vis "doubling". But we are not in 1970. And 100 years in the future, we won't be stuck with 2016 era programs.

I don't know how to predict what will happen as we approach perfection. I only know that if you wait long enough chess will eventually get there, assuming the sun continues to burn at the present rate, etc... I don't see any way to project to perfect play since we have no idea of what perfect play looks like..

in theory a program should be able to play perfectly if you increase the speed enough.

Practically it is not the case because stockfish cannot see more than 128 plies forward and I guess something similar is for komodo and there may be also hash collisions but if we change some constants we can prevent it with no big change in the rating(at the time control that kai test) so assume that we have a program that practically can play perfectly if it get enough time.

We can speculate what is the maximal possible elo by the limit of the sequence of the elo that we get from doubling the speed.

This speculation does not suggest significantly different numbers if we use some old program instead of using komodo or stockfish.

[rabbits23]

Robert; Just thinking about it whilst struggling to get to sleep on a very hot
night here in Sydney I realized of course that there are many examples of perfect moves. Mate in ...fit the bill perfectly. No doubt there are other examples.
Regards Allan

[bob]

None of this seems reasonable to me. The percentage of GM errors, for example. A GM makes far FEWER errors against a weak program than against a strong program, but not because he actually makes fewer errors, but because the opponent doesn't notice them and doesn't punish them.

I disagree here.
If the opponent does not play well it is easier not to make mistakes.

I clearly have games against humans when I did no significant mistakes based on computer analysis(no move reduce the evaluation by more than 0.2 pawn).

It is not because I am so strong but because it is easier not to make mistakes when the opponent does not play well.
If the opponent play well I expect myself to do more mistakes.

Uri

"no significant mistakes based on computer analysis" is meaningless when we are talking about PERFECT computer play. ANY mistake will be significant there. This extrapolation about what happens today is meaningless when we talk about a perfect chess opponent.

It is not meaningless because the computer is clearly stronger than me and find many mistakes.

If the computer find that I play more mistakes in games that I play against stronger opponents it means that it is easier to make mistakes against stronger opponents.

I do not have the 32 piece tablebases but my speculation is that there are many games when the winner did not do a mistake that change the theoretical result and also there are draws with no mistakes.
Of course the side who did no mistake could do mistakes in case of playing against a stronger opponent.

It is not easier to make mistakes against stronger opponents. It is simply more likely that they will understand your mistake and how they can exploit it. Your propensity for making mistakes is independent of the opponent, your brain cells don't suddenly change when you play a 2500 player vs a 1500 player.

As far as the "many games with no discernible mistakes" there are only "many games" because there are millions and millions of games that have been played. If your opponent spots the mistake and beats you only 1% of the time, that does NOT mean you played 99% of your games with no mistakes.

It is easier to make mistakes when the opponent help you to do mistakes.
If the opponent play weak moves I simply has good chances not to get positions when I do mistakes.

It is not that my brain change when I play against stronger player but
I simply need to solve harder problems that I cannot solve when
I do not need to solve hard problems against the weak player.

The opening moves I guess that I know to play perfectly because there are many drawing moves and after the opponent made a mistake it is easy not to make a mistake.

As an extreme example suppose that
I play against a player who make random moves.

I guess that part of the games are going to be 1.f3 e5 2.g4 Qh4 mate when I played perfectly because the opponent did not help me to do mistakes.

You can claim that maybe 1...e5 is a mistake(and I am not 100% sure because maybe 1...e5 draw when 1...d5 win) but my guess is that it is not a mistake.

Against non random player that is a very weak chess player something like the following can happen when again I cannot prove a mistake by white with today's software.

1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 Nf6 5.Nc6+ that win the queen can happen when after 5.Nc6+ I believe that it is easy to play perfectly(maybe I do not mate in the fastest way that is not important but I always play winning moves).

Your opponent can't help YOU "do mistakes". Not possible. He doesn't get to make moves for you. Both of you will still make mistakes at the same rate as always, but a stronger player will make fewer, and in particular fewer that the weaker player can grasp and punish. But he still makes mistakes just the same.

You are mixing playing skill with this mistake stuff. A player at a given level makes mistakes at a predictable rate. It just takes a stronger player to spot many of them. But just because they are not spotted, doesn't mean they are not there.

