[Premium Article] So Many Insane Plays: Improving Ichorid

[Elric]

Steve has a point sort of.  Basically if you always win game 1 then you only need to win 1 of the next two games.  For instance if you know you can win game 1 then you would much rather have a 100% chance of winning game 2 and a 0% chance of winning game 3 than having games 2 and 3 each be 50/50.

That's the exact point I was making in my first post.  That I'm in full agreement on.  What Steve indicated in his reply to you, though, seemed like something different- that being on the draw more often in actual play (because you win so many game 1s) makes Gemstone Caverns better in and of itself. 

See this quote:

The point is that if you have a super high game 1 win percentage, then the particular drawback that Caverns is only good on the draw isn’t a drawback because it will come into play (i.e. be useful) every match. 

My second post addresses why I believe that this logic is incorrect.  What I can't quite tell from the discussion here is what points Steve made in the article itself- I am pretty sure he made this second point and not sure if he made the first point.

[Smmenen]

There is a key piece of tech you guys are missing out on...

Phantasmagorian, 5BB
Creature - Horror (Uncommon)

When you play Phantasmagorian, any player may discard three cards. If a player does, counter Phantasmagorian.
Discard three cards: Return Phantasmagorian from your graveyard to your hand.
6/6

It fills the same role as Gigapede of letting you get Dredgers into your yard even if Bazaar gets nixed, also feeds Sutured Ghoul, plus it's black so it can feed Unmask or Ichorid too.

Which slots do you suggest cutting to make room for this card Stephen?

Part of the reason I didn’t mention this card is that I talked about it in my set review which was published yesterday.  Another reasons is that that card really only has a spot in Manaless Ichorid.   This thread is about mana ichorid.

As for Black Lotus, Mox Jet, Lotus Petal in Manaless Ichorid, it doesnt make sense to me...

Yes, Black Lotus can be powerful when it's in your opening hand.

But, it will be in your opening hand 11% of games. But what about the other 89% of games where instead of Dredging a black creature to feed Ichorid to into your graveyard, you'll be Dredging a Black Lotus with is basically a worthless slot.

But it’s not a worthless slot.   Remember, we aren’t simply using Black Lotus because of its game one value.   Let’s say that the benefits of Lotus include being able to hard cast Cabal Therapies and Shadows in game one, but that the costs are the loss of a dredger in that spot.   That’s not a full accounting of the benefits.   The benefit that I articulated as being the “clincher” was the fact that you open up a sideboard slot and will definitely be using Black Lotus in games 2 and 3 anyway.   In other words your “pre-sideboarding” Black lotus in Manaless Ichorid with ancillary benefits of being able to hardcast Therapies and Shadows.

Nevertheless, I strongly recommend testing Lotus in your opening hand as you stated. And when you do, please keep a record of the percentage of the games when Lotus in your opening hand proves to be very beneficial. If that percentage is under 50%, I would lean towards not playing Lotus, because that 5% of games where you have both Lotus and the ability to do something powerful without doesn't make up for the 60-95% of games where you will dredge Lotus into the yard where it does nothing rather than being a good black creature and feeding Sutured Ghoul or Ichorid or something.

It would be handy to list what broken plays Black Lotus enables...

Lotus, Cabal Therapy, Hardcast Golgori Thug, Sac Thug to Therapy, Bazaar, Dredge on the first turn!

Yes that is insanely broken, but it requires a lot of other cards to do (4, one of which is restricted), which means it will only happen a very small percentage of games. I would venture less than 1% of games if that.

Other than the above very unlikely play, what scenarios can you think of where Black Lotus is worth running?

As you can see from my explanation for why I’d run Lotus in Manaless Ichorid, these points aren’t directly relevant to it.   You are looking at Lotus’ game 1 utility and not its overall utilty in freeing up a sb slot and the fact that you will be using it every game 2 and 3. 

As for Mox Jet, that makes zero sense in this deck. Manaless Ichorid ONLY needs to stop Leyline/Planar Void. It can play through pretty much anything else.

I suggest you go read my previous, long reply on such assumptions.   

You can't stop those cards with Duress, you NEED to use Emerald Charm/Chain of Vapor. Black mana is useless for stopping the cards that this deck needs to stop. And thus, so is Mox Jet.

Duress is really subpar in this deck, as Meadbert stated well....

Duress is here to address one match: Combo.   Meadberts testing is completely opposite mine.   Pitch Long and the like annihilate manaless ichorid.   I have already harped on and on why: Ichorid must present two interactive spells of the non-Leyline variety by turn two or it will lose every combo game, and it will still lose a non-trivial number of games in which it does interact twice by turn two.   Any testing that suggests otherwise reflects, in my opinion, straw man combo opponents. 

I have tested Duress versus Leyline of the Void and I continue to find Leyline of the Void to be better. Basically, my goal is to keep Long from winning on turn 2. On turn 3 I can either win or use Therapies to disrupt Long till turn 4. The big question is do I survive Long's powerful turn 2. Leyline always comes down in time for turn 2. Duress does not. If I am on the draw then I either have to forgo playing Bazaar so I can play a manaland to get Duress mana up or I have to wait till turn 2 to Duress Long. A turn 2 Duress is too late. Also, once mana is added to the deck Duress competes with Cabal Therapy to a certain extent.

I wrote an article on skill some time ago in which I explained that decks with a low Average Skill Level are very likely to be straw manned.    This is because difficult decks to play have a greater distance from the average player to the top player and thus you aren’t accurately getting results that would reflect your match against a good player of that archetype.   

My testing of Pitch Long and combo matches is probably more accurate than almost anyone else could procure since I and my teammates (Pat Chapin, Paul Mastriano, Mike Herbig, etc) are among the top combo players around.   I assure that you that Manaless Ichorid has a very low win percentage versus Pitch Long.    I back this up by observing that most games Manaless Ichorid appears that it will win, but it ends up losing to careful play.   

As I said:

I assure that manaless Ichorid loses by a substantial margin to Pitch Long in my testing both two-fisted and against teammates of the calibre I referred to earlier. Unfortunately, I do not anticipate being put in a position where I can demonstrate this fact as all my future matchup articles will probably be using my mana list, which was motivated in part to not concede the combo match.   

If it helps clarify why Pitch Long wins, in my testing there were essentially three categories of games:

1) Pitch Long Blow outs - these are obvious

2) Manaless Ichorid blow outs- these were very rare and essentially and almost exclusively involved both Unmask and Chalice on turn one and often another card like Leyline and or turn two Cabal Therapy

3) Games in which Ichorid appeared like it was going to win, but Pitch Long ended up winning.   At least 50% of the games and probably upwards of 60% fit this category.    What this signaled to me is that these games required excellent technical play for the Pitch Long player to win.   It seems to me very possible that if your opponents ASL was anywhere from 7 or lower, they would probably lose a majority of these games.   

In other words, the difference between Pitch Long piloted at a top level and Pitch Long piloted at an average level is probably 50% swing in win ratio because most of the games fit into category 3.   Thus, if you are wining something like 70% of your games, you are wining 50% of the games you should be losing to a better Pitch Long player.

I will tackle this matchup in a future article. 

As for your remaining comments, I disagree with many of your base assumptions – assumptions I’ve already discussed previously in this thread.   

Game 2 and 3 are still symetrical.  If you have a 100% chance of winning games 1 and 3 then you have a 100% chance of winning the match.

I do see your point about usually only needing to win 1 out of the final 2 games so it is okay to use conditional cards like Gemstone Caverns.

But that’s really not the point.   It’s not about its conditionality so much as how a particular conditionality is negated by a particular feature of the deck.

The point is that if you have a super high game 1 win percentage, then the particular drawback that Caverns is only good on the draw isn’t a drawback because it will come into play (i.e. be useful) every match. 

The fact that you don’t understand this point explains why you said this:

Your winning percentage in game 1 has no impact on how good Gemstone Caverns is. 

This is flat out wrong.

Your winning percentage in game one is what justifies the use of Gemstone Caverns.  As I carefully explained in my article this week (which I thought you read), Gemstone Caverns has four particular drawbacks.   The drawback that requires you to be on the draw is not a problem if you have a super high win percentage.

As I said in the article, if we assume that we would be on the draw 0% of the time in a particular game, then obviously Gemstone Caverns is awful.  Conversely, if we were on the draw 100% of the time in a particular game, then that particular drawback is not at issue.  The question then becomes: what is the threshold level of times you need to be on the draw to make Caverns good?   I argue that the roughly 80% game 1 win percentage is above that threshold.   Do you see the connection now? Because it is clear from your statements that you didn’t get this before.

This argument isn't correct in general.  The short version, by way of an example:

Suppose that you win 100% of your game 1s and in a deck with Caverns you win 50% of game 2s (your opponent always chooses to go first in game 2) and you win 60% of your game 3s (if it gets there). 

Now, if you build your deck without Caverns you have a 40% chance to win game 2s and a 70% chance to win game 3s. 

In the first example you play an average of 1.5 total games post-sideboard, and an average of 1 of the post-sideboard games will be on the draw.  So 2/3 of your post-sideboard games in actual matches will be on the draw.

In the second example you play an average of 1.6 games post-sideboard, and an average of 1 of the post-sideboard games will be on the draw.  So 5/8 (about 60%) of your post-sideboard games in actual matches will be on the draw.  I've chosen these examples so your chance to win a post-sideboard game where you have a 50% chance of being on the play (and 50% chance of being on the draw) is the same, regardless of which version of the deck you have.

Given that in both cases you play the clear majority of your sideboarded games on the draw, and Gemstone Caverns makes your deck do comparatively better in post-board games when you're on the draw, this means you definitely prefer the Caverns version to the other one, right? 

That is not and has never been my argument.   

The discussion you quote above is not related to the overall utility of Gemstone Caverns, but was directed at a particular comment made by Meadbert in which he said:

Your winning percentage in game 1 has no impact on how good Gemstone Caverns is. 

All of the discussion you quote was aimed at explaining why there is an impact on how good Gemstone Caverns is based upon your game one winning percentage.

Now, I may have overstepped by using the word “justify” here:

Your winning percentage in game one is what justifies the use of Gemstone Caverns.

As you can see from the context around it, I did not mean justify in the sense of “this argument justifies its inclusion in terms of maximizing your ability to win the match” so much as, this argument negates the particular drawback at issue and thus prima facie justifies its inclusion in regards to that particular drawback.   

Meadbert says:

I stand by my original comments regarding how Gemstone Cavern's power is unrelated to your first turn win percentage.

This is logically refutable.   

Suppose:

1) There was a rule that you can run Gemstone Caverns in your sideboard and it would not count towards your 15 card limit. 
And
2) you lose 100% of your game ones

Now, Gemstone Caverns is practically useless as a game 2 sideboard card despite the space.   

Clearly, Gemstone Caverns power is increased if you win a large percentage of game 1s.  If you win 100% of game 1s, then that fact negates one of the four drawbacks that makes Gemstone Caverns a bad card because you can use it on the draw at least one game every match.   

What you guys are doing is conflating two separate arguments I’ve made.   One is a narrow argument regarding the specific drawback on Caverns.

Even supposing that Caverns’ particular drawback (that it is only good on the draw) is not relevant here, that does not address the question of whether Caverns is optimal.   That’s what your argument, Elric, is getting at here:

Wrong!  You have to look at your overall chance to win the match- the metric that you're thinking about if you answered "right" is the percentage of games that you will win in actual match play- that's not your goal!  It's pretty easy to see that in this example your chance to win the match is 1- (your chance to lose both sideboarded games), since you always win the first game. 

Your chance to lose both sideboarded games in the Caverns deck is 0.5 * 0.4= 20%.  So your chance to win each match with the Caverns deck is 80%.
Your chance to lose both sideboarded games in the non-Caverns deck is 0.6 * 0.3 = 18%.  So your chance to win each match with the non-Caverns deck is 82%.  The non-Caverns deck wins a higher percentage of its matches.

