Average Burn Damage

[Zuberi]

I was doing some messing around with my spellbooks and debating whether or not to make a Priest build. In my pondering, I came up with the question "How useful is Burn?" I wanted to know what kind of bang I was getting for my buck. I was pretty sure it is worth the 1 Mana, but exactly how valuable is it?

After twiddling around with the math a bit, I had my answer and I thought others may be interested in hearing about it, so I came here. I apologize if this is in the wrong forum as it regards a condition rather than a spell, but it seemed the most appropriate to me.

So first the Math. Thanks to Tarkin, I learned the formula for Average Burn Damage is Σ(2/3)^n where n = the number of rounds remaining in the game after Burn was applied minus 1, or Σ(2/3)^(r-1). Thus, to determine the Average Damage of the Burn Condition in general, you must determine how long a game is going to last. I created a chart in Excel that listed the Average Damage of the Burn Condition lasting from 1 Round to 100 rounds in one column, and in a second column I listed the Average Value of Burn assuming the Burn Condition could be placed every round remaining.

For example: If the game had 2 rounds remaining, the Average Damage of Burn would be 1.33. That is because a Burn placed immediately on your opponent would deal an Average of 1.66 Damage over the next 2 rounds, and a Burn placed the next round would deal an Average of 1 Damage. Taking the Average of both of these values yields a value of 1.33.
I'm not going to post the full chart, but I will post some milestones from it. All of these are rounded to the nearest tenths place.

In a game lasting 5 rounds, Burn deals an average of 2.0 damage. I believe most matches last longer than 5 rounds, but it shows that even in the most aggressive match, Burn has value.

In a game lasting 8 rounds Burn deals an average of 2.3 damage.

In a game lasting 11 rounds Burn deals an average of 2.5 damage.

In a game lasting 14 rounds Burn deals an average of 2.6 damage.

In a game lasting 18 rounds Burn deals an average of 2.7 damage.

In a game lasting 24 rounds Burn deals an average of 2.8 damage.

It takes a 40 round game before Burn reaches a value of 2.9 average damage.

The maximum value for Burn's Average Damage is 3. Although a Burn only requires 11 rounds to remain in the game for that specific Burn to deal an average of 3 damage, the Average Value of Burn over the course of an entire game doesn't reach 3 even within a 100 round game. It would require a truly epic game to reach that point.
Now, we need to account for your opponent removing Burn. I'm not sure how exactly to determine it's effect on Burn's Damage, but we can look at it's effects on the game.

The best way I could find to remove a single Burn Condition is with the Priestess' Restore ability. She can remove a Burn by simply paying it's removal cost (2 Mana) and spending a quick spell. This means that your burns might never do damage, but costs her twice as much mana as it cost you to apply it and costs her an action that you did not have to take. All together a pretty solid win.

The next best way, and THE best way for any other mages, looks to be with Geyser currently. It costs 4 Mana and also requires a quick spell, but it removes all Burn Conditions from the target. This means it gets better the more Burn Conditions there are before it is cast. However, if your opponent is letting Burns stack up before casting a Geyser, he's taking damage.

As an example, let us assume that your opponent decides to wait for 3 Burn Conditions to be applied before he casts Geyser. We'll also assume that you are only applying 1 Burn per round only using your Malakai's Fire ability on your Priest. You've spent a total of 3 Mana and dealt an average of 4.5 Damage (first burn has hit twice, 2nd burn has hit once, and the third burn was just applied so hasn't hit yet). All together, a 1 Mana advantage, 1 Action advantage, and 4.5 Damage advantage over your opponent. A good investment for 3 mana.

[DeckBuilder]

[Apologies in advance to everyone else, this is going to be a pretty sad spectacle of maths deconstruction]

That was a nice post, sir, for mechanics efficiency geeks like me! As for where to post this, I think a whole new Maths Geek category will be needed!

