Questions about the theoretical foundations of gambits

From what I've read in multiple sources about the Queen's Gambit Accepted (QGA) (https://en.wikipedia.org/wiki/Queen%27s_Gambit),  Black simply cannot hang onto White's gambited c-pawn indefinitely without losing.

In the King's Gambit Accepted (KGA) (https://en.wikipedia.org/wiki/King%27s_Gambit), however, it sounds like Black *can* safely hang onto White's gambited f-pawn indefinitely, and draw (at the least).

These are the two most important gambits. I also understand that in chess there is an inherent tradeoff between time gained and material sacrificed, which is why gambits work.

Now my questions...

(1) Is my understanding of the KGA correct, in that Black can in fact hold onto the gambited pawn safely and indefinitely?

(2) Are there any lines of the KGA that strongly suggest that White *must* lose, ultimately because of his lost pawn in the opening? If so, which lines are those?

(3) (The big question, where all this is leading:) What is the exact numerical relationship between the number of moves gained and the material sacrifice equivalent? In other words, does the one f-pawn sacrificed in the KGA equate to exactly 1.0 free moves, or maybe 1.5 free moves, or something else? (Mr. Muller might have to answer this one.)

My motivation here is to start to treat the game of chess like a physics problem, where there exist numerical constants that can be determined that characterize the system, numbers analogous to the speed of light, gravitational constant, and Planck's constant. I'll post more on this later, especially if I can get some answers to the above.

I've been thinking about these topics a lot for the past 2-3 days, and I came to the conclusion that chess differs from physics in that each position on a chessboard has its own characteristic constant values, as well as its own set of typical plans, typical endgames, typical tactical level, typical piece placement, etc. That would be analogous to the speed of light changing slightly when you travel to a different galaxy. That's why I mentioned only the two most important gambits and the specific pawn (and implied move number) where the sacrifice occurred: the KGA and QGA would have their own constants that aren't universal to all games, but even knowing which lines were critical would allow one to put an approximate upper and lower bound on specific constants, which would be almost as good as providing a specific constant.

I'm jumping ahead a little, but this is another case where one can use popularity statistics, say from 365chess, combined with a weighted average to estimate the average benefit gained for a given gambit pawn in a given position, with the drawback that one would have to go through many entire games to confirm that Black didn't change his mind and give the pawn back somewhere along the way.

Your comment reminded me of a related question: How many lost tempi in the opening would allow the opponent to force a win? For example: 1. e4 (pass) 2. d4 (pass) and now Black can respond to each of White's moves as usual. I claim that two lost tempi equals a lost game for the side that lost those tempi, so in the above example White could always force a win. Does anybody disagree with my claim of two moves?

I can't remember for certain but I believe the old idea was that two tempi equals a pawn in the opening. Though probably with computers, it's probably closer to 1.5 pawns now days.

That could be enough for an advantage, maybe even a sustainable advantage, but far from a forced win.

What usually wins for the gambit is when the opponent loses more time plus position in trying to keep the material. Maybe a three tempi gain, certainly four, plus position is probably winning. Though again not forced.

I guess that's why over time most gambits have been refuted by returning the material and gaining back at least a tempo with other plusses.

Checking out the Evans Gambit and was it Rubinstein's refutation might be one of the best illustrations of general gambit time vs. material considerations.

I should have been clearer. I'm referring to retaining the same *amount* of material for the rest of the game, without regard for which pawn (or piece) is being retained.

That's a good point about closed games, but in my experience two extra moves is just too overwhelming an advantage, even in a closed game. I played a number of computer games where I set up the board two tempi ahead, and there was just no way the computer's side could ever stay alive. I agree that it's hard to prove that since it takes a while for the defense to crumble, but in my experience it must.

If not 2, then how many lost tempi would you say *is* a forced loss to the side that lost the tempi? If 2 is not the magic number, and admittedly it's hard to prove that, then 3 certainly must work. This issue made me realize a better example may have been the Smith-Morra Gambit, rather than the QGA or KGA: 1. e4 c5 2. d4 cxd4 3. c3 dxc3 4. Nxc3. That gambits *two* pawns successively, so the outcome should be clearer. My guess is that nobody would be able to find any master game where Black survived while insisting on retaining two extra pawns of material for the entire game. I'll have to think about your tempo counting scheme to see how many tempi you think the Smith-Morra Gambit would be equivalent to.

Apparently the approximate answer to my question (3) is well-known but I didn't know it, though I vaguely remember somebody mentioning it recently. Per one thread updated today, and per an old post on chess.com, the relationship is:

1 tempo = .33 pawn

or

1 pawn = 3 tempi

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http://www.chess.com/forum/view/game-analysis/moving-the-king-to-corner-or-middle

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http://www.chess.com/forum/view/general/how-much-is-one-tempo-worth

The implication is that it takes *3* (not 2 as I believed) tempi gain in time to force a win, which answers the later question I had. As I suspected, it was HGMuller who knew the answer. Unfortunately, I don't believe it's possible to derive such results from sites like 365chess, only from specialized studies involving computer programs playing against themselves. But that gets my other studies moving again, at least in approximation. Two other studies I'm doing came to a halt without that information!

Good point. I erred based on a false assumption. I was using the 90% win statistic for K-P endgames with 1 pawn difference rather than the 50-60% statistic for all endgames with 1 pawn difference. Therefore at best, that 1 pawn = 3 tempi advantage would give a player only about a 10% winning advantage, statistically, not a 40% winning advantage.

"In endgames with pieces and pawns, an extra pawn is a winning advantage in 50 to 60 percent of the cases."

"In king and pawn endings, an extra pawn is decisive in more than 90 percent of the cases (Euwe & Meiden 1978:xvi)."

https://en.wikipedia.org/wiki/Chess_endgame