The stronger player make moves that lead to position when it is easier for the weaker player to make mistakes.

If you take a game that is solved like checker and 2 human non perfect players when one is significantly stronger but still usually lose against the perfect player then I believe that analysis can show that the winner often does not make mistakes based on computer analysis because the opponent made first a mistake and after the mistake it was an easy win.

I don't agree. We are not using the same definition of "mistake". A player of a given skill will make some predictable number of "outcome-altering" mistakes in a game. That's where his rating comes from in the first place. But his rating is not based only on the number of mistakes, it is based on the number of mistakes he makes that his opponent can take advantage of. The best GM on the planet might never lose a game, but that doesn't mean he doesn't make mistakes at his usual rate, it just means nobody can recognize and take advantage of them. But a perfect player will see every last mistake and win because of them.

As of today, nobody has a clue about how many mistakes a GM makes, because you have to first recognize the mistake, and that's often impossible in chess. Until you find a perfect player.

There was a time where some deep blue moves were labeled as "mistakes" based on GM analysis. Deep analysis showed that they were not mistakes, the humans just couldn't follow what was going on. 7 piece endings are already problematic for human understanding. What happens when there are 32 piece files in this hypothetical discussion?

I wonder what happens if you use chinook to analyze games of humans in checkers.

Does chinook find that a human player who is not the perfect player does relatively less mistakes when he play against significantly weaker player
relative to the case of playing against significantly stronger player?

I believe that chinook is going to find that not only that a player does less mistakes against weaker players but also that in part of the games the stronger player played perfectly because the opponent did a mistake in the beginning and after it winning is easy.

If we find that I am right then it support my speculation about chess when I believe that I play relatively less mistakes(moves that change the outcome of the game) against weaker players when in part of the games I do not do mistakes.

That was exactly the question I posed in this thread. I don't know the answer. I'll try to email Jonathan to see if this was ever done...

[bob]

Robert: I only have a couple of comments to make. You mention the word perfection on several occasions apropos chess play and concede that it is difficult to pin down. My question is whether the the term 'perfect' can be justifiably applied to chess? The aim of the game is to get checkmate right? And, as a number of people have pointed out, to do this your opponent has to make mistakes. As far as I can see even the best chess programs make moves which an opponent can exploit. But does perfect play ever enter the equation and will it ever feature in the game? I mean what is it? Chess has so many variables that I doubt whether a human or a chess program could find a "so-called" perfect move for every possible position that arises during the course of a game. Now of course 1e4 is perfect, correct? And 1e4 e5 is perfect as well. Why? Because neither move leads to the downfall of their position. But after that it gets quite tricky. What if white plays 2f4? Is this an imperfect move and how can you tell?? Some people would say yes others no But but there is no way of knowing.There are too many factors involved I would argue. And this is why I love chess so much The uncertainty and unpredictability of it.

My definition of perfect is a little relaxed from true perfection. IE for a KR vs K ending, perfection is ALWAYS playing the move that reduces the mate-in-N value by one. Where my definition of perfection is always making a move that leads to a win. IE not making a move that allows the 50 move rule or insufficient material escape by the weaker side.

32 piece egtbs would play truly perfect chess. But so long as the human keeps a won game once he has reached one is good enough here. I don't believe humans can do this.

As far as the handicap for perfection goes, one might discover that one can not give any handicap at all and still win. Or even draw. But only the computer will be perfect in this discussion, humans remain human.

Do I believe perfection is possible? No. I've stated that too many times to count. We won't ever have 32 piece endgame tables. We won't ever be able to search to the end of the game from the starting position. So this was hypothetical from the beginning, as in the original posts at the beginning of the thread.

Won't ever happen, I agree.

[bob]

When you give a knight advantage to a 2000 player, how do you expect him to fare against a 3000 player? Do you believe that knight advantage is "so durable" there?

> 75% - I'd happily take an extra knight vs stockfish (which is certainly beyond 3000). a 2000 player reasonably knows to play chess. He should convert a full knight edge regardless of the strength of his opponent most of the time. So yes i believe it's so durable there. I would say it's a feature of the game that it's extraordinarily materialistic and given such a large material edge with no compensation off the start, a reasonable player should expect to win against any opponent.

