A Measure of the Importance of Winning the Coin Flip

[Elric]

Here's a way to think about the importance of winning the coin flip.  To make the notation easier I'm going to assume that being on the play is always at least as good as being on the draw (and that players who can choose to play first do so for this reason).  The Importance of winning the coin flip is (your chance to win the match given that you won the coin flip) - (your chance to win the match, given that you lost the coin flip).

Assume the following are all constants.
W1= chance to win an unsideboarded game when going first ("on the play").
W2= chance to win an unsideboarded game when going second ("on the draw").
S1= chance to win a sideboarded game when going first.
S2= chance to win a sideboarded game when going second.

If you win the coin flip, your chance to win the matchup is (math omitted):

W1* (1-(1-S1)(1-S2)) + (1-W1)*S1*S2

If you lose the coin flip, your chance to win the matchup is:

W2 * (1- (1-S1)(1-S2)) + (1-W2)*S1*S2

So the importance of winning the coin flip is the difference between these two terms:

(W1-W2)*[ 1-(1-S1)(1-S2)] + (W2-W1)*S1*S2, and this simplifies to

Importance= (W1-W2)*(S1 + S2 - 2*S1*S2).

Plugging in a few numbers, let's say that W1= 0.55, W2= 0.45, S1= 0.6, S2= 0.45.

Then you can calculate that the Importance of winning the coin flip is 5.1%.

Anyone want to take a guess at what these numbers for typical (and maybe also a coin flip dependent?) matches in Vintage?  Note that this Importance of the coin flip measure has no impact whatsoever on how you should build decks since your chance to win the coin flip is always 50/50.  It just tells you how much that initial flip matters.

I remember that the Tinker/FCG/Belcher Extended Pro Tour had the winners of the coin flip winning about 60% of the matches, which means that the losers of the coin flip won 40% of the matches.  So the Importance of winning the coin flip was around 20%.  That seems high to me- I'd guess that most estimates would put Vintage as significantly lower.  Thoughts?

Edit: typos

[Machinus]

I think data on this has only been collected a few times. It would be really interesting to analyze this information if it was possible to get players to record this data like on their matchslips or something, but it hasn't been successful. However, I think most players would agree that the die roll isn't as crucial in Vintage as it might be in less broken formats (Legacy, Extended).

[Vegeta2711]

However, I think most players would agree that the die roll isn't as crucial in Vintage as it might be in less broken formats (Legacy, Extended).

Surely this is a typo. You can't honestly believe a more broken format would make the die roll LESS crucial.

[Anusien]

I think data on this has only been collected a few times. It would be really interesting to analyze this information if it was possible to get players to record this data like on their matchslips or something, but it hasn't been successful. However, I think most players wouldn't agree that the die roll isn't as crucial in Vintage as it might be in less broken formats (Legacy, Extended).

Fixed.
Formats where you're guaranteed to see turn 1 place less importance on the die roll.

[Gabethebabe]

A lot depends on the type of decks that are paired against eachother.
If they are two control decks, the outcome of the dieroll will affect the result less than in a match between Belcher and Stax.

[Machinus]

Well, I should have said that differently.

I think that Vintage-legal cards are so powerful that they can mitigate the effect of going second. Remember the coin-flip stigma about our format? Well that's still the exception to the rule. Obviously, everybody plays first in Vintage, but until I see a lot of data on this, I am not going to be convinced that going first directly leads to winning the game more often. (You don't always play first just because that means you are going to win faster. Disruption is a key part of the format, and in fact going first to cast disruption slows down the game. Going first gives different advantages to different strategies. While it is second nature to us as magic players and as Vintage players, going first is not an obvious choice. Trying to say why going first is right is equivalent to trying to explain tempo. We have an understanding of these ideas, and an intuition about them, but they are not easy or obvious.)

What I am trying to point out here is not that going first isn't important - I play first just like everyone else - but rather that it's less consistently a detriment in Vintage. In formats with speed in between Vintage and Standard, you can't play "catch-up" like you can in Vintage, but the decks are much faster than they are in Standard. What I am suggesting is that going first in those formats can have a stronger correlation to winning the game than in Vintage. In some way, this is related to the quantization of mana acceleration and the power limit of the cards in those formats. Going first provides a boost that is uncomparable to the cards in the format, whereas cards in Vintage often overpower the rules of the game and basically dismiss them.

