Mathematics question about winning chances

When I was a kid, I used to ask my dad to crunch numbers in his head for me when I was thinking about dumb questions and needed actual quantification. He would say, "I do mathematics, I don't do arithmetic."... then he'd make me go do it myself. Usually badly.
So what I have, is a Mathematics question. Do you consider all plays that increases your winning chances by some amount, to be equally good, regardless of where the percent to win ends up, so long as the amount the chance to win has increased is the same?... If you are about to lose, then you make a play that saves you and sets the game state at one where you're 25% to win... is that just as good as if you are at 75% to win and you make a play that just sets your chances at 100% and you win..?
(What the hell is he talking about?... ok examples.) Exhibit A: You are about to die to Storm combo. They have a lot of mana out and cards in hand. You play Git Probe and see that when they untap you are just going to die. So you tap out and play Time Twister, which is normally a bad play, but it's your only play now since you know you are about to die. (Lets pretend that we can somehow know that you have exactly a 25% chance of winning after the Twister.)... You can maybe hit a broken 7 off your Twister and win. Or you can hope that your opponent just draws total garbo and you can win the game later.... so winning chances went from 0% to 25% with your line. +25
Exhibit B: You are locked in the dreaded Vintage monored burn mirror, and each player is in full topdeck mode at 3 life with 2 cards left in their deck. Each player has 1 bolt left in deck and 1 land and your turn is starting. So you are at 75% to win from here. And you topdeck your Bolt and kill the opponent. (Other than this not being a choice, your odds of winning went from 75% to 100%...) +25
Are plays that take you from 0% winning, to unlikely to win but not dead, just as valuable mathematically as plays that take you from very likely to win to actual, consummated victory?

said:
ll plays that increases your winning chances by so
Well, those plays are called "optimal plays". Most of the time, when having different options (unless you can do absolutely nothing), you are making choices that bring you to win or lose the game. I'm pretty bad understanding when to take a breath and analyze the whole state of the game. Most of the time I play by intuition, since I don't play enough to have real automatisms, and tend to switch deck more than I should.
When the situiation is close, measuring the mana and guessing opponent's cards (or which potential cards can make a real impact) is key. And finally, when odds are really good have in mind what could do your opponent to win you. And when odds are really bad, play like you were going to draw the few cards that save you. The problem is where there are 2 or 3 different cards that save you, since you will have 50% of making the right play.

@xouman My question is... since I've just noticed this in my own play. Do you intuitively value the play that actually finishes a game that you are almost certainly going to win, much more than you value a play that preserves a slim chance to win in a game that you will almost certainly lose anyway... even though they are mathematically identical... aren't they?

said:
Well, those plays are called "optimal plays". Most of the time, when having different options (unless you can do absolutely nothing), you are making choices that bring you to win or lose the game. I'm pretty bad understanding when to take a breath and analyze the whole state of the game. Most of the time I play by intuition, since I don't play enough to have real automatisms, and tend to switch deck more than I should.
I think the issue with this line of logic is that the game has so much imperfect information that you can almost never know when this will come up, so you will almost never know what the actual optimal play is in many situations. There have been a few points in my gameplay where I knew, certifiably, that I was dead unless I did something, in which case you go for any hail mary you have and if you have multiple of them you likely go for the one that has the best chance of winning you the game.
In most situations you can't even look at both players hands and gamestates and even know, since both players are acting with imperfect information what you can see playing out with ideal information may not be what happens since they may make the "wrong" plays. So I don't know you can ever evaluate these situations to begin with to understand the value of them in a real way.
I would argue the cards in the middle are typically the most important though, since most match ups in magic are won or lost on margins. A card that brings you from 010% vs a card that brings you from 90%100% is probably not as relevant as a card or play that brings you from 45% to 55%, since those will be the vast majority of plays where as the ones that certify wins or pull you from the brink are the outliers.