So you give it to a 2800 player, and pit him against a paltry 3800 player. Still "so durable". What about a player that is absolutely perfect? Who knows what that rating will be? Still "so durable"? Yes it is possible, but it is not so probable that one can say "a knight is enough, based on how programs and humans play today, Q.E.D."
That is my point. I'm not taking either size of the knight argument. I am sticking in the middle as "absolutely unknown today."

I don't know where the asymptote is, that much is true, but i'm pretty sure a knight is beyond it. But then i'm not so sure on for instance double-exchange that by traditional bean counting is beyond a knight. The slow speed at which rooks begin to exert their influence on the game might allow the stronger player to develop sufficient compensation for being two exchanges down to make a game of it. I'd still hope to win it, but it's plausible that it'd make a 'fair' fight against a strong enough opponent.

It is technically true that it's 'unknown' as to how bad humans are at the game but one can still make reasonable guesses about the bounds of what's possible in the confines of the game.

I think the most reasonable guess is that you can't give ANY material away and win. Assuming perfect play by both sides. But when you KNOW one side is going to make mistakes, it is impossible to predict what can be given away that will be compensated for by opponent mistakes. We don't know the human mistake rate since we have no way to measure it against perfection. Therefore it is simply speculation whatever we say. My take is to not say anything, until the answer is known.

[bob]

700-1500 Elo is certainly meaningless. Where does that come from? 20 years ago the assumption was that 2800 was the upper bound on Elo. That seems to have bitten the big banana. The only thing that bounds Elo is that the best player will be hard-pressed to get more than 800 above the second-best player. But then the second best can get to 800 below the 3rd.

Well, I will show how that is "meaningless". Some 2-3 years ago I computed the estimate of the rating of the perfect engine as limiting ELO of Houdini 3 to infinite number of nodes (time control). Duncan refers to that post of mine. Today I repeated with thousands of games this estimate with Komodo 9.3 to infinite time control (doubling in time). I guess you consider ELO gain per doubling as "close to 70 ELO points", which is totally misleading, and the diminishing returns are thought by you as "hard to measure".

My today's tests:
First doubling: 10s+0.1s vs 5s+0.05s -- 148 ELO points
Second doubling: 20s + 0.2s vs 10s+0.1s -- 128 ELO points
Third doubling: 40s+0.4s vs 20s+0.2s -- 110 ELO points

Other data:
Close to eighth doubling -- CEGT 40/20' -- 65 ELO points
Close to ninth doubling -- CCRL 40/40' -- 55 ELO points

Diminishing returns are very visible here. I fitted these results with the relevant curve. Here is the plot of ELO gain per doubling in time:


The red dots are not data, they are predictions, first for Larry's 45'+15'' on 24 cores level against humans in odds matches, for which Larry got as Komodo performing at 3250 FIDE ELO level, second red dot shows TCEC level of 150'+30'' on 24 faster cores. One sees that at those high levels the gain per doubling is below 40 ELO points.

The fit: the fitting curve chosen is relevant as I show. It is a/(1+b*(number of doublings)^c). {a,b,c} are parameters to fit, and {c} is the relevant exponent here. If {c} is between 0 and 1, then we do have diminishing returns, but the total ELO is unbounded for higher and higher number of doublings, therefore the rating of perfect engine is undetermined and high. If c>1, then the perfect engine has a definite limited rating which can be derived.

It turns out that the best fit is c=1.56, which is significantly larger than 1, and the ELO of the perfect engine can be computed and is not very high. I actually derive this, it is not assumed.

Do you have a more relevant fitting expression? I assume only that diminishing gains are going to 0 gain per doubling to infinite number of doublings, and this is related to the draw ratio going to 100% to infinite time control (doublings). Do you have a more plausible model which shows that the draw ratio has in fact lower limit than 100%?

Having the fit, I can compute the ELO of the perfect engine by summing up all gains from doublings starting from established by Larry FIDE ELO 3250 (close to 13th doubling in time). It is ~1300 ELO points above Larry's 24 core Komodo. Therefore, Komodo 9.3 shows a FIDE ELO of the perfect engines at about 3250+1300 = 4550 ELO points. CCRL rating would be 100-200 ELO points higher, as their rating is computer rating. And very close to 4800 CCRL I got 2-3 years ago with Houdini 3.