However, without data to discuss, I'm not sure this is a really constructive debate. Let me restate that I think it's the right choice to play first in Vintage, in case anyone missed that.

[Vegeta2711]

Again, in any format where you have the very real possibility of losing the game on turn 1 (Either by combo kill or massive card advantage via stax) I can't see how the die roll isn't more important than other formats. Your side of being broken doesn't really matter if you're crippled or dead.

[Moxlotus]

Well, I should have said that differently.

I think that Vintage-legal cards are so powerful that they can mitigate the effect of going second. Remember the coin-flip stigma about our format? Well that's still the exception to the rule. Obviously, everybody plays first in Vintage, but until I see a lot of data on this, I am not going to be convinced that going first directly leads to winning the game more often. (You don't always play first just because that means you are going to win faster. Disruption is a key part of the format, and in fact going first to cast disruption slows down the game. Going first gives different advantages to different strategies. While it is second nature to us as magic players and as Vintage players, going first is not an obvious choice. Trying to say why going first is right is equivalent to trying to explain tempo. We have an understanding of these ideas, and an intuition about them, but they are not easy or obvious.)

What I am trying to point out here is not that going first isn't important - I play first just like everyone else - but rather that it's less consistently a detriment in Vintage. In formats with speed in between Vintage and Standard, you can't play "catch-up" like you can in Vintage, but the decks are much faster than they are in Standard. What I am suggesting is that going first in those formats can have a stronger correlation to winning the game than in Vintage. In some way, this is related to the quantization of mana acceleration and the power limit of the cards in those formats. Going first provides a boost that is uncomparable to the cards in the format, whereas cards in Vintage often overpower the rules of the game and basically dismiss them.

However, without data to discuss, I'm not sure this is a really constructive debate. Let me restate that I think it's the right choice to play first in Vintage, in case anyone missed that.

True that the level of brokeness can negate going second, but Vintage is the only format where turn 1 kills turn into complete and utter shit going second.  I have started to note in testing how often games are entirely dependent on the coinflip, and the number is staggering.  And this is just playtesting Oath vs. Stax and Gifts.  So many games are won by Oath going first and dropping Chalice to fuck them, or Stax going first and laying a Sphere which slows me down from my turn 1 Oath and then I get wasted out. 

I'm assuming once combo is even more coinflip dependent.

[Elric]

True that the level of brokeness can negate going second, but Vintage is the only format where turn 1 kills turn into complete and utter shit going second.  I have started to note in testing how often games are entirely dependent on the coinflip, and the number is staggering.  And this is just playtesting Oath vs. Stax and Gifts.  So many games are won by Oath going first and dropping Chalice to fuck them, or Stax going first and laying a Sphere which slows me down from my turn 1 Oath and then I get wasted out. 

I'm assuming once combo is even more coinflip dependent.

Even if the percent of games where who goes first decides the outcome is high, that doesn't mean that the Importance of winning the flip is that high (because Importance is for a best 2 out of 3 match, not a single game).

For example, if W1= 60%, W2= 40%, S1= 60% and S2=40% (you have a 60% chance to win any game in which you go first and a 40% chance to win any game in which you go second) the Importance is 10.4% (you have a 55.2% chance to win the match if you win the flip and a 44.8% chance to win the match if you lose the flip).

I agree that in general the stronger the decks in a format the more important the coin flip should be (Force of Will helps negate this somewhat).

[Anusien]

No winning the coinflip is that important.  Let's say you're playing combo versus control.  If you lose the flip, you'll probably lose the game, but if you win the flip you'll win the game (like those nuts Belcher versus Stax hands).  If you lose game 1, win game 2 on the play, then you still have to win game 3 on the draw, where if you win the coin flip, you're favored going into game 3 because you'd be on the play after losing game 2 on the draw.

[dandan]

Despite players using a wide variety of decka and having vastly different play styles, just about everyone plays first. This alone tells you that there is a significant advantage to going first. This advantage has a stronger attraction than keeping to a 60 card deck or even not putting in 'fun' cards so again we can assume it is a more significant advantage than running a 60 card deck over a 61/62 card deck or say running a Top/Juggernaut/etc over more standard choices.

I'd expect to see a 60/40 split if someone did some number crunching but wouldn't be at all surprised if it were higher. Note that some decks are capable of drawing (Dragon) or drawing out game 2 (control) if they won game 1, making the importance of game 1 higher and therebye making the coin toss more important than you might otherwise expect.