@Topical_Island
You are comparing apples to oranges here, because in one of the scenarios, scenario A, you are comparing two choices that you have, and opting for the one that increases your probability of winning from 0% to 25%. In scenario B, you haven't actually made a choice. I mention this because I think I know where you are going with this (Are plays that take me from large favorite to larger favorite worth more, less or the same as plays that take me from dead to large underdog). but Ive altered the form of your scenarios a bit, and this is why.Scenario B represents a 33% increase in expected value, whereas scenario A represents an indeterminate increase. If we simplify a bit and say in scenario A your win % goes from 1% to 25% then this represents a 2400% increase in expected value! So clearly A is more valuable right, since it represents a larger proportional increase in your EV? Maybe. In the vaccuum of your scenarios, I would say A is adding more value. But the last dimension of this problem, and one that isn't given here, is how often you are going to be in a position like scenario A and how often you will be in scenario B.
That would be information you would need, because depending on the frequency of each scenario, it could wind up being that one or the other (probably B) comes up so much more often that even if it may be adding a smaller amount of value, its higher frequency means that it is the one you should be paying more attention to, because over time it will cost you more.
@Topical_Island said:
@xouman My question is... since I've just noticed this in my own play. Do you intuitively value the play that actually finishes a game that you are almost certainly going to win, much more than you value a play that preserves a slim chance to win in a game that you will almost certainly lose anyway... even though they are mathematically identical... aren't they?
The play that ends the game is a function of all the plays made up until that point. I read this post after I wrote the above, so it looks like I didn't answer your question with it. But to answer this directly, the answer is no. If I am playing storm, the play that wins the game is putting tendrils on the stack with enough storm for lethal. There is nothing interesting at all about the play that ends the game. The interesting plays, and the most valuable ones, are the ones that lead up to it.

I think what you are hinting at is resultsoriented thinking. In the first case you have 25% percent chance to win by making the appropriate decision. In the second case, you have a 100% chance of winning making that decision. In terms of gratification, the second one always pays off while the first one only does so 1/4 games. This is a problem with resultsoriented thinking  the results only coincide with the actual "correctness" of a decision a certain percentage of the time and that only becomes apparent over the course of many, many games.
Removing the results from the equation, the two choices are equal but even that isn't a great way to think about it  you shouldn't be judging those lines against each other but against other lines you could take at each juncture. And your goal should be to maximize your chances of winning, which again can be independant of the actual results and there can be many different lines at each point, some building off of each other. In the case where I am facing a 0% or a 25% choice, I would backtrack and look for what (if anything) I could have done to put myself in a better position. For me I derive gratification in the process of going through these choices and examining them  I don't normally keep track of wins and losses when I test as I don't really care about it.
TLDR; I value them equally, which is to say, not much at all.

@Topical_Island
I think the answer is "yes, they are the same." I haven't spent a ton of time thinking about this, but I strongly suspect they both impact your Expected Match Wins the same. That should be the only number you actually care about.

@mmcgeach Assuming you have a fixed winrate in each game (which is false because of play/draw, but whatever) then this is correct. If R is your win%, then each increases your odds of winning the match by R/2*(1R).
Although I do agree with @ChubbyRain that this is kind of a silly hypothetical, since there's no real life situation where you are choosing between these two options.

This is one of those scenarios where I think you can only look at the current situation and evaluate options and assumed percentages, since you may never get into that scenario again. I don't think you can judge a card or build against these scenarios since they are so specific.

You've said you're asking a math question, but you're actually asking a philosophy/semantics question.
What do you mean when you ask if two plays are "equally good" ? You ask if two plays are "just as valuable mathematically" ... but math has no concept of value like this. You're really asking something like "which of these would you rather have happen", which obviously has an answer like "that's kind of up to you"
If you ask a more specific question, like "which of these plays increases your chance to win more", then you'll get a specific answer ... in this case, obviously "they're the same", but then you're begging the question.
What are you actually asking here?

said:
What do you mean when you ask if two plays are "equally good" ? You ask if two plays are "just as valuable mathematically" ... but math has no concept of value like this. You're really asking something like "which of these would you rather have happen", which obviously has an answer like "that's kind of up to you"
He's asking if 2 plays are mathematically identical (something we can never quantify, just theorycraft) and raise your win percentage by a fixed number, which is more important, one that prevents you from losing or one that ensures you win. The issue with the question is that the scenarios they come up in are the extremes of your gameplay. Winning and losing typically only happens once a game, and its all the nominal plays before that which get you into whatever situation that are really the important ones.
That being said, while I do think the plays in the middle are the most important overall, a play that brings you from 75% win to 100% win is likely no where near as important as a play that prevents you from losing. Logic being that if you do not win on your 75% play then the game continues and you will be afforded more opportunities to win, where as if you lose the game is over and you will not have other options.