However, if you go to a tournament, you can expect to come home having won 72% of your games with the Caverns deck, and only 70% of your games with the non-Caverns deck (because you stop after 2 games if you've won both of them), even though the above calculation shows that the non-Caverns deck is actually the better deck!
 

The key assumption that makes this argument hold is your assumption that not running Caverns raises your game 3 win percentage.   

I would contest this assumption.   There are only so many cards you can and should be willing to sideboard in post-board without digging out too many core components.   I completely disagree that you would gain 10% in game 3 simply on the fact that you don’t have caverns.   Most of your post board games are going to be decided on the strength of cards I’ve already included: Chain of Vapor and Grudge/needle (whichever you choose to run).  My view is that I’m increasing the chances of winning game 2 dramatically with virtually no impact on game three.  If you think otherwise, prove it.   

[Elric]

This is logically refutable.   

Suppose:

1) There was a rule that you can run Gemstone Caverns in your sideboard and it would not count towards your 15 card limit. 
And
2) you lose 100% of your game ones

Now, Gemstone Caverns is practically useless as a game 2 sideboard card despite the space.   

Clearly, Gemstone Caverns power is increased if you win a large percentage of game 1s.  If you win 100% of game 1s, then that fact negates one of the four drawbacks that makes Gemstone Caverns a bad card because you can use it on the draw at least one game every match.

This is the exact logical flaw that I am pointing out.  The idea of a "game 2 sideboard card" doesn't mean anything.  Whether a card is good "in game 2" or "in game 3" is irrelevant.  Instead you should be thinking in terms of "going first" and "going second" sideboard cards.  Under the assumption that given the chance to chose whether to play or draw, everyone wants to go first, you can think about a match (excluding time limits and draws) exactly as follows:

1) You flip a coin to determine who goes first in an unsideboarded game 1. 
2) After game 1, you play a sideboarded game going first if you lost game 1 and going second if you won game 1.
3) After game 2, you play a sideboarded game going first if you won game 1 and going second if you lost game 1.
4) If you have won two or more of these three games, you win the match.

That you stop in actual match play once you've won two games doesn't matter.  So when you think about whether a card is good, you should assume this structure to a match.  This means that you should assume that you're going to have one game post-board going first, and one game post-board going second, regardless of what happens in actual match play (where you often stop after two games).

Wrong!  You have to look at your overall chance to win the match- the metric that you're thinking about if you answered "right" is the percentage of games that you will win in actual match play- that's not your goal!  It's pretty easy to see that in this example your chance to win the match is 1- (your chance to lose both sideboarded games), since you always win the first game. 

Your chance to lose both sideboarded games in the Caverns deck is 0.5 * 0.4= 20%.  So your chance to win each match with the Caverns deck is 80%.
Your chance to lose both sideboarded games in the non-Caverns deck is 0.6 * 0.3 = 18%.  So your chance to win each match with the non-Caverns deck is 82%.  The non-Caverns deck wins a higher percentage of its matches.

However, if you go to a tournament, you can expect to come home having won 72% of your games with the Caverns deck, and only 70% of your games with the non-Caverns deck (because you stop after 2 games if you've won both of them), even though the above calculation shows that the non-Caverns deck is actually the better deck!
 

The key assumption that makes this argument hold is your assumption that not running Caverns raises your game 3 win percentage.   

I would contest this assumption.   There are only so many cards you can and should be willing to sideboard in post-board without digging out too many core components.   I completely disagree that you would gain 10% in game 3 simply on the fact that you don’t have caverns.   Most of your post board games are going to be decided on the strength of cards I’ve already included: Chain of Vapor and Grudge/needle (whichever you choose to run).  My view is that I’m increasing the chances of winning game 2 dramatically with virtually no impact on game three.  If you think otherwise, prove it.   

This wasn't an argument about the merits of running Caverns itself- I made those numbers up.  This was an argument about the logic that winning a high percentage of game 1s means that Caverns is better because you're going to be on the draw in a lot of game 2s.  Obviously if Caverns helps your deck in games going second and there's no card you could replace it with in your sideboard that helps your deck in any games, then you should run Caverns.  But this would be true regardless of the percentage of game 1s that your deck wins.  What this example showed is that your logic that a high game 1 win percentage leads to a lot of game 2s going second, and thus makes Caverns better, isn't correct.   

[InfinityCircuit]

One issue I've come up with Mana Ichorid is that it becomes much easier to fight through graveyard hate.  However, the lower black creature count and the low dredge count have come up as a serious issue against decks that pass both graveyard hate and creature hate (Swords/Darkblast/Lava Dart/Fire/Ice).  How much testing have you done against combinations of graveyard and non-graveyard hate?  I find Manaless Ichorid superior at handling non-graveyard hate.

[Smmenen]

Elric, I’ve read through your comments and your points and I’ve been able to isolate the reason why you continue to make the arguments you have been advancing. 

Let me explain, briefly, why your argument is wrong and then I will elaborate.   

The short answer is this:

The opportunity cost of Gemstone Caverns necessarily rises by some amount as Ichorid’s game one win percentage decreases.   Put another way: The utility of Gemstone Caverns necessarily decreases as Ichorid’s game one win percentage falls.

This is logically and necessarily the case.   This is why both you and meadbert are wrong.

In order to comprehensively explain this argument, let me open with a segment of the article that prompted this thread.  Pay particular attention to the “magic theory aside.”

 Take a look at this card:

Gemstone Caverns.

There are four drawbacks:

1) It has to be in your opening hand to make good use of it
2) You have to pitch a card to play it
3) It is legendary
4) It is only good on the draw

Now for Ichorid, the first doesn't matter. Generally Ichorid doesn't play cards that aren't in its opening hand or drawn on the first turn. That's why Leyline of the Void is an arguable maindeck inclusion into Ichorid variants.

The second is equally irrelevant. Ichorid is used to pitching cards. Most of its cards end up in the graveyard. It is also used to removing its cards to Unmask and the like. Ichorid is full of bad cards it loves to remove when called to do so.

The third is not problematic either. Even if Ichorid had two Caverns in its opening hand, there is really no reason it would want to pitch another card to put another Caverns into play, even if it could.

That narrows our focus to one question: What percentage of the time would you have to be on the draw to make this card worthwhile to run? Certainly, if you were on the draw 100% of the time, then there would be no drawback in running it in the maindeck assuming the two other drawbacks we mentioned are an issue.

But what if it were a sideboard card?

That narrows the question even further:

What percentage of game 1s would you have to win to make Gemstone Caverns good?

Ichorid wins about 80%+ of its game 1s.

If ever Gemstone Caverns would have a home, it is in this deck. Thanks to Patrick Chapin for the idea.

Gemstone Caverns is a helpful sideboard cards. The plan goes like this:

Turn 0
Your opponent: Put Leyline of the Void into play.

You put Gemstone Caverns into play.

Turn 1
You play Chain of Vapor on the Leyline.

On your turn, you play Bazaar of Baghdad and activate it. You tap your Caverns to play Duress / Cabal Therapy.

You proceed to win the game.

You can substitute Leyline here for any hate card and you can achieve a similar effect. There are a number of permutations of this basic line of play, but the advantages of Caverns over regular land are fairly substantial.

Magic Theory Aside

Whether you think about it or not, every card that is selected into a deck is chosen on the basis of cost / benefit decision-making.

In economic terms, each card has an opportunity cost. The opportunity cost is the cost of the next best card. Let's say you want to run Counterspell in your mono-Blue control deck. The opportunity cost of running that Counterspell is that you could be running Mana Drain that slot. Since Mana Drain is a better card and would do all of the work of Counterspell, Counterspell will be substituted for Mana Drain. Another way of putting it is that the opportunity cost of running Counterspell in lieu of Mana Drain is too high. Why run Counterspell when you could have Mana Drain?

Constructed Magic is defined by sixty available card slots. When designing decks, we typically begin by putting in the most powerful cards. Most Vintage decks begin with Black Lotus and Ancestral Recall because they contribute so powerfully to winning the game.

That is essentially the bare measure of utility that goes into deck construction.

The utility of a card is equal to the amount by which it contributes to game wins:

U(card)=contribution to game wins

The goal of deck construction is to maximize the summative value of U(card) for all sixty cards. That is, by definition, what is meant by “optimal decks.”

Sometimes this can't be clearly measured since synergies are such a powerful element of deck construction. For example, the value Animate Dead is virtually nil without a target for you to Animate. Thus, one without the other has a utility approaching zero. But together, they have very high utilities. So you can make some assumptions about synergies when calculating card utility, and we do.

The bold point I want to make is that deck construction is cost / benefit decision-making with the goal to maximize your game wins.

End Theory Aside

Now the good stuff:

Let’s scale utilities.   Let’s say that the maximum utility for any given slot is 10.   

Now, utility, as we said, is ability to contribute to the game win – it is the summative value of a card in all game states in which it contributes to the win weighted by probability of arising.   

The implication of this point is critical: The opportunity cost of Gemstone Caverns rises by some amount as your game one win percentage decreases. 

To explain my point, let’s add some numbers to this. 

Let’s assume that if Ichorid has a game 1 win percentage of 100%, then Caverns would definitely see play in every match.  Thus, its probability of arising would simply be a function of its chances of being in your opening hand that you keep.   Let’s assume that under that math, Caverns has a utility of 9 with an opportunity cost of 7, meaning that it is better by a measurement of 2, than any other card in that slot.

Now, assuming that we will win some non-zero amount of game twos, every single amount that we drop the game 1 win percentage, its utility decreases and its opportunity cost rises.    That’s because more and more matches it will never be used and therefore have a utility of zero in those matches.   

My argument about its inclusion, simply put, is this: I believe that Ichorid has a high enough game 1 win percentage that Caverns provides the highest utility in that slot.

The question I posed in the article and in this thread is: what is the percentage of game 1s you would have to win so that Caverns becomes a legitimate competitor for a slot in the Ichorid SB?   

I would argue that it has to be pretty damned high, but that our turn one win percentage justifies it.   

In other words, it’s a question of utility maximization.   

Let’s take a look at what you say:

This is logically refutable.   

Suppose:

1) There was a rule that you can run Gemstone Caverns in your sideboard and it would not count towards your 15 card limit. 
And
2) you lose 100% of your game ones

Now, Gemstone Caverns is practically useless as a game 2 sideboard card despite the space.   

Clearly, Gemstone Caverns power is increased if you win a large percentage of game 1s.  If you win 100% of game 1s, then that fact negates one of the four drawbacks that makes Gemstone Caverns a bad card because you can use it on the draw at least one game every match.

This is the exact logical flaw that I am pointing out.  The idea of a "game 2 sideboard card" doesn't mean anything.  Whether a card is good "in game 2" or "in game 3" is irrelevant.  Instead you should be thinking in terms of "going first" and "going second" sideboard cards.  Under the assumption that given the chance to chose whether to play or draw, everyone wants to go first, you can think about a match (excluding time limits and draws) exactly as follows:

1) You flip a coin to determine who goes first in an unsideboarded game 1. 
2) After game 1, you play a sideboarded game going first if you lost game 1 and going second if you won game 1.
3) After game 2, you play a sideboarded game going first if you won game 1 and going second if you lost game 1.
4) If you have won two or more of these three games, you win the match.

That you stop in actual match play once you've won two games doesn't matter.  So when you think about whether a card is good, you should assume this structure to a match. 

"Good" in this context has many possible meanings, but I interpret it to mean what it denotates.   

Thus, my quarrel with Meadbert.  He says "Your game 1 win percentage has no impact on how GOOD Caverns is."  This is clearly incorrect in practical sense of the word "good” for reasons I explained at the beginning of this post.   

What's clear to me is that talking about whether a card is "good" is just being too damned vague to really understand what we are talking about.   

