Comparing Courage Effects

[Hannibal]

So this is where I, the evil economist, comes in Grin.  I show you a Reroll +1 card and a preemptive Courage +3 card.  Which one do you want?  "Reroll +1!"  Ok, now I bump up the Courage to +4.  Now which one do you want?  I reduce the Reroll card to a straight reroll.  Now which one?  What if the preemptive Courage card affects two units?  I can keep messing with the numbers.  Now which card do you want?

And this is where I say its not an apples-to-apples comparison.  To truly get the value of a card based on the bonus it provides, I think you should assume they are all played before the roll.  Then I guess you can determine a value of being played post-roll.

Perhaps that's what you're getting to with conditional probability.  I'm way out of my element there, so I guess I'll step back from this conversation.  Perhaps my thinking falls afoul of the Monty Hall syndrome (where people make the mathematically wrong choice based on intuitive reasoning), but to me I look at the outcome where both cards mean the unit will be there 91% of the time after its all said and done.  That to me says that the courage effect on the cards is equal.

The timing makes it a different situation, of course.  For purely Courage purposes you'd much prefer Cold Blooded over Rune of the Wary.  Which is why Rune of the Wary also provides a Def cookie.  Does that make up for it?  Well, that's a question of personal preference...

[Zelc]

Maybe here's a weak attempt to quantify how Courage fits in with other Command Cards.  Kevin, I'm not sure this really matters for your table since all it should do is move everyone up or down the same amount.

Suppose I Kill You Meself and Devotion of Courage (auto-pass version) are both average Courage cards (I Kill You Meself is probably better than average and Devotion of Courage is probably worse than average).  Now let's say there's 1000 units that need to make a rout check, let's say on 11 (38% fail chance).  If we want all of them to pass, we'd need 1000 Devotions of Courage and on average 380 I Kill You Meselfs.

So each I Kill You Meself is worth roughly 2.63 Devotions of Courage.  If we normalize the value of I Kill You Meself at 1 (matching it with the table), we get that Devotion of Courage is worth .38.  (Hey, that's the average of proactive autopass on the table!  That's basically what the table is trying to get at.)

The 1 point of damage is worth 62% points on the chart (failed rout checks turned into successful rout checks).  So an average (faction-specific) Command Card is worth 62% points on the table.

Some things that skew this:

• I Kill You Meself is better than an average Courage card, and Devotion of Courage is probably worse than an average Courage card.  That means an average Command Card should be worth less courage.
• I assumed the average faction-specific card is worth 1 damage, but it should be worth slightly more.  This should offset the above and means an average Command Card should be worth more courage.
• 440 copies I Kill You Meself still carries a variance (sometimes you won't have enough, sometimes you'll have extras), while 1000 copies of Devotion of Courage guarantees everyone passes.  That's worth something, so the value of I Kill You Meself should go down relative to the value of Devotion of Courage.  That means an average Command Card should be worth slightly less courage (this is a scenario where there's no scarcity, so in a real game the bias isn't very large).
• I assume the Courage checks are at 11.  Depending on where you think a more realistic assumption is, that should also influence the value of an average Command Card.

Let's fudge this and say that these effects net to zero if I Kill You Meself was 1.33 points of damage (this is really ad-hoc).  Divide 62 by 1.33 and we get ~46.5% points.  So that's how much a normal faction-specific card should be worth in Courage.

Now, let's look at Rune of the Wary.  It cancels about 1/2 of an Accuracy on a standard offense vs. standard defense attack, so let's call the +1 defense half of what a faction-specific blue card should do (it cancels 2/3 of Strike, but faction-specific cards should be above average).  So this is worth about ~24% points.

For a courage 11, the C+3 boost is +28%.  However, it has an additional failure chance since you might not take any damage this turn, thus wasting the Courage.  On a straight (5) 5/5 vs. 2/2, the chance is 40% after you play your Rune, but with a Strike or Accuracy, it goes down to an average of 15%.  Let's split the difference and call it 27.5%.  Now the Courage boost is worth about 20.5% points on the table.

Together, it's worth about 44.5%.  So, slightly below the worth of an average faction Command Card.  Not bad, except these are the Dwarves' best cards .

[RushAss]

Let's fudge this and say that these effects net to zero if I Kill You Meself was 1.33 points of damage (this is really ad-hoc).  Divide 62 by 1.33 and we get ~46.5% points.  So that's how much a normal faction-specific card should be worth in Courage.