@Protoaddct I calculated the change in the likelihood of winning the match under some reasonable simplifying assumptions and found they were identical. I'm not sure why you are arguing with intuition when the question is about "mathematics".
@BrassMan I guess maybe your intuition is better than mine, but it was not obvious to me that these 2 plays lead to an identical increase in match win %.

oh, no, I didn't do any math, I just took him at his word when he said "one play brings you from 75100, the other brings you from 025" ... if we assume that's true, then the rest of the question is just "which of those do you like more", which is unanswerable in the general case

said:
Are plays that take you from 0% winning, to unlikely to win but not dead, just as valuable mathematically as plays that take you from very likely to win to actual, consummated victory?
No, I'd say a play that takes you from 0% chance of winning to anything at all is a better play than any play that takes likely to win to actual win. A condition in which you are likely to win is like when you have Yawgmoth's Bargain or a Necropotence in play. You didn't actually win the game yet, but you resolved Bargain/Necro, so the rest of your plays aren't really that important, even putting Tendrils on the stack; you could wait until the next turn to cast the Tendrils and it probably wouldn't matter much. In the case of the Timetwister, you have to cast the Twister to disrupt your opponent's hand, or you are going to lose. You HAVE to make that play.

This is an incredibly complicated question. First off, I wonder why you want to compare two plays that come in defferent scenarios. In a normal game of Magic you will never be forced to choose between those two plays you presented. So I guess your question is purely theoretical (although it is still hard to define what the goal of such a comparison would be).
Basically you are asking "is the expected win percentage (EWP) in a match after doing a certain play minus the EWP after not doing that play a good metric for comparing Magic plays?". And whether a metric is good depends on what you want to know.
Obviously the most useful comparisons are between two plays in the same scenario. In that case we can probably agree that the proposed method is good, because it finds what we call "the optimal play" (the play after which you have the highest EWP). We still have little to no means of calculating it though. When you have different scenarios, things get more complex. Do you want to learn one play and know which one gives you the highest overall expected MWP? Then you have to also look at things like the metagame presence of decks and the likelihood of a certain scenario in a given matchup. You would even have to check your chances of playing a certain deck (because after you choose a deck, a maximum of one of these plays stays open to you).
Or maybe you want to choose a deck? And you have your spreadsheet with all of the possible scenarios that can happen in a game of Magic. So you look at these two plays and you think: is the Burn play better for my Burn vs the field strategy or the Twister play for my whateverplaystwister vs the field strategy? To know that, you need to, again, know the chances of that scenario coming up. Then you can see how much your percentages vs the field changed and you can choose a deck based on updated values.
Or maybe you came up with some crazy heuristic that will revolutionize Magicplaying bots. And somewhere under the hood you just NEED to know the difference in EWPs that you mentioned. Here obviously this information is very useful (the scenario itself is purely theoretical, but who knows, right?).
So I think you see the pattern. In a real world it is hard to imagine a situation when you want to compare two plays in completely different scenarios. If you want to just be theoretical, then you need to define what are you talking about, as "mathematically identical" is not very precise (depends what is used to measure identity). Different goals can lead to different answers.
However, we still have to remember that even should this kind of measure ever proof useful, we don't have any ways to reasonably estimate it (other than learning from the data, but I don't think we will ever have enough data to analyze the very rare and specific scenarios).
PS: Let's say both scenarios are equaly likely (which is usually just false). Then from the perspective of overall expected MWP those two plays are the same. BUT. Maybe your matchup vs Storm is great and against Burn you basically just lose. And you only have time to learn one play before the tournament. Also, you want the best chance of a money finish. In this case, notice that if you learn the Storm play, your MWP has more variance than it would have if you learned the Burn play. If your expected finish isn't too high before choosing which play to learn, you might want higher variance! (Your expected finish is the same with learning either play, but your range of likely finishes is wider with more variance. So provided that you don't care about a finish unless it is a money finish and money is being given only to the very best finishes, more variance  the Storm play  is better).
Alright, now I'm done