I am guilty for using the term and for using the term "justify" (as I already explained) in my reply to Meadbert.  Because that's not what I meant.   

Nonetheless, taking Meadbert's statement:

The card Gemstone Caverns is clearly better if you are on the play 100% of the games.  Why?   Because a card is better if it is useful, as the discussion of utility explains.   

If you win 0% of game 1s, then there is a chance you'll never use Caverns, making it, by definition "less good," meaning less useful as in producing less utility.   Thus, Meadberts point is refuted.   And yours. 

In either case, my argument DEPENDS UPON THE context of a match, contrary to your assertion above.

You explicitly said that the fact that a match stops after two games if one player has won both is irrelevant.  It is not irrellevant because a card's inclusion is always the answer to the question about utility maximization.    Utility maximization necessarily involves a question about possible usage.    Thus, the higher the game one win percentage, necessarily weighs in our utilty maximization.

This means that you should assume that you're going to have one game post-board going first, and one game post-board going second, regardless of what happens in actual match play (where you often stop after two games).

Not at all.    Gemstone Caverns has a particular disability.  Its merit in terms of being an inclusion depends upon its utility adding power and its opportunity cost in the sideboard.   If we assume that we are going to be playing one game post sideboard first, then we are forced to conclude that that game will be game two, necessarily.   

The logic looks like this:

1) We will be playing first at least one post board game

2) there are at most two post board games

3) whoever loses game one will be playing first game two. 

Under those rules, there are two possibilities:

4) We win game 1 and are on the draw in game 2.   

5) Win lose game 1 and play first game 2.

6) Some matches will be two games some of which we will win and some of which an opponent will win. 

Premises 1, 4, and 6 produce an inconsistency that makes them illogical:

For premise 1 and 4 to be logically consistent, we would have to lose every game 2.   Since we will not lose every game 2, as premise 6 suggests, your suggestion that we make assume premise 1 is refuted.   

All we should be doing is this: Maximizing our chances of winning the match.   The card choices we make are designed to do that.

Wrong!  You have to look at your overall chance to win the match- the metric that you're thinking about if you answered "right" is the percentage of games that you will win in actual match play- that's not your goal!  It's pretty easy to see that in this example your chance to win the match is 1- (your chance to lose both sideboarded games), since you always win the first game. 

Your chance to lose both sideboarded games in the Caverns deck is 0.5 * 0.4= 20%.  So your chance to win each match with the Caverns deck is 80%.
Your chance to lose both sideboarded games in the non-Caverns deck is 0.6 * 0.3 = 18%.  So your chance to win each match with the non-Caverns deck is 82%.  The non-Caverns deck wins a higher percentage of its matches.

However, if you go to a tournament, you can expect to come home having won 72% of your games with the Caverns deck, and only 70% of your games with the non-Caverns deck (because you stop after 2 games if you've won both of them), even though the above calculation shows that the non-Caverns deck is actually the better deck!
 

The key assumption that makes this argument hold is your assumption that not running Caverns raises your game 3 win percentage.   

I would contest this assumption.   There are only so many cards you can and should be willing to sideboard in post-board without digging out too many core components.   I completely disagree that you would gain 10% in game 3 simply on the fact that you don’t have caverns.   Most of your post board games are going to be decided on the strength of cards I’ve already included: Chain of Vapor and Grudge/needle (whichever you choose to run).  My view is that I’m increasing the chances of winning game 2 dramatically with virtually no impact on game three.  If you think otherwise, prove it.   

This wasn't an argument about the merits of running Caverns itself- I made those numbers up.  This was an argument about the logic that winning a high percentage of game 1s means that Caverns is better because you're going to be on the draw in a lot of game 2s.  Obviously if Caverns helps your deck in games going second and there's no card you could replace it with in your sideboard that helps your deck in any games, then you should run Caverns.  But this would be true regardless of the percentage of game 1s that your deck wins.  What this example showed is that your logic that a high game 1 win percentage leads to a lot of game 2s going second, and thus makes Caverns better, isn't correct.   

As I explained, utility maximization does depend upon the percentage of game 1 wins.   

Rephrased: If you lose 100% of game 1s, then unless your game 2 win percentage is 100%, there is some chance that you won't be getting to use Caverns.    That means that Caverns will produce absolutely no utility in those matches.    If U = Summative value of all possible utilities that Gemstone Caverns will produce, Gemstone Caverns utility is maximized the more you win game 1 because it will necessarily see more play. 

[Elric]

This means that you should assume that you're going to have one game post-board going first, and one game post-board going second, regardless of what happens in actual match play (where you often stop after two games).

Not at all.     Gemstone Caverns has a particular disability.  Its merit in terms of being an inclusion depends upon its utility adding power and its opportunity cost in the sideboard.   If we assume that we are going to be playing one game post sideboard first, then we are forced to conclude that that game will be game two, necessarily.   

The logic looks like this:

1) We will be playing first at least one post board game

2) there are at most two post board games

3) whoever loses game one will be playing first game two. 

Under those rules, there are two possibilities:

4) We win game 1 and are on the draw in game 2.   

5) Win lose game 1 and play first game 2.

6) Some matches will be two games some of which we will win and some of which an opponent will win. 

Premises 1, 4, and 6 produce an inconsistency that makes them illogical:

For premise 1 and 4 to be logically consistent, we would have to lose every game 2.   Since we will not lose every game 2, as premise 6 suggests, your suggestion that we make assume premise 1 is refuted.   

All we should be doing is this: Maximizing our chances of winning the match.   The card choices we make are designed to do that.

I think the part I put in bold is the crux of the matter here.  So, in defense of why I said that you should assume my structure for a match, here's the math.  Note that I'm not using a concept like "utility" or "opportunity cost" since the only thing of interest- your probability of winning the match- can be calculated directly. 

Assume that everyone who has the chance to go first does so, and that there are no draws (or time issues).  Also assume the following are all constants.
W= chance to win an unsideboarded game.
S1= chance to win a sideboarded game when going first.
S2= chance to win a sideboarded game when going second.

The chance that you win the match equals:
W* (S1 + S2 - S1*S2) + (1-W)*S1*S2

How did I get this?  You win the match if the following separate events happen:
Win Game 1, Win Game 2 (game 3 not played)
Win Game 1, Lose Game 2, Win Game 3
Lose Game 1, Win Game 2, Win Game 3

If you win game 1, the chance that you win game 2 is S2.  So the chance that the first combination happens is: W*S2
If you lose game 2 the chance that you win game 3 is S1.  So the chance for the second combination is W*(1-S2)*S1= W*(S1 - S1*S2)
If you lose game 1, the chance that you win game 2 is S1.  If you win game 2, the chance that you win game 3 is S2.  So the chance for the third combination is (1-W)*S1*S2.

Add these together to get your chance to win the match equal to W*(S2 + S1 - S1*S2) + (1-W)*S1*S2. 

Now let's consider my example match structure (using the same notation):

1) You flip a coin to determine who goes first in an unsideboarded game 1. 
2) After game 1, you play a sideboarded game going first if you lost game 1 and going second if you won game 1.
3) After game 2, you play a sideboarded game going first if you won game 1 and going second if you lost game 1.
4) If you have won two or more of these three games, you win the match.

Your chance to win this match is the chance that you win two or more of these three games.  Breaking this into separate events:
Your chance to win all three games: W*S2*S1
Your chance to win exactly the first and third games: W*(1-S2)*S1= W(S1- S1*S2)
Your chance to win exactly the first and second games: W*S2*(1-S1)= W*(S2- S1*S2)
Your chance to win exactly the second and third games: (1-W)*S1*S2

Add up these events to get your chance to win the match=
W (S1*S2 + S1- S1*S2 + S2 - S1*S2) + (1-W)*S1*S2= W(S1 + S2- S1*S2) + (1-W)*S1*S2

From these calculations, you can see that the chance to win the match is exactly the same in both cases.  So when you go to think about your chance to win a match, the way that you should think about it must be in accord with the idea that you're going to play both one post-sideboard game going first and one post-sideboard game going second since, as I've just shown, that structure is exactly the same (from the perspective of each player's chance to win) as an actual tournament match.

[Smmenen]

"Both Cases"?  What do you mean by both cases? You mean, we should be assuming that we will be on the play and the draw post board because your win percentage remains the same whether we make the assumption that we will play at least one game post board or not? 

Well DUH!   If you win game 2, then there will be no game 3!   Thus, assuming a game on the play in which we've won game 2 on the draw won't change our win percentage one iota and will make your math correct.   

Yes, you've proven that the win percentage remains the same whether 1) we assume we are going to be drawing post board for sure or 2) playing and drawing post board.   But that doesn't prove anything because it is axiomatic.  If we win game one and two, assuming we are playing a third game on the play doesn't change our win percentage.  We still win the match even if we assume a game 3 loss.  Vice versa, if we lose game 1 and 2, assuming a third game on the draw doesn't change our win percentage.   We still lose the match even if we assume a game 3 win.

Nor does that help us address the question of whether Caverns is better if your game one win percentage goes up.

You cannot ignore the fact that if your game one win percentage goes down, the summative value of Caverns utility decreases.   It Must!  Because it will see less play.

Here are the possibilities:

Caverns only sees play in two situations:

1) You win game one (and thus is used in game 2 on the draw)
2) You lose game one, win game 2, and are on the draw in game 3 and thus use it.

As your game one win percentage falls, the chances that you won't use it or have a chance to use it rise.

Thus, as your game one win percentage goes down, it necessarily means that Caverns is worse.   The opposite is also true: as your game one win percentage goes up, Caverns utility goes up because it will see more play (the set of games in which you lose game 1 and lose game 2 are the only times you won't be playing with it).

[Elric]

"Both Cases"?  What do you mean by both cases? You mean, we should be assuming that we will be on the play and the draw post board because your win percentage remains the same whether we make the assumption that we will play at least one game post board or not? 

Well DUH!   If you win game 2, then there will be no game 3!   Thus, assuming a game on the play in which we've won game 2 on the draw won't change our win percentage one iota and will make your math correct.   

Yes, you've proven that the win percentage remains the same whether we assume we are going to be drawing post board for sure or playing and drawing post board.   

But that doesn't help us address the question of whether Caverns is better if your game one win percentage goes up.

You cannot ignore the fact that if your game one win percentage goes down, the summative value of Caverns utility decreases.   It Must!  Because it will see less play.

Here are the possibilities:

Caverns only sees play in two situations:

1) You win game one (and thus is used in game 2 on the draw)
2) You lose game one, win game 2, and are on the draw in game 3 and thus use it.

As your game one win percentage falls, the chances that you won't use it or have a chance to use it rise.

Thus, as your game one win percentage goes down, it necessarily means that Caverns is worse.   The opposite is also true: as your game one win percentage goes up, Caverns utility goes up because it will see more play (the set of games in which you lose game 1 and lose game 2 are the only times you won't be playing with it).

No- utility has nothing to do with it.  This "Because it will see less play" argument is not correct.  By both cases I mean 1) Assume that you always play out both a sideboarded game going first and a sideboarded game going second vs. 2)what you do in actual match play (where the loser of game 1 goes first in game 2, then you stop if someone has won both games, and if the games are split the loser of game 2 goes first in game 3).  In terms of the percentage of the time you win the match, these scenarios are exactly equivalent. 

So when you write:

Caverns only sees play in two situations:

1) You win game one (and thus is used in game 2 on the draw)
2) You lose game one, win game 2, and are on the draw in game 3 and thus use it.

I think of the following thought experiment: Suppose that tournament Magic was played according to the rule that the winner of game 1 got the choice of whether to go first in game 2, and the loser of game 1 had the choice of whether to go first in game 3, should games 1 and 2 be split.  The usual assumptions: everyone would always rather go first than go second and matches never go to time (you'd probably also want "no really surprising sideboards" here).