[cosineme]

Well, I should have said that differently.

I think that Vintage-legal cards are so powerful that they can mitigate the effect of going second. Remember the coin-flip stigma about our format? Well that's still the exception to the rule. Obviously, everybody plays first in Vintage, but until I see a lot of data on this, I am not going to be convinced that going first directly leads to winning the game more often. (You don't always play first just because that means you are going to win faster. Disruption is a key part of the format, and in fact going first to cast disruption slows down the game. Going first gives different advantages to different strategies. While it is second nature to us as magic players and as Vintage players, going first is not an obvious choice. Trying to say why going first is right is equivalent to trying to explain tempo. We have an understanding of these ideas, and an intuition about them, but they are not easy or obvious.)

Here you defeat your own argument. Sure, the power level of the cards are so high that they can negate the effect of going second. However, this is completely neutralized by the fact that your opponent is ALSO playing type 1, and his card's power levels are probably ridiculous as well. When your opponent is playing the same cards as you, equivalent in power level, and playing first, then...then going second with a strong hand doesn't mitigate the problem, it just puts you in the same position but a turn slower...

[Elric]

No winning the coinflip is that important.  Let's say you're playing combo versus control.  If you lose the flip, you'll probably lose the game, but if you win the flip you'll win the game (like those nuts Belcher versus Stax hands).  If you lose game 1, win game 2 on the play, then you still have to win game 3 on the draw, where if you win the coin flip, you're favored going into game 3 because you'd be on the play after losing game 2 on the draw.

Note that as (S1-->100% and S2-->0%) you have a situation when the outcome of the first game almost solely determines the outcome of the match. The Importance of the coin flip in this scenario is going to be higher than it will be for a situation in which S1 and S2 are closer together (i.e., when going first matters less in post-sideboarded games).

Again, though, for the Importance of the coin flip to be something close to 20% going first has to make a very large difference in each individual game.  For example, when W1=S1= 2/3 and W2=S2= 1/3 (you have a 2/3 chance to win any game in which you go first and a 1/3 chance to win any game in which you go second) then the Importance of the coin flip is 18.5% (you win 59.25% of the time if you win the coin flip and you win 40.75% of the time if you lose the coin flip). 

[xrobx]

First, I'm going to say that I didn't really follow your logic to clearly in your deduction of the argument of W1, W2, S1, S2.  That said, you came up with what seems to be a rational inference as to getting a 5% "better chance" or whatever of winning.  However, my argument is that this is arbitrary information.

Certain decks work towards certain goals, and depending on how the deck is built, and the way it plays its spells, turn one (as in, winning the coin flip) can prove most crucial.  It often proves the difference between a match win or loss in many given matchups.

Since certain decks function fundamentally different from other decks, this affects their ability to "go broken" or perform on the first turn.  Since all decks are not the same, and function fundamentally different on the first turn, it seems quite irrational to apply a mathmatical approximation equally to these unequal decks.

Hence, the coinflip is relevant dependant on matchups; which is a random factor.  Since this random factor is random, and not controlled, the coinflip percentile win rate seems quite arbitrary.

[PucktheCat]

xrobx:  I think the main significance of the format wide coin-flip factor would be to speak to the health of the format as a whole.  If to many games were won on the flip we might think that the game was a bit to random for our tastes.  Theoretically, we might even want some change in the rules to help even it out (like the play or draw rule was originally).  Obviously, since none of us are likely to choose to let our opponent play first when we win the flip this information has little direct practical value, but it is still something that people care about for a number of reasons.

Leo

[Elric]

xrobx:  I think the main significance of the format wide coin-flip factor would be to speak to the health of the format as a whole.  If to many games were won on the flip we might think that the game was a bit to random for our tastes.  Theoretically, we might even want some change in the rules to help even it out (like the play or draw rule was originally).  Obviously, since none of us are likely to choose to let our opponent play first when we win the flip this information has little direct practical value, but it is still something that people care about for a number of reasons.

Leo

Well said.  Here's what I said in my initial post:

Note that this Importance of the coin flip measure has no impact whatsoever on how you should build decks since your chance to win the coin flip is always 50/50.