As you see, 700-1500 ELO points estimate for improvement over Komodo I gave previously is not meaningless at all, my model here is simple, robust, and consistent with earlier results (including one or two by Don Dailey). If you want to dismiss it, it's surely not by "we don't know" mantra, because it seems it is mostly you who "doesn't know".

Again, your "diminishing returns" applies to programs of today and earlier. Forget the "doubling" stuff, we are not talking about doubling. We are talking about an infinite improvement, all the way to perfect play. If you take programs from 1970, you would get a completely different perspective vis a vis "doubling". But we are not in 1970. And 100 years in the future, we won't be stuck with 2016 era programs.

I don't know how to predict what will happen as we approach perfection. I only know that if you wait long enough chess will eventually get there, assuming the sun continues to burn at the present rate, etc... I don't see any way to project to perfect play since we have no idea of what perfect play looks like..

in theory a program should be able to play perfectly if you increase the speed enough.

Practically it is not the case because stockfish cannot see more than 128 plies forward and I guess something similar is for komodo and there may be also hash collisions but if we change some constants we can prevent it with no big change in the rating(at the time control that kai test) so assume that we have a program that practically can play perfectly if it get enough time.

We can speculate what is the maximal possible elo by the limit of the sequence of the elo that we get from doubling the speed.

This speculation does not suggest significantly different numbers if we use some old program instead of using komodo or stockfish.

How much better does a program play with 6 piece endgame tables than with 5? For those positions that reach 6 pieces it would play perfectly with them, not so perfectly without them. Back away another ply and it plays even worse without the 6's. The point is, that when you get 'em all done, the rating may take a quantum leap over just having N-1 piece tables, because it is much easier to find the correct answer instantly than to have to search up to 100 plies deep to hit that 31 piece EGTB.

Schaefer certainly saw this effect with Chinook. As it began to reach the endgame tables, its rating took a huge jump, and once it could reach them for every move, it became perfection.

So the diminishing returns + ultimate perfection are not necessarily tied to each other via some sort of prediction formula.

[bob]

Robert; Just thinking about it whilst struggling to get to sleep on a very hot
night here in Sydney I realized of course that there are many examples of perfect moves. Mate in ...fit the bill perfectly. No doubt there are other examples.
Regards Allan

The danger with that thinking is that it is highly restrictive. I've seen MANY GM games where they did not play the shortest mate. But they still mated the opponent. I don't try to include that level of tedium in my definition of "perfect". Simply, "if the game is won, the GM wins the game." Note that if he makes one losing mistake it is over since a perfect player will spot it.

The only question might be in the details. For example, suppose the first move in a knight-odds game is mated in 1200 moves. Absolutely perfect play would be mate in 1199 on the next move. By my definition, any mate will do. And then there is the case having to hope that the deepest mate is the hardest. Suppose the GM manages to drop a piece, but still is winning. If capturing the piece is mate in 1100, and not capturing is mate in 1198, which should the perfect player choose? Not so simple, since there is no way the human sees either mate anyway.

[syzygy]

Robert; Just thinking about it whilst struggling to get to sleep on a very hot
night here in Sydney I realized of course that there are many examples of perfect moves. Mate in ...fit the bill perfectly. No doubt there are other examples.
Regards Allan

Perfect moves are played all the time, but for most moves we lack the computational resources to verify their perfection.

I'm reasonably sure that many perfect GM-GM games exist. But we lack the computational resources to figure out which games are perfect and which aren't.

If the opening position is a win for white (unlikely, but not inconceivable), then 1.e4 f6 2.d4 g5 3.Qh5# is probably a perfect game. Otherwise, 1.d4 f6 2.e4 g5 3.Qh5# seems a good candidate (if the opening position wins but 1.e4 draws, then the winning opening move is probably 1.d4). Or simply: 1. e4 1-0 {black resigns}

If the opening position is a draw, then the most boring GM-GM game you can find might be a perfect game. Or simply: 1.e4 1/2-1/2 {drawn by agreement}

In a knight odds game, it will still be difficult for us humans (and engines) to know which moves or whole games are "perfect", but the moves that a good GM will come up with should normally be perfect and as long as the GM manages to play perfect moves, he will win that game. Now and then he might mess up, that is human.