Then would your argument about Caverns being better if your game one win percentage goes up be reversed in this case?  Here Caverns only sees play in two situations:
1) You lose game 1 (and thus it is used in game 2 on the draw)
2) You win game 1, lose game 2, and are on the draw in game 3 and thus use it. 

Would you argue that in this case Caverns is worse if your game one win percentage goes up?  I contend that this situation is no different than your example above and my conclusion is that just like your line of reasoning above isn't an argument in Caverns' favor, this isn't an argument against Caverns.

[Jacob Orlove]

Your game 1 winning percentage *does* matter when evaluating Caverns:

You win more matches with skewed postboard odds than with even ones, if your G1 percentage is good. If you have 50% odds of winning G2 and G3, you will go win-lose-lose 25% of the times where you win G1. If your odds are instead 70%/30%, you will go win-lose-lose only 21% of the time where you win G1 (of course, your odds of going lose-win-win drop from 25% of lost G1s to 21% of lost G1s).

So, if you win 100% of game 1s, having your postboard games each skewed by 20% (40% apart) gives a flat 4% increase in match wins. This decreases as you win fewer G1s, to the point where if you win 50% of game 1s, you gain no advantage whatsoever, and if you win less than 50% of G1s, the skewed distribution actually hurts your chances.

So, Caverns improves our odds of winning the match even if it provides no net increase in winning %s, provided that:
1) our game 1 win % is very high, AND
2) Caverns increases, rather than reduces, the skew between games 2 and 3.

Of course, Caverns also actually boosts our chances of winning games, but we have to compare that against the boost we'd get from some other SB card that could replace it.

[Elric]

Your game 1 winning percentage *does* matter when evaluating Caverns:

You win more matches with skewed postboard odds than with even ones, if your G1 percentage is good.

In fact, this was my argument in my first post in this thread (on page 2).  But this is clearly not what Smmenen is arguing here.  I am not arguing that your game 1 win percentage is irrelevant in evaluating a card like Caverns- I am arguing that the following logic is incorrect:

Caverns only sees play in two situations:

1) You win game one (and thus is used in game 2 on the draw)
2) You lose game one, win game 2, and are on the draw in game 3 and thus use it.

As your game one win percentage falls, the chances that you won't use it or have a chance to use it rise.

Thus, as your game one win percentage goes down, it necessarily means that Caverns is worse.   The opposite is also true: as your game one win percentage goes up, Caverns utility goes up because it will see more play (the set of games in which you lose game 1 and lose game 2 are the only times you won't be playing with it).

[Jacob Orlove]

Well, the point is that if you're winning more game 1s, then you get the benefit of Caverns in game 2 without any penalties in game 3 (because you don't play game 3). If you're relying on Caverns to help you in matches where you need to win both SB games, it is worse, because regular SB cards are (at least in theory) good on the play and on the draw, whereas Caverns, while helping in the postboard game on the draw, hurts when you are on the play.

The opportunity cost of Gemstone Caverns necessarily rises by some amount as Ichorid’s game one win percentage decreases.   Put another way: The utility of Gemstone Caverns necessarily decreases as Ichorid’s game one win percentage falls.

[Elric]

Well, the point is that if you're winning more game 1s, then you get the benefit of Caverns in game 2 without any penalties in game 3 (because you don't play game 3)

I'm assuming you meant "you don't play game 3 if you win both game 1 and game 2."  This is the exact logic that I am arguing is incorrect.  The fact that you might stop after playing only games 1 and 2 (and wouldn't play game 3) is irrelevant when evaluating how good Caverns is.  Yes, in actual match play Ichorid will play more post-sideboard games going second than most decks will because of its high game 1 win percentage.  No, this argument (more post-sideboard games going second in actual match play) doesn't constitute an argument in favor of Caverns.

Consider my thought experiment from the post right before your first response.  Would you argue that in this thought experiment case a higher game 1 win percentage makes Caverns worse (or at least, comparatively worse than it was in actual tournament magic)?

Think of the following thought experiment: Suppose that tournament Magic was played according to the rule that the winner of game 1 got the choice of whether to go first in game 2, and the loser of game 1 had the choice of whether to go first in game 3, should games 1 and 2 be split.  The usual assumptions: everyone would always rather go first than go second and matches never go to time (you'd probably also want "no really surprising sideboards" here).

Then would your argument about Caverns being better if your game one win percentage goes up be reversed in this case?  Here Caverns only sees play in two situations:
1) You lose game 1 (and thus it is used in game 2 on the draw)
2) You win game 1, lose game 2, and are on the draw in game 3 and thus use it. 

[policehq]

What is the point of arguing for the effectiveness or ineffectiveness of Gemstone Caverns outside of match-play? I don't get why you're creating a different tournament scenario to present a new logical case against Caverns.

Steve, I like the mana list with -4 Leyline of the Void, -1 other for +5 mana sources the most in testing. The pre-emptive sideboarding strategy of building the maindeck is probably one of the most important but least talked about discussions of this thread.

-hq

[Elric]

What is the point of arguing for the effectiveness or ineffectiveness of Gemstone Caverns outside of match-play? I don't get why you're creating a different tournament scenario to present a new logical case against Caverns.

It's not a logical case against Caverns.  I'm not saying that Caverns should or shouldn't be played in Ichorid.  I'm trying to point out that the logic that Steve uses for one of his reasons why Caverns is good in Ichorid is not correct. 

The point that I am trying to make is that my different tournament structure is equivalent in all ways that matter to the tournament structure that we currently have.  If the argument that Steve is making for why increased game 1 win % makes Caverns better is correct, then this tournament structure would reverse that argument- Caverns would be worse the higher % of game 1s you win.  But if you don't believe that in the tournament structure I've invented, Caverns is worse the higher % of game 1s you win based on the reasoning that Steve has articulated, then you also shouldn't believe that Steve's argument is correct.

Further Edit- I should mention that the following match structures, I am arguing, are all equivalent to the normal tournament match structure (provided the usual no time-issues and everyone wanting to play first assumptions apply).  This means that if you consider the effect on Caverns of your game 1 win % in these various tournament match structures, it should be the same in each case.  But Steve's logic would imply that these cases are different from each other, which tells you that his logic isn't correct.

1) You flip a coin to determine who goes first in an unsideboarded game 1. 
2) After game 1, you play a sideboarded game going first if you lost game 1 and going second if you won game 1.
3) After game 2, you play a sideboarded game going first if you won game 1 and going second if you lost game 1.
4) If you have won two or more of these three games, you win the match.

Suppose that tournament Magic was played according to the rule that the winner of game 1 got the choice of whether to go first in game 2, and the loser of game 1 had the choice of whether to go first in game 3, should games 1 and 2 be split (if the same player wins both game 1 and game 2, you don't play game 3). 

1) You flip a coin to determine who goes first in an unsideboarded game 1. 
2) After game 1, you play a sideboarded game going second if you lost game 1 and going first if you won game 1.
3) After game 2, you play a sideboarded game going second if you won game 1 and going first if you lost game 1.
4) If you have won two or more of these three games, you win the match.

[Neonico]

Say just for the sake of the argument that Meadbert is correct in his assessment that Leyline is superior to Duress in all your problem matchups and that Duress is just not worth running, and you don't want to play more than 1 Ashen Ghoul. Is Black Lotus still worth it then. Is the difference so marginal then that it no longer matter?

Yes, Black Lotus can be powerful when it's in your opening hand.

But, it will be in your opening hand 11% of games. But what about the other 89% of games where instead of Dredging a black creature to feed Ichorid to into your graveyard, you'll be Dredging a Black Lotus with is basically a worthless slot.

What percentage of the games when Lotus in your opening hand prove to be very beneficial? If that percentage is under 50%, I would lean towards not playing Lotus, because that 5% of games where you have both Lotus and the ability to do something powerful without doesn't make up for the 60-95% of games where you will dredge Lotus into the yard where it does nothing rather than being a good black creature and feeding Sutured Ghoul or Ichorid or something.

It would be handy to list what broken plays Black Lotus enables...

Lotus, Cabal Therapy, Hardcast Golgori Thug, Sac Thug to Therapy, Bazaar, Dredge on the first turn!

Yes that is insanely broken, but it requires a lot of other cards to do (4, one of which is restricted), which means it will only happen a very small percentage of games. I would venture less than 1% of games if that.

Other than the above very unlikely play, what scenarios can you think of where Black Lotus is worth running?

If meadbert assertion that Duress isn't worth it is correct, I really can't think of any other than that one scenario where I would prefer Black Lotus over another dreadger or something.

As far as i tested the deck , and performed in great french tournaments with it, i must say that i found really funny that you mention this, as i get bashed saying that lotus is total out of place in the deck... This deck need permanent mana sources and nothing else. Once in a while, youll have a very strong T1, Such as lotus land study, discarding dredgers, Tral, imp brainstorm.... Once in a while... And in 99% of the games, you'll allways prefer a basic swamp or an underground sea.... Lotus petal and black lotus are bad in ichrid IMO.

Duress belong to the deck i think, or at least to the maindeck, more than leyline of the void. When playing against ichorid, no deck rely really on a graveyard strategy (hard to pull off a lethal will in less than 4 turns) and decks such as stax doesnt get any weapon to fight ichorid game one. The only deck that leyline could be usefull against is slaver, because its able to snake a fast slaver with T1 welder T2 Thirst T3 bring slaver back and activate. I think that Duress is better to beat this sequence than leyline.

[meadbert]

Leyline is in there for Combo decks that can race you such as Dragon and Long.

I am slowly coming around to Steve's claim that Leyline is not good enough.  Steve has been saying that Leyline is not good and I have been saying that Duress is not good and basically I think we are both right.  Either is good enough on the draw and neither is good enough on the play.

Duress is bad on the draw because Long and Dragon get two turns to Goldfish before you hit them with your Duress.  This is too long.

Leyline's advantage is that it comes out at Force of Will speed on turn 0.  Also it cannot be countered or Duressed which matters versus Pitch and Grim Long respectively.  The problem is that your opponent has 3 turns to win around Leyline of the Void assuming that your first Therapy is not till turn 3.  This is tough for Dragon, but Grim Long has little trouble winning without its graveyard on turn 3.  The Baubles might make some sense if they can allow turn 2 Therapies with Nether Shadow.

[Smmenen]

Now, I’m certainly open to the possibility here that I’m missing something or not getting something.  But I’ve read through all of the most recent posts and with the exception of part of Jacob’s first post, I believe I understand what everyone is saying.

I think some of the confusion could have been alleviated, however, if Elric had quoted my edited post last night – I edited it just after I finished writing it, but he quickly quoted and replied.   

I will quote the revised parts in just a moment.

That you stop in actual match play once you’ve won two games doesn’t matter.   So when you think about whether a card is good, you should assume this structure to a match.   This means that you should assume that you’re going to have one post-board game going first, and one game post-board going second, regardless of what happens in actual match play (where you often stop after two games.

I hope you aren’t ignoring the case in which your opponent wins the next two games.

The edited part of my post that didn’t get caught in your reply (and possibly was missed by you) was this:

Yes, you’ve proven that the win percentage remains the same whether 1) we assume we are going to be drawing post board for sure or 2) playing and drawing post board.    But that doesn’t prove anything because it is axiomatic.   If we win game one and two, assuming we are playing a third game on the play doesn’t change our win percentage.   We still win the match even if we assume a game 3 loss.  Vice versa, if we lose game 1 and 2, assuming a third game on the draw doesn’t change our win percentage.  We still lose the match even if we assume a game 3 win.  

Nor does that help us address the question of whether Caverns is better if your game one win percentage goes up.

In other words, I was explaining that, as I understand it, your math is correct but it is a truism that doesn’t get at the point I was making: which is that the utility of Caverns falls as your game 1 win percentage falls, necessarily, since it will see less use. 

It doesn’t matter that assuming a post-board game on the draw and the play doesn’t change your match win percentage – because it won’t.   