The reason why you shouldn't care about the coin flip when building decks isn't that you don't know the matchups you will face: the reason why you shouldn't care about the coin flip is that you have a 50% chance to win the flip and a 50% chance to lose the flip and you can't do anything to change that.  So you should just maximize your expected chance to win a matchup, which is the average of (the chance that you win the matchup given that you win the coin flip) and (the chance that you win the matchup givne that you lost the coin flip)

A deck that wins 90% of its matches when it wins the coin flip and 30% of its matches when it loses the coin flip is equivalent to a deck that wins 80% of its matches when it wins the coin flip and 40% when it loses the coin flip.  It's also equivalent to a deck that wins 70% of its matches when it wins the coin flip and 50% of its matches when it loses the coin flip, etc.

The fact that Importance is greater than 0 (since it's always an advantage to win the coin flip), by the way, is the reason why you should alternate going first when you are testing decks- whenever there is an advantage to winning the coin flip doing this reduces the variance of your testing results (although you can't say how much it reduces the variance without knowing W1, W2, S1, and S2 in advance). 

Edit: reducing the variance of your testing results means that the results you get tell you more about the matchup- statistically, it is like you played more games. That's another reason why it would be nice to have estimates of W1, W2, S1 and S2 in a typical matchup- it would tell you approximately how much you can lower the variance with this method.

[pyr0ma5ta]

Too much math makes my head hurt 

Um, as a primarily Limited player (1801 rating), I guess I have an interesting view here.  In Limited, you're not only guarenteed to see your turn 1, but in fact, you're almost guarenteed to get to see turn 4 even if you just go land, go every turn.  In fact, I often choose to go SECOND just to make sure I hit my drops if my deck is a little slower and has bigger creatures with bigger costs.

Of course, Limited is the format where the power level of the decks is the absolute lowest.  In standard constructed, it almost doesn't matter how fast or slow your deck is, the extra card is not worth playing second except in very strange decks/matchups.  In extended, I can't even think of a deck that wants to go second.  I don't play legacy, but I imagine nobody chose to go second in any of the games at the recent Legacy GPs.  I hope you don't need this humble noob to tell you that nobody chooses draw in Vintage.

This trend should tell you something.  The higher the power level of the cards is, the more important tempo is over the extra card.  I'd imagine the go-first=win correlation is quite high in Vintage, although highly matchup dependant.  The only and obvious reason is that every deck has something stupidly powerful it wants to do in its turn 1 or turn 2, and if your opponent gets to go first, he might have something to answer, and your stupidly strong play turns into poopoo.

[Prometheon]

That's why I think a cool rule would be for the player who goes first to draw 7 cards and the player who goes second to draw 8, and then neither draws a card in the first turn. This is a plus and a minus. It makes mulligan decisions much easier on the second player (as he or she can mull to 7), gives them a better chance to have Force, but also makes effects like Duress potentially more potent.

[cosineme]

That's why I think a cool rule would be for the player who goes first to draw 7 cards and the player who goes second to draw 8, and then neither draws a card in the first turn. This is a plus and a minus. It makes mulligan decisions much easier on the second player (as he or she can mull to 7), gives them a better chance to have Force, but also makes effects like Duress potentially more potent.

nice idea, but i don't think any advantage needs to be given to the first player. maybe something like your idea, but the 8th card is viewable but not in your hand, or in your hand, but RFG so they can't duress or something.

[Godder]

I remember a poll was done at PT: Tinker (Extended 2003 PT: New Orleans), and 51% of players drawing (as opposed to playing) first won. That said, with Drain combo-control and Artifact prison decks running around, going first strikes one as very useful - beating Drain (whether actually winning or just getting into a winning position) before they reach Drain mana is crucial, while going before the artifact prison deck means getting stuff on table before Chalice becomes a factor.

[pyr0ma5ta]

I remember a poll was done at PT: Tinker (Extended 2003 PT: New Orleans), and 51% of players drawing (as opposed to playing) first won.

eh?  51% of the time, the player who drew first won?  That is, it was better to draw first than to play first?  I was actively not playing MTG in 2003 having quit, but I can't think of a format other than Sealed Deck where drawing first can conceivably be correct.  Especially if you're playing with Tinkers and other broken stupidity in your deck.

[Godder]

Eh?  51% of the time, the player who drew first won? That is, it was better to draw first than to play first?  I was actively not playing MTG in 2003 having quit, but I can't think of a format other than Sealed Deck where drawing first can conceivably be correct. Especially if you're playing with Tinkers and other broken stupidity in your deck.