OK, first of all I'd like to commend you on these charts and all of the research.  This is some pretty good stuff here Bohan.

But I think you are over thinking this from a purely mathematical standpoint.  A vital component of courage cards is how they fit into the context of the faction itself and that is far more difficult to quantify using charts and straight math.  Looking at I Kill You Meself, it's no doubt a pretty good card.  What makes it awesome is that it exists as the only courage help in the deck of a faction that generally has awful courage.  If you took I Kill You Meself and plopped it into the command deck of a high courage faction like Dark or High Elves it wouldn't be quite as awesome since there will normally be fewer opportunities to play it since the average High Elven unit has 1 more point of courage than the average Orc/Goblin unit.  And even though Outhbound is maligned by many since it is a pre-played +2 courage, the context of the faction it is in makes it better than if it was in another faction.  If we put Oathbound into the Orc command deck it would be pretty lame.  In fact, you'd have to make it a +3 courage bonus to have it behave the same way it behaves in the HE deck.  See my further thoughts on Oathbound here: http://yourmovegames.com/forum/index.php/topic,4851.30.html (rules team forum members only).

Turning this to the Dwarven cards, none of them certainly stack up to I Kill You Meself for courage help.  But it's really nice when you are playing Dwarves and you know that 1 out of every 3 cards you draw is going to at least give you some help.  When playing Orcs you are probably just going to draw 1 I Kill You Meself.  Context context context.

[Zelc]

Rush, that's exactly backwards.  The value of a card beng reactive increases with the unit's courage and the value of a card being proactive decreases with the unit's courage.  Intuitively, let's consider the limits.  If you have a 0 courage unit, the value of a reactive autopass is no more than the value of a proactive autopass.  On the other hand, if you have a 17 courage unit, playing proactive cards on it is horrible.  They'll be wasted 99% of the time.  However, that 1% when you fail, you will really wish you had the reactive card.

I Kill You Meself is better in the high elf deck because their base courage is higher, and Oathbound is better in the orc deck because their base courage is lower.  The table captures this.

[RushAss]

Again, I think you are over thinking a card's value within a faction from a strictly mathematical standpoint.  Mathematically you are totally correct concerning the value of these cards, but we are discussing apples and oranges when it comes to the context of the cards themselves within their respective factions.  A numeric courage bonus is always more help for units with lower courage.  I get that.  But I'm looking at the end result.  Oathbound in the Orc deck where the average line unit has a base courage of 12 means that these units will need a 13 or lower to succeed a check in the yellow if the card was played.  Oathbound in the High Elven deck where the average line unit has a base courage of 13 means that these units will need a 14 or lower to succeed a check in the yellow if the card is played.  A courage of 14 is always better than 13, right?  Sure +2 courage increases the odds of a successful courage check for lower courage units better than it does for higher courage units, but the higher courage units don't need the help as much and hence their card fits in with the faction better IMO. 

To sum it up:
Orc courage sucks and they need courage help, so they get an auto pass card.  Keep in mind that this card only effects 1 unit and it comes at a price.
High Elven courage is superb and they generally don't need a ton of help, so they get a proactive card with a modest boost.  And this card can effect multiple units.

Now let's go a step further and say each of these cards are standard cards like Might or Parry and they are included in every deck.  I'd be willing to bet that players would prefer to use 1 of them for certain factions and the other one for other factions.  But taken out of the context of a specific faction I'm thinking folks would be happier drawing I Kill You Meslelf than Oathbound when playing factions with average or poor courage.  After all, their units are more likely to rout.  I know you play the High Elves, but have you played the Orcs much?  Rolling courage checks for them is a totally different experience.

I'm not knocking your calculations, they are fine.  I'm simply trying to point out that in my line of thinking you cannot completely quantify the effectiveness of each card without considering the context of the faction those cards exist in.