Think about this:

If we have a 100% game 1 win percentage and a 100% game 2 win percentage in actual practice, we would never play a game 3.   But your system would show that our win percentage is exactly the same whether we assume a game 3 or not.    Well, duh.   But that just shows the irrelevancy of the argument you’ve made.   

  I am not arguing that your game 1 win percentage is irrelevant in evaluating a card like Caverns

But Meadbert is:

Your winning percentage in game 1 has no impact on how good Gemstone Caverns is.

Well, the point is that if you're winning more game 1s, then you get the benefit of Caverns in game 2 without any penalties in game 3 (because you don't play game 3)

I'm assuming you meant "you don't play game 3 if you win both game 1 and game 2." 

I think what he meant by “penalties” was costs generally, including the cost that you have 15 sb slots and are devoting 4 to Caverns, which could be used to play a card that will see play regardless of whether you are on the play or the draw.   

This is the exact logic that I am arguing is incorrect.  The fact that you might stop after playing only games 1 and 2 (and wouldn't play game 3) is irrelevant when evaluating how good Caverns is.  Yes, in actual match play Ichorid will play more post-sideboard games going second than most decks will because of its high game 1 win percentage.  No, this argument (more post-sideboard games going second in actual match play) doesn't constitute an argument in favor of Caverns.

This is essentially a rearticulation of what I quoted you as saying at the beginning of this post, but with an important difference.   Here you aren’t making a distinction between you winning the first two games and your opponent winning the first two games. 

That makes a big difference.

If your opponent wins the first two games, you never get a chance to use Caverns.   That is the crux of my argument regarding its utility.   

As I said: Utility is the summative value of a card’s contribution to winning games in all possible game states weighted by the possibility of those games states arising.

As your G1 win percentage decreases, your chance of ever seeing a game 3 decreases and thus there is an increase chance that Caverns is dead weight – i.e. more game states in which Caverns produces no utility and thus decreases the summative value of its utility.   

Your only counter argument appears to be this:

There is no mathematical difference in the chance of winning the match between the two “cases” you articulated: 1) assuming that you always play out both sideboarded games one going first and a sideboarded game going second versus 2) what you do in actual play.

My counter argument to that is that your mathematical proof is axiomatic but irrelevant to the logic I’ve articulated.    There is no mathematical difference in terms of your match win percentage between the two cases, but that doesn’t address the individual utility of a particular card with a particular drawback whose individual utility depends upon a particular disability being removed a threshold amount of the time to make it good enough to run.    Your implied counterargument seems to be that we are assuming we will be using Caverns at least once every post board because it doesn’t affect our win percentage to do so.    The fact that it doesn’t affect your win percentage to make that assumption is true, but ridiculous to the practical reader because those game 3s aren’t actually played. 

As I said:

If we win game one and two, assuming we are playing a third game on the play doesn’t change our win percentage.   We still win the match even if we assume a game 3 loss.  Vice versa, if we lose game 1 and 2, assuming a third game on the draw doesn’t change our win percentage.  We still lose the match even if we assume a game 3 win.

Nor does that help us address the question of whether Caverns is better if your game one win percentage goes up.

Consider my thought experiment from the post right before your first response.  Would you argue that in this thought experiment case a higher game 1 win percentage makes Caverns worse (or at least, comparatively worse than it was in actual tournament magic)?

Think of the following thought experiment: Suppose that tournament Magic was played according to the rule that the winner of game 1 got the choice of whether to go first in game 2, and the loser of game 1 had the choice of whether to go first in game 3, should games 1 and 2 be split.  The usual assumptions: everyone would always rather go first than go second and matches never go to time (you'd probably also want "no really surprising sideboards" here).

Then would your argument about Caverns being better if your game one win percentage goes up be reversed in this case?  Here Caverns only sees play in two situations:
1) You lose game 1 (and thus it is used in game 2 on the draw)
2) You win game 1, lose game 2, and are on the draw in game 3 and thus use it. 

I’ll be honest: I do not fully grasp the purpose of this thought experiment.   I’m not sure which way my answer will cut, but here is my answer:

A higher game 1 win percentage would indeed make Caverns worse under your thought experiment because it would increase the opportunity cost of that slot.   There are only 75 card slots, each of which is very valuable.   Other cards become more valuable relative to Caverns under your thought experiment.   So, yes, my argument about Caverns being better if your game one win percentage goes up would be reversed in this case.

Note, however, that I would never play a deck that has such a poor game 1 win percentage.   In order to reach the threshold level where Caverns actually becomes competitive for the slot its fighting over, you’d have to lose such a high number of game 1s that your deck would suck. 

  In short, you’re trying to get around my argument by suggesting that your match win percentage doesn’t change if we assume we always play a game 3.   Well, that’s a ridiculous argument because, although it is true our match win percentage doesn’t change, Caverns won’t be contributing to the overall match win percentage in game 3s in which we are forced to assume its usage because one player has won both game 1 and 2. 

[Elric]

  I am not arguing that your game 1 win percentage is irrelevant in evaluating a card like Caverns

But Meadbert is:

Your winning percentage in game 1 has no impact on how good Gemstone Caverns is.

No- Meadbert did acknowledge after my very first post in this thread that the "skewness argument" (the one recently articulated by Jacob Orlove) can be an argument in favor of a card.  That's not the point you're making here, though.

Well, the point is that if you're winning more game 1s, then you get the benefit of Caverns in game 2 without any penalties in game 3 (because you don't play game 3)

I'm assuming you meant "you don't play game 3 if you win both game 1 and game 2." 

I think what he meant by “penalties” was costs generally, including the cost that you have 15 sb slots and are devoting 4 to Caverns, which could be used to play a card that will see play regardless of whether you are on the play or the draw.   

This is the exact logic that I am arguing is incorrect.  The fact that you might stop after playing only games 1 and 2 (and wouldn't play game 3) is irrelevant when evaluating how good Caverns is.  Yes, in actual match play Ichorid will play more post-sideboard games going second than most decks will because of its high game 1 win percentage.  No, this argument (more post-sideboard games going second in actual match play) doesn't constitute an argument in favor of Caverns.

This is essentially a rearticulation of what I quoted you as saying at the beginning of this post, but with an important difference.   Here you aren’t making a distinction between you winning the first two games and your opponent winning the first two games. 

That makes a big difference.

If your opponent wins the first two games, you never get a chance to use Caverns.   That is the crux of my argument regarding its utility.   

As I said: Utility is the summative value of a card’s contribution to winning games in all possible game states weighted by the possibility of those games states arising.

As your G1 win percentage decreases, your chance of ever seeing a game 3 decreases and thus there is an increase chance that Caverns is dead weight – i.e. more game states in which Caverns produces no utility and thus decreases the summative value of its utility.

By this definition of utility, Caverns' utility is decreasing as it is used less in actual match play.  But this is not a good definition from the perspective of what really matters- your chance to win a match.  Consider a deck with a 100% chance to win game 1, a 100% chance to win a post-sideboard game going first, and a 50% chance to win a post-sideboard game going second (in regular match play).  Then you might think that cards that increase your chance to win the second game have some high "utility" because yoiu always go second in game 2, but in fact it's clear that your game 2 win % has no effect on your match win %- it's 100% in any case.  On the other hand, if you were in my thought experiment case (go first in game 2 if you won game 1; go first in game 3 if you lost game 1) then you'd correctly identify that your win % in post-board games going second doesn't matter at all (since in this case you never get to game 3 since you always win the first two games). 

And I think you'd agree that this is the correct answer here- if you always win preboard games and always win post-board games going first, it doesn't matter what percentage of post-board games going second you win.  So this definition of utility isn't meaningful because as you can see, changing the structure of the match in a way that doesn't affect each player's chance to win the match can affect (substantially) the "utility" that you would place on a card.

[Smmenen]

By this definition of utility, Caverns' utility is decreasing as it is used less in actual match play.  But this is not a good definition from the perspective of what really matters- your chance to win a match.  Consider a deck with a 100% chance to win game 1, a 100% chance to win a post-sideboard game going first, and a 50% chance to win a post-sideboard game going second (in regular match play).  Then you might think that cards that increase your chance to win the second game have some high "utility" because yoiu always go second in game 2, but in fact it's clear that your game 2 win % has no effect on your match win %- it's 100% in any case. 

Not at all! My logic would suggest, on the contrary, that game 2 is as irrelevant in that model as game 3 is in your model in which one player has won the first two games!

  Then you might think that cards that increase your chance to win the second game have some high "utility" because yoiu always go second in game 2,

Quite the contrary, under my model Caverns has absolutely ZERO utility because it adds absolutely nothing to contributing to the match win.

Look:
In the example you just came up with:

G1: 100% win percentage
G2: (always on the draw) 50% game win percentage
G3: 100% win percentage

Put this way: the utility of a card is its ability to contribute to game wins with one critical caveat: Provided, that those game wins are relevant to winning the match

In the example you just came up with, Caverns provides absolutely no utility at all.  In fact, the only rationale thing to do is to immediately concede every game two to save time.    Therefore, Caverns should see absolutely no play whatsoever.   

On the other hand, if you were in my thought experiment case (go first in game 2 if you won game 1; go first in game 3 if you lost game 1) then you'd correctly identify that your win % in post-board games going second doesn't matter at all (since in this case you never get to game 3 since you always win the first two games). 

And I think you'd agree that this is the correct answer here- if you always win preboard games and always win post-board games going first, it doesn't matter what percentage of post-board games going second you win.  So this definition of utility isn't meaningful because as you can see, changing the structure of the match in a way that doesn't affect each player's chance to win the match can affect (substantially) the "utility" that you would place on a card.

On the contrary, this simply reveals that you have no idea what utilty means.

I’ll try to explain it once more.

There are 75 possible slots to play with – 60 in the maindeck and 15 in the sideboard.

To advance in magic, the only thing that matters is winning the match.   The way you win matches is by winning games.

Thus, the selection of cards into your 75 open slots is a quest to find those 75 cards that will maximize your chances to win games within the context of a match.   

In other words, it’s cost/benefit decision making.   An ‘optimal’ decklist is a decklist that maximizes your RELEVANT game win percentage.     Irrellevant game wins are game 3s in which one player has already won the first two games OR game 2s in which you have a 100% game 1 and game 3 win percentage.

The cost of a particular card is the opportunity cost that you could be running a different card in that slot.   Thus, the card chosen must have the lowest opportunity cost since it is better than any other card you could run in that slot.

The way that we figure out if a card is best is by looking at the cost of the slot.   We figure out how valuable a card is by its contribution to game wins.  This is utility.   We do this by summing up all of the game states in which this card is used weighted by the possibility that this card will arise.   We then select the card that maximizes utility.

What you don’t understand is that utility only looks at RELEVANT games.   If a card contributes to game wins in games that are irrelevant, that provides zero utility to the relevant metric: winning the match.

We typically use game wins as a proxy to measure utility in the match because in the vast majority of cases each game is relevant.  But under your models, there are several instances in which games are irrelevant:

1) Game 2 in which you have a 100% game 1 and game 3 win percentage
2)  Game 3 in which one player has won the first two games.

That’s why your model is ridiculous and doesn’t refute my logic at all.   The fact that you can show the same match win percentage with or without post board games both on the draw and on the play is axiomatic, but as I have said before:

But that doesn’t prove anything….  If we win game one and two, assuming we are playing a third game on the play doesn’t change our win percentage.   We still win the match even if we assume a game 3 loss.  Vice versa, if we lose game 1 and 2, assuming a third game on the draw doesn’t change our win percentage.  We still lose the match even if we assume a game 3 win. 

  I am not arguing that your game 1 win percentage is irrelevant in evaluating a card like Caverns

But Meadbert is:

Your winning percentage in game 1 has no impact on how good Gemstone Caverns is.