It can depend on the exact match - in Sligh mirrors, it was usually correct to draw first rather than play first, for example. That aside, reading what I wrote earlier, I see that it could be a little misleading, so I will restate for clarity.

Players and coverage authors were asked to keep track of who played first (seeing as pretty much everyone chose to play first, as they do in Constructed) and the results. The player going first won 49% and the player going second won 51%, so the DCI was happy, because it meant that the coin flip wasn't as important as people had complained about beforehand (even at PT: Tinker).

[Elric]

Godder- do you have a link to that PT New Orleans poll result? I'd be interested in seeing it, since those types of polls are so rare.  I wouldn't expect the Importance of winning the coin flip at PT: New Orleans to be approximately 0- that seems like the type of tournament that would have a high Importance to winning the flip. 

Also, was this a poll about who won the coin flip for each match and who won that match, or a poll about who went first in each game and who won the game?

[bebe]

There are factors here that are not being assesed. If I'm going second against a combo deck i often feel compelled to mulligan more aggressively. In fact going second i will mulligan more aggressively against most decks. Going first I fear FoW if I'm looking to play a key spell but I do not have to worry about being Duressed, having a Mana Leak played against me or facing a Chalice for 0 or 1. Having broken cards in my hand is not enough to equalize the table. This forces me to mulligan unless I feel have a few options on my turn to negate my opponents possible openings. I've kept one land Brainsorm hands only to have my Brainstorm duressed away.  I've kept hands with one land and moxen only to see a Chalice dropped after my opponent drops his moxen and land.
I know that I lose more games going second and any statstiics that show otherwise are not looking at all the variables. Numbers lie all the time.

[Elric]

There are factors here that are not being assesed...

I know that I lose more games going second and any statstiics that show otherwise are not looking at all the variables. Numbers lie all the time.

I'm not sure what you mean by "factors that aren't being assessed."  If you are talking about the formula for the Importance of winning a coin flip in a match, then W1, W2, S1 and S2 capture everything that you are talking about.  If (W1>W2 and S1>S2) then in each game you would rather go first than second (no big surprise there).  One of the few reasons why the above formula doesn't always work is that matches can go to time.  Otherwise, it should be true in general.

Let's suppose that there is really a statistic that says "at Pro Tour New Orleans, we had players write who won the initial coin flip on the match result slips: the player that won the coin flip won 49% of the matches."  This is an unbiased and consistent measure of the Importance of winning the coin flip as long as players have been making the correct choice about whether to play or draw in a match (which isn't very hard if the correct answer is always play).  Put in normal English, this means that given a "large" sample size this will tell you the correct answer- with a smaller sample size you're going to have some margin of error to your results (depending on how confident you want to be in your conclusions).

If you had data on every single match from all of PT: New Orleans (a large sample size), then this would be strong evidence that winning the coin flip didn't matter much at the tournament.  It wouldn't be perfect, because even if the deck that went first was expected to win 60% of the time decks that went first at the tournament could still win 49% of the matches at the tournament due to chance alone, this would just be very unlikely.

[Scott_Limoges]

Force of Will is a good example to show why the die role is important in T1.  Force of Will is so good and considered a staple in T1 because it reduces the impact of the die role. Otherwise, trading two cards for one card would not be so good.

[jcb193]

This wouldn't be that hard to track if the format was truly interested.  With all of the SCG and Waterbury tournaments, it would be very easy to keep these stats in an official manner.  I don't know what they would accomplish, unless the Vintage format needs an adjustment to the play/draw rule.

Now, what would you do if the player going second got to start with 9 cards?

[pyr0ma5ta]

Wow.  9 cards is huge.  I'd have to think reallllll hard about play/draw....

[Nastaboi]

Remember, if rules are changed so that it becomes more beneficial to draw than to play, it does not reduce the importance of coin flip, because winner of the flip gets to choose whether to play or draw.

[jcb193]

Remember, if rules are changed so that it becomes more beneficial to draw than to play, it does not reduce the importance of coin flip, because winner of the flip gets to choose whether to play or draw.

Yeah, but if the players actually has to think about it, that is much better than "default go first."  Not to mention, certain matchups might warrant different options.  Control might want 9 cards, combo might want to go first.  Personally i don't know if this is a good thing (becoming closer to Vanguard), but it's an option.