[Zelc]

Again, I think you are over thinking a card's value within a faction from a strictly mathematical standpoint.  Mathematically you are totally correct concerning the value of these cards, but we are discussing apples and oranges when it comes to the context of the cards themselves within their respective factions.  A numeric courage bonus is always more help for units with lower courage.  I get that.  But I'm looking at the end result.  Oathbound in the Orc deck where the average line unit has a base courage of 12 means that these units will need a 13 or lower to succeed a check in the yellow if the card was played.  Oathbound in the High Elven deck where the average line unit has a base courage of 13 means that these units will need a 14 or lower to succeed a check in the yellow if the card is played.  A courage of 14 is always better than 13, right?  Sure +2 courage increases the odds of a successful courage check for lower courage units better than it does for higher courage units, but the higher courage units don't need the help as much and hence their card fits in with the faction better IMO.

Actually, the High Elves need plenty of help with Courage.  Since their units are so expensive, one failed Courage check means disaster because the faction has no way of dealing with it (i.e. with backup units).  Since the units are checking on 12, you're going to fail rout checks.  That's the reason Niko hates playing a normal High Elves line army, and is one of the primary reasons he didn't play High Elves in the last tournament.  I'm pretty sure Niko would argue that I Kill You Meself is just as good in the High Elf deck as it is in the Orc deck, if not better.

14 courage is better than 13 courage, but going from 11 courage to 13 courage is better than going from 12 courage to 14 courage.  And that's what we're trying to measure here.  Looking only at the end Courage result is probably committing a version of the Post Hoc fallacy, where you attribute the full value of passing a rout check to playing the Command Card, when your unit might have passed without it.

Again, we can think about your argument at the limit.  Suppose there's a faction filled with cheap 9 Courage line units, and a faction filled with expensive 17 Courage line units.  The first faction has a card that only gives a unit +8 Courage (they check on 16 in the yellow), and the second faction has a card that only gives a unit +1 Courage (they check on 17 in the yellow).  Would you say the second card is better than the first card when you consider the contexts of the factions?  I think we can clearly say the first card is better, regardless of which faction it's in.


EDIT: Here's another way to think about it.  On average, I Kill You Meself in the Orc deck guarantees 3 units pass their rout checks.  On average, I Kill You Meself in the High Elf deck guarantees 4 units pass their rout checks.  On average, Oathbound (if you play it on 3 units) in the Orc Deck guarantees 2.5 units passes their rout checks.  On average, Oathbound (if you play it on 3 units) in the High Elf deck guarantees 2.7 units passes their rout checks.

[Niko White]

I hate Oathbound in general, but it actually gets even worse if you look at it within the context of the High Elf army.  As Zelc pointed out, checking on 14 vs. 12 is less of a % boost than 13 vs. 11, and the fact that High Elf units tend to cost a billion points and can't afford backup means that failing a check is a catastrophe.  In addition, the whole faction has incredible defense profiles, so it is really unlikely that you'll ever end up rolling three checks in the same turn.  (About the only time it can happen is if someone gets pinched in the same turn a couple of battlesquads are likely to go into the yellow but that's a wild corner case.)  I'm actually not sure I've ever played an Oathbound that wouldn't have been strictly better as that Carthage card that gives two guys +3.

Meanwhile, the value of any reactive card (IKYM, Cold Blooded, w/e) goes up significantly the more likely you are to pass, since having to commit a card ahead of time when you've already got good odds is so annoying.  Even given the high value of High Elf health boxes, I would snap-switch Oathbound with IKYM.  In the Undying Lands of the Utmost West, the High Elf decks all contain a Cold Blooded reprint; that's actually why the Elves go to the Havens.  Little known fact.

[RushAss]

I'll stand corrected on this one then.  I still do believe that faction context has an effect on the value of a card, though.

And maybe it's because I play a lot of the bigger games, but I actually have seen Oathbound used on 3+ HE units in a single turn multiple times. 

[gull2112]

First off, I never find Orc courage to be a problem. In fact, I usually finish a game with IKYM in my hand unused.

Secondly, and seriously, I have never thought of HE courage as fragile or glass cannony. Putting fixes in for them because they can't afford failed courage checks is just wrong. HE have a certain feel, and that is part of it, and the skill in using any army is in trying to minimize its weaknesses. I've said this before and I'll say it again, if you want completely always equal chances how about a nice game of chess? Certain tweaks do improve the game, but this is getting to the point where every army is going to have several tweaks and band-aids and once you start you never stop. It seems like some people just don't like being out-played and so if they lose it must be a game flaw. Making abominations elite was a needed fix. The recent shooting rules improved the game IMHO. But this endless tweaking is too much.