No- Meadbert did acknowledge after my very first post in this thread that the "skewness argument" (the one recently articulated by Jacob Orlove) can be an argument in favor of a card.  That's not the point you're making here, though.

Then why did he say this, after your first post:

http://www.themanadrain.com/index.php?topic=31723.msg458515#msg458515

I stand by my original comments regarding how Gemstone Cavern's power is unrelated to your first turn win percentage.

[Elric]

On the other hand, if you were in my thought experiment case (go first in game 2 if you won game 1; go first in game 3 if you lost game 1) then you'd correctly identify that your win % in post-board games going second doesn't matter at all (since in this case you never get to game 3 since you always win the first two games). 

And I think you'd agree that this is the correct answer here- if you always win preboard games and always win post-board games going first, it doesn't matter what percentage of post-board games going second you win.  So this definition of utility isn't meaningful because as you can see, changing the structure of the match in a way that doesn't affect each player's chance to win the match can affect (substantially) the "utility" that you would place on a card.

On the contrary, this simply reveals that you have no idea what utilty means.

I’ll try to explain it once more.

There are 75 possible slots to play with – 60 in the maindeck and 15 in the sideboard.

To advance in magic, the only thing that matters is winning the match.   The way you win matches is by winning games.

Thus, the selection of cards into your 75 open slots is a quest to find those 75 cards that will maximize your chances to win games within the context of a match.   

In other words, it’s cost/benefit decision making.   An ‘optimal’ decklist is a decklist that maximizes your RELEVANT game win percentage.     Irrellevant game wins are game 3s in which one player has already won the first two games OR game 2s in which you have a 100% game 1 and game 3 win percentage.

The cost of a particular card is the opportunity cost that you could be running a different card in that slot.   Thus, the card chosen must have the lowest opportunity cost since it is better than any other card you could run in that slot.

The way that we figure out if a card is best is by looking at the cost of the slot.   We figure out how valuable a card is by its contribution to game wins.  This is utility.   We do this by summing up all of the game states in which this card is used weighted by the possibility that this card will arise.   We then select the card that maximizes utility.

What you don’t understand is that utility only looks at RELEVANT games.   If a card contributes to game wins in games that are irrelevant, that provides zero utility to the relevant metric: winning the match.

We typically use game wins as a proxy to measure utility in the match because in the vast majority of cases each game is relevant.  But under your models, there are several instances in which games are irrelevant:

1) Game 2 in which you have a 100% game 1 and game 3 win percentage
2)  Game 3 in which one player has won the first two games.

That’s why your model is ridiculous and doesn’t refute my logic at all.   The fact that you can show the same match win percentage with or without post board games both on the draw and on the play is axiomatic, but as I have said before:

But that doesn’t prove anything….  If we win game one and two, assuming we are playing a third game on the play doesn’t change our win percentage.   We still win the match even if we assume a game 3 loss.  Vice versa, if we lose game 1 and 2, assuming a third game on the draw doesn’t change our win percentage.  We still lose the match even if we assume a game 3 win. 

Ok, time to simplify:

The way I would phrase my position is: Provided the usual no-time limits and everyone wants to go first (and maybe "no surprising sideboards") assumptions apply, the effect on "how good Caverns is" of your game 1 win % can only be evaluated by its effect on your match win % as given by the formula: W(S1+S2 - S1*S2) + (1-W)*S1*S2, and any other reason why game 1 win % has an impact on "how good Caverns is" is wrong.

But I think you are arguing a point that I don't agree with: "Ichorid's high game 1 win % means that you're going to be going second post-board more often than going first post-board in actual match play and for this reason Caverns is (except in extreme examples like my 100% win percentage one where you always win every match with or without Caverns) better than it would otherwise be if your game 1 win % was lower."

Is this a fair summary of your position?

[Smmenen]

Ok, time to simplify:

The way I would phrase my position is: Provided the usual no-time limits and everyone wants to go first (and maybe "no surprising sideboards") assumptions apply, the effect on "how good Caverns is" of your game 1 win % can only be evaluated by its effect on your match win % as given by the formula: W(S1+S2 - S1*S2) + (1-W)*S1*S2, and any other reason why game 1 win % has an impact on "how good Caverns is" is wrong.

But I think you are arguing a point that I don't agree with: "Ichorid's high game 1 win % means that you're going to be going second post-board more often than going first post-board in actual match play and for this reason Caverns is (except in extreme examples like my 100% win percentage one where you always win every match with or without Caverns) better than it would otherwise be if your game 1 win % was lower."

Is this a fair summary of your position?

I don’t think so.

Here’s how I would state my position and simply and tersely as possible:

“Gemstone Caverns is more useful (higher utility) the higher your game 1 win percentage.”

The problem with your summary of my position is that it is not simply a matter of “going second post board” more often than going first.   The order matters because it reflects on the success of game 1.   The reason order matters is because if you lose game 1s over a certain amount of the time, Gemstone Caverns cannot be justified as taking up one of the precious sideboard slots because the opportunity cost of that slot is too high for the benefit that Caverns provides.

Here is the logic of my position put into three short bullets:

1) Optimal deck design is design that maximizes the overall utility of the 75 card slots.

2) Utility is usefulness as measured by contribution to relevant game wins.  This is done by summing all possible game states weighted by possibility of arising. 

3) The lower the game 1 win percentage the less games states will arise in an actual match in which Caverns will be used (and thus it has lower utility and higher opportunity cost).  The contrapositive of this premise is that Gemstone Caverns will be used more in relevant games the higher the game 1 win percentage.   (This is critical because I think this is part of what you are disagreeing with.)   

That’s my position.

Your position, as I understand it, is this:

You have algebraically calculated the chances of winning the match and found that the match win percentages are the same regardless of whether we assume that there will be two post-board games or not (meaning a post-board game both on the play and on the draw).   You are using this fact to undermine the relevancy of the claim that the G1 win percentage matters on how good Caverns is because we can assume the use of Caverns post board with no change in our match win percentage.

Is that a fair summary of your position?

[Elric]

Your position, as I understand it, is this:

You have algebraically calculated the chances of winning the match and found that the match win percentages are the same regardless of whether we assume that there will be two post-board games or not (meaning a post-board game both on the play and on the draw).   You are using this fact to undermine the relevancy of the claim that the G1 win percentage matters except in so far as how it relates directly to win percentage by the formula Chance to Win Match= W(S1+S2 - S1*S2) + (1-W)*S1*S2 on how good Caverns is because we can assume the use of Caverns post board with no change in our match win percentage.

Is that a fair summary of your position?

Yes.  I added the part in bold to clarify that a card's value can depend on game 1's win percentage- but not in the way that you've claimed in your number 3).  You're definitely right that this is at the heart of our disagreement:

3) The lower the game 1 win percentage the less games states will arise in an actual match in which Caverns will be used (and thus it has lower utility and higher opportunity cost).  The contrapositive of this premise is that Gemstone Caverns will be used more in relevant games the higher the game 1 win percentage.   (This is critical because I think this is part of what you are disagreeing with.)

I'll add that if I put your 3 points into my points, it would look like (inventing some new terminology):
1) Optimal deck design maximizes the expected chance to win a match for an expected metagame (this isn't quite right- in a tournament, you know that metagames change over the course of multiple rounds, and other people might be doing this step at the same time, but it's the intuition that I'm going for).  When I talk about it, I'll assume that the metagame consists of a representative deck.

2) The underlying measure of comparing two versions of a deck is the (expected) match win percentage.  "Usefulness" doesn't have any meaning outside of your chance to win a match.  If you want a measure of "usefulness" based on the contribution of games in actual play, you must weight these games by the possibility of arising, and the chance that these games are "probabilistically decisive."  How Probabilistically decisive a game is equals (the chance that you win the match, given that you win this game)- (the chance that you win the match, given that you lose this game).   Note that what you'd call an "irrelevant game", I'd call a match that is probabilistically decisive 0% of the time. 

In actual match play, any game 3 that is reached is probabilistically decisive 100% of the time (the winner wins the match with probability 1).  How probabilistically decisive a game is can be thought of as its "importance" to winning the match.  Note that except for pathological examples, game 2s are not probabilistically decisive 100% of the time- game 3s, though reached less often, are comparatively "more important".
 
3) When you think about things based on how "probabilistically decisive" cards are in actual match play, if the usual assumptions (no time issues, no "surprise sideboards", everyone wants to go first) hold this is equivalent to the assumption that you play all three games, one unsideboarded, then one post-board going first, and then one post-board going second (where the post-board games can come in either order), and you can simply use the formula match win %= W(S1+S2 - S1*S2) +(1-W)*S1*S2 to capture everything that is relevant. 

[Smmenen]

 
3) When you think about things based on how "probabilistically decisive" cards are in actual match play, if the usual assumptions (no time issues, no "surprise sideboards", everyone wants to go first) hold this is equivalent to the assumption that you play all three games, one unsideboarded, then one post-board going first, and then one post-board going second (where the post-board games can come in either order), and you can simply use the formula match win %= W(S1+S2 - S1*S2) +(1-W)*S1*S2 to capture everything that is relevant. 

So, we are in rough accord on my points 1 and 2.   It is point 3 that you and I part ways.   Can you elaborate a bit more on this?   What is the specific point of contention?   

It isn't that you've grafted your own point onto my point three, but you've completely displaced it.

I was making the point that as your game one win percentage goes down, Caverns will see less play and therefore have a higher opportunity cost in that slot.    How are you refuting that?   You have math, but explain the math so that we can understand why it works the way it does.  If you can't explain it, then the math doesn't prove anything.   Math is symbolic logic.   Put the math into the English language and explain why you disagree with my contention that as your game 1 win percentage falls, Caverns utility decreases (that is, it's ability to be "probabilitistically decisive diminishes.) 

BTW: I have now come around the view that you are indeed correct, and I'll explain why, but I want to see what you say first. 

[Elric]

 
3) When you think about things based on how "probabilistically decisive" cards are in actual match play, if the usual assumptions (no time issues, no "surprise sideboards", everyone wants to go first) hold this is equivalent to the assumption that you play all three games, one unsideboarded, then one post-board going first, and then one post-board going second (where the post-board games can come in either order), and you can simply use the formula match win %= W(S1+S2 - S1*S2) +(1-W)*S1*S2 to capture everything that is relevant. 

So, we are in rough accord on my points 1 and 2.   It is point 3 that you and I part ways.   Can you elaborate a bit more on this?   What is the specific point of contention?   

It isn't that you've grafted your own point onto my point three, but you've completely displaced it.

I was making the point that as your game one win percentage goes down, Caverns will see less play and therefore have a higher opportunity cost in that slot.    How are you refuting that?   You have math, but explain the math so that we can understand why it works the way it does.  If you can't explain it, then the math doesn't prove anything.

BTW: I have now come around the view that you are indeed correct, and I'll explain why, but I want to see what you say first. 

We're not in accord on the points before 3) because, in particular, although you have the word "relevant" in your description, you haven't fully considered all of the implications of extending this to "more or less relevant."  This is a difference of more than just terminology, and I should have spelled it out in my previous post.

2) The underlying measure of comparing two versions of a deck is the (expected) match win percentage.  "Usefulness" doesn't have any meaning outside of your chance to win a match.  If you want a measure of "usefulness" based on the contribution of games in actual play, you must weight these games by the possibility of arising, and the chance that these games are "probabilistically decisive."  How Probabilistically decisive a game is equals (the chance that you win the match, given that you win this game)- (the chance that you win the match, given that you lose this game).   Note that what you'd call an "irrelevant game", I'd call a match that is probabilistically decisive 0% of the time.

2) Utility is usefulness as measured by contribution to relevant game wins.  This is done by summing all possible game states weighted by possibility of arising.