In a game of infinite length, mean damage from 1 burn = 3. I can't remember the infinite series equation (Wikipedia!) but this can be easily demonstrated in Excel

Dmg   /   Prob
0   =   0.333333
1.5   =   0.222222
3   =   0.148148
4.5   =   0.098765
6   =   0.065844
7.5   =   0.043896
9   =   0.029264
10.5   =   0.019509
12   =   0.013006
13.5   =   0.008671
15   =   0.005781
16.5   =   0.003854
18   =   0.002569
19.5   =   0.001713
21   =   0.001142
22.5   =   0.000761
24   =   0.000507
25.5   =   0.000338
27   =   0.000226
28.5   =   0.000150
30   =   0.000100
31.5   =   0.000067
33   =   0.000045
34.5   =   0.000030
36   =   0.000020
37.5   =   0.000013
39   =   0.000009

If you SUMPRODUCT damage array with probability array, your mean damage = 2.999. In an infinite series, this will hit its asymptote of 3.

However, when you have an infinite series like burn, surely median damage is a more reliable indicator?
In this case 3/2 damage = 5/9 probability.
Therefore median = 3/2 x 9/5 x 1/2 = 27/20 damage = 1.35 median damage.

The reality is, if you test 100 burn counters repeatedly in a Monte Carlo simulation, half the time, they will deal less than 135 damage and the other half of the time, they deal more. With the ability to deal a significant amount in outlier cases. Median identifies its 50th percentile, unbiased by the high damage of those outlier observations which raises the mean.

I look at cards like Lash of Hellfire (7-10 1 Burn, 11+ 2 Burn) as dealing (3/6 + 1/6) x 27/20 = 27/30 extra direct damage.

All those effects that deal 7-10 1 Burn, 11+ 2 Burn really only deals roughly 1 more direct damage on a 50/50 case basis. But in a game where you have plentiful burn, I would look at this d12 roll as 2 direct damage (mean is slightly more than double median) as the mean becomes more relevant the more times the mechanic is used. This is why everybody remembers "that Warlock match" when that burn lasted the entire remaining game. We remember the outliers and this distorts our evaluation of its potency.

i don't know if mean or median is better. I try not to rely on d12 rolls (the dice hate me) so I guess because I use effects sparsely, I prefer relying on the conservative median. Maybe a maths expert reading this will know how best to quantify burn?

Coming from ultra-analytical games, I think Mage Wars is meant to be played far more laid back (look at some of these rules fuzziness still unresolved), certainly no removal of the game's mystique by deconstructing it down to maths mechanics. Hey, the next set has you literally fighting with Flower Power - you can't get more hippy than that!

Nice post though, Zuberi. Hopefully, it may get a few more players looking at the game in a similarly analytical fashion.

For everyone else, sorry for this lapse, I couldn't resist. I feel dirty and ashamed revealing this side of me...

[tarkin84]

Here's the exact formula:

Σ(2/3)^n

Where n goes from 0 to any number you might want to use. In an ideal world n would tend to infinity and the series' result would be 3 as posted DeckBuilder, but, as Charmyna once pointed out, we do not live in that kind of world and you can use the number of rounds you might expect the game to take for n.

For example, if n=5 the you might expect each burn marker to deal 2,74 damage

[DeckBuilder]

we do not live in that kind of world and you can use the number of rounds you might expect the game to take

Good point Tarkin.
Strategy - burn becomes slight less valuable to include in your book the shorter the game length of your game plan.
Tactics - burn becomes slightly less valuable to play the longer the game progresses.

I like the fact you can't quantify its value easily as a source of delayed direct damage of indeterminate length and intensity.

[Wiz-Pig]

we do not live in that kind of world and you can use the number of rounds you might expect the game to take

Good point Tarkin.
Strategy - burn becomes slight less valuable to include in your book the shorter the game length of your game plan.
Tactics - burn becomes slightly less valuable to play the longer the game progresses.

I like the fact you can't quantify its value easily as a source of delayed direct damage of indeterminate length and intensity.