Look at my pathological example, you win 100% unsideboarded, 100% post-board going first, and X% of the post-board games going second (all game 2s).  Then, as you said, post-sideboarded games going second aren't relevant because your chance to win the match is 100% no matter what the outcome of these games is- and when you think about how much to weight post-sideboard games going second by, the answer is 0.  But now consider a similar example:

You win 100% unsideboarded games, X% post-board games going second (all game 2s), and 99% of post-sideboard games going first.  Now post-board games going second are relevant- if you win a post-board game going second it improves your chance to win the match (pretty obvious).  But how much does it improve your chance to win the match?  Not very much, because pre-board games and post-board games going first are so favorable already- you're still very likely to win the match (if you win a post-sideboard game going first you have a 100% chance to win the match and if you lose a post-sideboard game going second you have a 99% chance to win the match).  So it might be called unimportant. 

Certainly, it is unimportant compared to situations in which you win 50% of the unsideboarded games and 50% of the post-sideboarded games going first- in this case, that post-sideboarded game going second makes a big difference!  But the way that you've set things up, by not considering this, doesn't take this "less important" weighting into account.  Game 2s are "less important" than game 3s because game 3s decide the match every time, even though they are reached less frequently. 

Your matchup win % is the relevant consideration when you evaluate cards with respect to their effect on matchup winning percentage.  So that's all that you have to think about.  The reason I stuck "probabilistically decisive" in there was to show how your concept of relevance wasn't quite the correct way to think about things.    But it isn't even needed if you start from the idea that matchup win % is what counts.  Then you evaluate a card on only its merits as it impacts your chance to win the matchup.  The matchup win % expression is just a bunch of algebra that I did before.  So you can simplify it down to the following.

1) Optimal deck design maximizes the expected chance to win a match for an expected metagame (this isn't quite right- in a tournament, you know that metagames change over the course of multiple rounds, and other people might be doing this step at the same time, but it's the intuition that I'm going for).  When I talk about it, I'll assume that the metagame consists of a representative deck.

2) The underlying measure of comparing two versions of a deck is the (expected) match win percentage.  Edit- in so far as things don't affect the underlying match win percentage (like the "reversed order of playing after game 1"), they can't affect your evaluation of cards.  By symmetry, then, if flipping the post-game 1 order doesn't affect your match win percentage, then the argument: "less games states will arise in an actual match in which Caverns will be used (and thus it has lower utility and higher opportunity cost)" isn't an argument for or against Caverns.

3) This is all that counts- if you do some algebra, you can get the exact chance to win a match: match win %= W(S1+S2 - S1*S2) +(1-W)*S1*S2.

[Smmenen]

 
3) When you think about things based on how "probabilistically decisive" cards are in actual match play, if the usual assumptions (no time issues, no "surprise sideboards", everyone wants to go first) hold this is equivalent to the assumption that you play all three games, one unsideboarded, then one post-board going first, and then one post-board going second (where the post-board games can come in either order), and you can simply use the formula match win %= W(S1+S2 - S1*S2) +(1-W)*S1*S2 to capture everything that is relevant. 

So, we are in rough accord on my points 1 and 2.   It is point 3 that you and I part ways.   Can you elaborate a bit more on this?   What is the specific point of contention?   

It isn't that you've grafted your own point onto my point three, but you've completely displaced it.

I was making the point that as your game one win percentage goes down, Caverns will see less play and therefore have a higher opportunity cost in that slot.    How are you refuting that?   You have math, but explain the math so that we can understand why it works the way it does.  If you can't explain it, then the math doesn't prove anything.

BTW: I have now come around the view that you are indeed correct, and I'll explain why, but I want to see what you say first. 

We're not in accord on the points before 3) because, in particular, although you have the word "relevant" in your description, you haven't fully considered all of the implications of extending this to "more or less relevant."  This is a difference of more than just terminology, and I should have spelled it out in my previous post.

Part of the problem I think with your posts, in general, is that you are often unclear in stating what you mean.  I may use terms like "utility" or "opportunity cost," but I go out of my way to expain what they mean.   In those instances where you are vague, unclear or seemingly obfuscate, you rarely go out of your way to clarify.   You leave the confusion as it is.    Consider your sentence above: "we're not in accord..."    It is far from evident what you mean by "more or less relevant" let alone the meaning of this sentence at all.   You need to spend more time explaining and less time stating.   

You'll notice that I go out of my way to explain even basics because it is necessary to understanding.   I went out of my way to do that with my definition of utiliy tand its importance after it became evident that you had no idea what I meant by it.   Effective writing begins with clarity.   

This is more than just semantics or semiotics. 

In point 2, I said:

2) Utility is usefulness as measured by contribution to relevant game wins.  This is done by summing all possible game states weighted by possibility of arising.

In a previous post I said that "relevant" means games that affect whether you win the match or not. 

What you don’t understand is that utility only looks at RELEVANT games.  If a card contributes to game wins in games that are irrelevant, that provides zero utility to the relevant metric: winning the match.

Hence, I already spelled out the metric that defined relevance: winning the match.

You either forgot or gloss over that.   There is no such thing as "more or less" relevant.  A card either contributes to the match win or it is unnecessary to the match win.   

Furthermore, do not assume that I haven't considered implications.

Remember the context here: it was you that asked for us to "simplify."

Now, oddly, it seems that you are calling me out on things that "didn't fully consider the implications of."

Do you realize the absurdity of that?

You said:

Ok, time to simplify:

[....]

Is this a fair summary of your position?

Now that I went out of my way to simplify my positions (omitted minor qualifications like the assumptions you stated in your point 1). 

But now, it seems, you are saying that based upon my statemnts in my "simplified summary" my points are flawed because I haven't fully considered some implications.

Quite frankly, I'm not sure how you came to that conclusion, since, by your own request, my summary was just that - a summary that omitted many qualifications, left implicit many assumptions, and other things unstated. 

2) The underlying measure of comparing two versions of a deck is the (expected) match win percentage.  "Usefulness" doesn't have any meaning outside of your chance to win a match.  If you want a measure of "usefulness" based on the contribution of games in actual play, you must weight these games by the possibility of arising, and the chance that these games are "probabilistically decisive."  How Probabilistically decisive a game is equals (the chance that you win the match, given that you win this game)- (the chance that you win the match, given that you lose this game).   Note that what you'd call an "irrelevant game", I'd call a match that is probabilistically decisive 0% of the time.

2) Utility is usefulness as measured by contribution to relevant game wins.  This is done by summing all possible game states weighted by possibility of arising.

Look at my pathological example, you win 100% unsideboarded, 100% post-board going first, and X% of the post-board games going second (all game 2s).  Then, as you said, post-sideboarded games going second aren't relevant because your chance to win the match is 100% no matter what the outcome of these games is- and when you think about how much to weight post-sideboard games going second by, the answer is 0. 

But also recall, for the moment, that you raised this example - an interesting example that I will be sure to cite in the future, for hte purpose of suggesting that utility wasn't a relevant measurement in this debate because you thought that in game two Caverns would have a high utility.   I showed you that it would not.  In fact, the correct play in your hypo is to immediately concede game two to save time.   

But now consider a similar example:

You win 100% unsideboarded games, X% post-board games going second (all game 2s), and 99% of post-sideboard games going first.  Now post-board games going second are relevant- if you win a post-board game going second it improves your chance to win the match (pretty obvious).  But how much does it improve your chance to win the match?  Not very much, because pre-board games and post-board games going first are so favorable already- you're still very likely to win the match (if you win a post-sideboard game going first you have a 100% chance to win the match and if you lose a post-sideboard game going second you have a 99% chance to win the match).  So it might be called unimportant. 

As I said: a card either contributes to a match win or is unnecessary to it.   The scenario you constructed does not mean that we ignore game 2.   What you need to do is figure out exactly how to maintain that 99% game 3 while maximizing your X% game 2 win percentage.   It's relatively unimportant, but not irrellevant.   Even the smallest things that might take away from your game 3 will need to be considered.  For example, every single second that you spend in game 2 could produce costs for game 3.   Thus, depending on how much time has elapsed in the match when game 1 is over, the rational decision may be to concede game 2 and move onto game 3.   

Certainly, it is unimportant compared to situations in which you win 50% of the unsideboarded games and 50% of the post-sideboarded games going first- in this case, that post-sideboarded game going second makes a big difference!  But the way that you've set things up, by not considering this, doesn't take this "less important" weighting into account.  Game 2s are "less important" than game 3s because game 3s decide the match every time, even though they are reached less frequently. 

The way I've set things up?  I don't remember there being a rule that said that I couldn't scoop a game 2 in the situation you just came up with.   The difference between this situation and the 100/0/100 situation is this: In the 100/0/100 situation, game 2 is absolutely irrellevant to your match win %.   In the 100/X/99 situation, game 2 is relevant, but the costs of playing any game two are very steep.   Even a couple of seconds of play could detract from your very high game 3 win percentage.   That doesn't mean my rule doesn't take account of this.   Take a look at it again:

1) Optimal deck design is design that maximizes the overall utility of the 75 card slots.

2) Utility is usefulness as measured by contribution to relevant game wins.  This is done by summing all possible game states weighted by possibility of arising.

Relevant game wins are defined as contributing to match wins.   It is true that because of the time limitations we will probably scoop every game 2 and thus never play them, but that doesn't make it irrellevant.  What it does is make it, as you say, relatively unimportant.  The consequence is that any card which is used for game 2 will have a lower utility (perhaps a better term is "higher opportunity cost.").   Thus, since we will, in the vast majority of matches, never play game 2, the opportunity cost of cards for the game 2 are too high and therefore would be cut for something else.   

Your matchup win % is the relevant consideration when you evaluate cards with respect to their effect on matchup winning percentage.  So that's all that you have to think about. 

How we frame things is important.  I agree that I am guilty of placing too much emphasis on individual games perhaps - but that is because in the vast majority of all practical instances, games become match wins.   This an instance where a discussion of Ichorid has burst normal operating assumptions about magics operation.   I discussed much earlier in this thread how the assumption about card advantage generally works from an analysis of cards that move from the library or GY to the hand.    Ichorid bursts this assutmpion since so many of its cards are flashback cards or automatically recur from the graveyard. 

Similarly, game wins can often stand in as a proxy for match win.   In other words, when we talk about "utility" or "opportuinty" cost and I talk about the utility of a card in terms of its contribution to game wins, the vast majority of the time, that thereby extends to match wins.   

As you have shown, if you stretch any rule long enough or apply it in unusual circumstances it bends and breaks.  It literally breaks in your hypo of: 100/0/100 and bends in your 100/X/99 hypo.   However, it should have been clear if not from context then from implication that I was summarizing and that my attempt to use "relevant" was a way of trying to address your point.    Restated I might say:

2) Utility is usefulness as measured by contribution to match win percentage.  This is done by summing all possible game states weighted by possibility of arising that contribute to match wins.

Perhaps I've gone down a perilous path by talking about utility when it may have just been simpler to frame things in terms of their "opportunity cost."   

In other words: you want to run the card with the lowest opportunity cost. 

So, restated with that language, here would be my first two points:

1) Optimal deck design is design that miniimizes the opportunity cost of each of the 75 card slots.

2) Utility is usefulness as measured by contribution to match wins.  This is done by summing all possible game states weighted by possibility of arising and those game states contribution to match wins.

The reason I stuck "probabilistically decisive" in there was to show how your concept of relevance wasn't quite the correct way to think about things.   

It wasn't my use of the word "relevance" so much that I think you find objectionable so much as the fact that I emphasized and mentioned "games" as opposed to "matches."   As I said, I used the word "relevant" to try and take account of that and exclude those games that don't contribute to match wins.   

But it isn't even needed if you start from the idea that matchup win % is what counts.  Then you evaluate a card on only its merits as it impacts your chance to win the matchup. 

Exactly.   However, there is a danger.   Magic is a volitile game.   You will never have a 100% game in percentage because of mulligans, if nothing else.   We shouldn't miss the forest for the trees, but the 100% of the time in practical play - not the realm of theoretical magic and possible hypotheticals we currently inhabit - a focus on games will never lead you astray in terms of deck design.   

The matchup win % expression is just a bunch of algebra that I did before. 

I realize that.  However, math is explicable in lingual terms.    Economics before it became thoroughly engulfed by mathmatics was explainable in terms of logic and reasoning.   Math can be converted into paragraph form.   Since you came up with that equation, I was hoping you might take a few paragraphs to explain its operation in language instead of mathmatical terms. 

So you can simplify it down to the following.

1) Optimal deck design maximizes the expected chance to win a match for an expected metagame (this isn't quite right- in a tournament, you know that metagames change over the course of multiple rounds, and other people might be doing this step at the same time, but it's the intuition that I'm going for).  When I talk about it, I'll assume that the metagame consists of a representative deck.

2) The underlying measure of comparing two versions of a deck is the (expected) match win percentage.  Edit- in so far as things don't affect the underlying match win percentage (like the "reversed order of playing after game 1"), they can't affect your evaluation of cards.  By symmetry, then, if flipping the post-game 1 order doesn't affect your match win percentage, then the argument: "less games states will arise in an actual match in which Caverns will be used (and thus it has lower utility and higher opportunity cost)" isn't an argument for or against Caverns.

3) This is all that counts- if you do some algebra, you can get the exact chance to win a match: match win %= W(S1+S2 - S1*S2) +(1-W)*S1*S2.

All in all, your explanation is immensely unsatisfying.   

I phoned some teammates tonight and in conversation with them I discovered in the flaw in my reasoning.   

I was hoping you'd be able to explain it since you identified it in mathmatical terms.

While I'm impressed that you found the flaw, I'm dissapointed that you struggled (and ultimately were unable) to explain it in short, coherent terms. 

The basic flaw in my analysis is this: 

My concept of utility was too narrow. 

This sentence is wrong:  3) The lower the game 1 win percentage the less games states will arise in an actual match in which Caverns will be used (and thus it has lower utility and higher opportunity cost).

That is wrong because I'm equating Utility with usage.   My assumption regarding Caverns was that its utility is derived from maximzing its usage.

You were right: the game one win percentage is completely irrellevant to how good Caverns is. 

(Edits to clarify)

That's because I was confusing "usage" with "boost."    When in fact Utility is a careful measure that combines the two into a single metric.   A card could see very low usage but produce a huge boost and therefore have equally high utility to a card that sees very high usage, but a relativeloy modest boost.   Thus, Tormod's Crypt might provide a modest boost but is very good against lots of decks.   Comparatively, consider this hypo"

Suppose you are in a metagame with 1% deck X in the field.

You have a 60% win percentage against everything in the field except deck X, to which you lose 100% of the time.

If you put card Y in your sideboard - just one copy of this card - you make deck X a 100% match in your favor.   That is, the addition of one card in your sideboard swings the match from 0% to 100%.   

Now, the risk that you'll face deck X is quite slim.  But if you do face it, you will lose.    Thus, the gravity of the harm is great if that event comes about.   

Caverns boost doesn't depend on usage alone but has to take account for how good it does when it does come up. 

If you win game 1, you'll be using Caverns.

But if you lose game 1, you'll want the Caverns boost in game 3. 

If someone had simply said: the problem with your definition of utility is that you are assuming that usage = utilty and I would have gone: Oh shit, you're right.  That's not correct.   

That's why I kept saying to you: the utility of Caverns goes down as your game 1 win percentage goes down.   That was wrong.  It isn't that the utility goes down, but that the usage goes down.   The utility remains the same because if you lose game one, you'll still use it in game 3.   The fact that there are some situations in which you'll never use it does not mean that it doesn't still have a hgh utility.  In fact, if you lose game 1, it's importance arguably is greater since you'll need it to win game 3, should you get there. 

Anyway, this post took me nearly an hour to write and I'm heading to bed soon.   If there is more to say on this subject, my replies will have to wait until later this weekend, and probably next week.   

[Jacob Orlove]

Here's some illustrative numbers. Let's assume that Ichorid wins 80% of game 1s, 28% of SB games on the draw, and 55% of SB games on the play, without Caverns, and that Caverns shifts that to 80/40/54, respectively (80/54/40 if you lose game 1). These numbers were, incidentally, at least somewhat close to reality, but let's not get into a debate on them here.

Without Caverns, you have a 68% chance to win the match if you win game 1, and 15% chance if you lose game 1, for a total match win % of about 57%.

With Caverns, you have a 72% chance to win if you win game 1, and a 22% chance if you lose game 1, for a net total of 62% match wins. Caverns provides about a 5% boost in total wins.

Now, let's drop your game 1 to 50%, and see how much Caverns boosts you up.

Without Caverns, you still have a 68% chance to win the match if you win game 1, and 15% chance if you lose game 1, but the total is now 42% match wins, because far more games will be under the latter case.

With Caverns, you still have a 72% chance to win if you win game 1, and a 22% chance if you lose game 1, but the net total is now 47%.

Even with a dramatically lower game 1 win %, Caverns still provides exactly the same boost to your overall match win chances. You can put in different numbers and see basically the same thing happen; the boost from Caverns is independent of your game 1 win percentage.

[Elric]

That's why I kept saying to you: the utility of Caverns goes down as your game 1 win percentage goes down.   That was wrong.  It isn't that the utility goes down, but that the usage goes down.   The utility remains the same because if you lose game one, you'll still use it in game 3.   The fact that there are some situations in which you'll never use it does not mean that it doesn't still have a hgh utility.  In fact, if you lose game 1, it's importance arguably is greater since you'll need it to win game 3, should you get there. 

Anyway, this post took me nearly an hour to write and I'm heading to bed soon.   If there is more to say on this subject, my replies will have to wait until later this weekend, and probably next week.   

Well, I'm sorry if this wasn't clear, but I was really trying to convey the point that you were counting the card being used more as making it more valuable.  

See my attempt to summarize your position above (bolding added)

But I think you are arguing a point that I don't agree with: "Ichorid's high game 1 win % means that you're going to be going second post-board more often than going first post-board in actual match play and for this reason Caverns is (except in extreme examples like my 100% win percentage one where you always win every match with or without Caverns) better than it would otherwise be if your game 1 win % was lower."

Economics before it became thoroughly engulfed by mathmatics was explainable in terms of logic and reasoning.   Math can be converted into paragraph form.   Since you came up with that equation, I was hoping you might take a few paragraphs to explain its operation in language instead of mathmatical terms.

Guilty as charged! (I'm an economics graduate student)  So that equation is the chance to win a match.  You win a match when (and only when) the following separate ("disjoint") events occur (since these events are disjoint- they don't overlap with each other so if you're looking for the probability that one of more of them happens you just add the probability that each happens on its own):

Notes: Since I've excluded draws, 1- the chance that you win a given game= the chance that you lose that game.  The "everyone wants to go first" rule is very helpful here because it means that, even in advance, you know who is going to go first in each game based on the results of the previous game.  "No surprise sideboards" captures the fact that if you could really surprise your opponent, it might be better for you to play a particular sideboarded game before a different one, so your chance to win each game wouldn't be constant across games- because after you use it the first time it changes your future game win percentage.  "No matches go to time" means that you don't have to worry about the fact that a deck that can win game 1 doesn't have to even try to win game 2 if you can run the time out and win 1-0- but obviously you wouldn't use this strategy having lost game 1 (and who knows whether you'd want to use it in game 3)

Assume the following are constants. W= the chance to win an unsideboarded game (aka "game 1")
S1= the chance to win a postboard game going first
S2= the chance to win a postboard game going second

You win game 1 and then win game 2 going second.  Probability of this happening= chance to win game 1 * chance to win a post-sideboard game going second= W*S2
You win game 1 and lose game 2 going second, and win game 3 going first.  Probability of this happening= Chance to win game 1 * chance to win a post-sideboard game going first * (1- chance to win a post-sideboard game going second)= W*S1*(1-S2)
You lose game 1 and win game 2 going first, and win game 3 going second.  Probability of this happening= (1- chance to win game 1) * chance to win a post-sideboard game going first * chance to win a post-sideboard game going second= (1-W)*S1*S2

Add these three probabilities up and you get the equation above: Match Win %= W*(S1 + S2 - S1*S2) + (1-W)*S1*S2

But I too have work to do, so anything else can probably wait for the weekend.

[diopter]

Here's some illustrative numbers. Let's assume that Ichorid wins 80% of game 1s, 28% of SB games on the draw, and 55% of SB games on the play, without Caverns, and that Caverns shifts that to 80/40/54, respectively (80/54/40 if you lose game 1). These numbers were, incidentally, at least somewhat close to reality, but let's not get into a debate on them here.

Without Caverns, you have a 68% chance to win the match if you win game 1, and 15% chance if you lose game 1, for a total match win % of about 57%.

With Caverns, you have a 72% chance to win if you win game 1, and a 22% chance if you lose game 1, for a net total of 62% match wins. Caverns provides about a 5% boost in total wins.

Now, let's drop your game 1 to 50%, and see how much Caverns boosts you up.

Without Caverns, you still have a 68% chance to win the match if you win game 1, and 15% chance if you lose game 1, but the total is now 42% match wins, because far more games will be under the latter case.

With Caverns, you still have a 72% chance to win if you win game 1, and a 22% chance if you lose game 1, but the net total is now 47%.

Even with a dramatically lower game 1 win %, Caverns still provides exactly the same boost to your overall match win chances. You can put in different numbers and see basically the same thing happen; the boost from Caverns is independent of your game 1 win percentage.

This isn't quite correct.
Label your game 1 win % as W (this is independent of whether or not Caverns is in your board).
Label your match win% without Caverns as M1.
Label your match win% with Caverns as M2.

The quantity we're interested in is M2 - M1.

By your numbers, we have:

M1 = 0.68W + 0.15(1-W)
     = 0.53W + 0.15
M2 = 0.72W + 0.22(1-W)
     = 0.5W + 0.22

M2 - M1 = (0.5W + 0.22) - (0.53W + 0.15)
            = -0.03W + 0.07

For W = 0.8 (80%), M2 - M1 = 0.046 (4.6%)
For W = 0.5 (50%), M2 - M1 = 0.055 (5.5%)

The boost to your match win % from using Caverns absolutely depends on your game 1 win %.

[Jacob Orlove]

You actually can't take those numbers to 3 decimal places, because I was rounding them. If we assume that the 80/28/55 and 80/40/54 numbers are exactly correct, then we get the following complete numbers:

M1 = 0.676W + 0.154(1-W)
     = 0.522W + 0.154
M2 = 0.724W + 0.216(1-W)
     = 0.508W + 0.216

M2 - M1 = -0.014W + 0.062

For W = 0.8 (80%), M2 - M1 = 0.0508 (5.1%)
For W = 0.5 (50%), M2 - M1 = 0.055 (5.5%)
For W = 0.3 (30%), M2 - M1 = 0.0578 (5.8%)

But you're still correct. I'm pretty sure that's just because we're going against the skew of the two games, though.

[policehq]

Algebra aside, I'm just going to say this:

I prefer Gemstone Mine infinitely over Gemstone Caverns because in actual playtesting, Gemstone Mine has saved me from situations where Caverns would've been RFG to Leyline or produced colorless mana for no significant reason. Bazaar of Baghdad awards you with two extra cards a turn for the first few turns of the game, and Gemstone Caverns simply can't take advantage of that. It is reliant purely on opening hands that contain it, an answer to your opponent's threat, a Bazaar, and a card to remove from game in your opening hand as well as playing on the draw. Despite the answers you can bring in game 2, you may lose, and then it's really crunch time for those 4 sideboard slots to come through for you in game 3. Gemstone Mine in combination with Ancient Grudge, Chain of Vapor, and possibly Duress is better at winning matches than Gemstone Caverns; no formula can argue with this particular playtesting experience.

-hq