There is a countervailing factor here that has not been mentioned when considering the value of burn. There seems to be the assumption that damage is damage but I would have to argue that immedicy is an import factor. Because burn damage is strung out over time it is potentially less impactfull on the balance on tempo of the game. In fact its potential as calculated here assumes that you will now use another action to kill your opponent's creatures sooner to avoid there continuing attack, the chafed of burn's failure or there hinderance/hairs effects on the board. For these reasons I would argue that damage immediately dealt is of significantly greater value then damage dealt over time unless the net result of when a creature is removed from play is the same or better somehow of some action advantageis gained allowing for more damage give applied elsewhere.

[DeckBuilder]

Yes, totally agree, Wiz. Very good point. This is why people carry 2x Hurl Boulder. I want the range 2 burst damage now, not later! Even why burst healing is sometimes better than passive regeneration or vampirism. If I don't QC Heal now, he can kill me with his 2 Hurl Boulders.

So what you are effectively saying is there should be a third column, "Tempo Benefit" to the SUMPRODUCT array. So relevance of that first burn damage roll is 1.0000 and this degrades over time, reducing the value of burn further.

In addition, we have damage thresholds. During upkeep, the creature is on 1 life with 1 burn. I couldn't care about any other result, now or future burns, just that it isn't a blank. Excess damage is wasted (as is excess healing). This effectively means there is yet another tactics consideration, burn being less effective in damage terms against lower life remaining targets. But as the end benefit is killing that target, you could argue it the other way too, burn quickens its demise further.

Considering all these factors, I think I am going to stick with my conservative median evaluation of burn.

Whether by design or accident (I suspect both), they have created a deceptively deep and complex game in the clothes of an intuitive duel.

[tarkin84]

You made a big point here Wiz-Pig.

Just for the fact, I have to face an opponent next week in my league match up and I think he'll gonna use a Jokhtari BM + Steelclaw hyper aggressive build (I have many spies in my realm: he defeated my girlfriend playing that book). As a MW and maths enthusiastic I have lots of fun thinking what to play against every match-up. At first I thought that an Adramelech build out of the Warlock would be the perfect counter rush as burn tokens makes the demon attack slightly more powerful than the grizzly's. However, when I started doing some maths I realized that time is an important issue here. In a short game, burn is not such of a factor and my first assumption (buffed Adramelech  winning over grizzly with ease) was not so true: the demon has the edge if every roll sticks to the average, but if the dice don't behave as they should I would lose the match because of not having enough time for the fire tokens to do their job.

So, yeah. Fire needs time to deal all the damage it is supposed to deal. If the game is about to end, burn markers are not so great as other damage sources.

However, we must not forget that fire deals direct damage, avoiding armor and defense. This is important in a metagame where high-armored mages are heavily played.

[ringkichard]

The thing that keeps down burn damage's median value is infant mortality. 1/3 of burn tokens never do any damage. Another 1/3 of the remainder die after only the first round, dealing an expected value of only 1.5 damage.

1/3 + 2/9 = 5/9.

More than half of burn tokens never live past their second birthday.

This is why, today, the warlock foundation is announcing a very special pledge drive to help at risk burn tokens get through those first critical turns so they can thrive and meet their full mean potential. Just two complete rounds is all it takes for a burn token to do (1.5 * 2) = 3 expected damage. When a warlock comes to your door with her lash of hellfire, we do hope you'll give generously.

[DeckBuilder]

I have to face an opponent next week in my league match up and I think he'll gonna use a Jokhtari BM + Steelclaw hyper aggressive build

This would be outrageously unfair but... Play Earth Wizard with 4 Iron Golems and 4 Hurl Boulders, ideally via 2 Wands and 2 Towers. His Wounded Prey is no benefit. Your Hurl Boulders have faster tempo and range 2. Once you Dissolve the Bow a few times, Sleep the Grizzly. Earth Wizard can be played aggro-control with your Golems guarding your Idol of Pestilence, Deathlock and Suppression Orb. Deathlock may be key as Nature mages rely on passive healing to outlast counter-aggro.

This is why I never play the same book twice in a row. The game is lovely as there will always be some counter to every build (thankfully). Even Golem Pit should lose to Priestess turtle, as Koz highlighted.

However, if Earth Wizard seems cheesy (outrageous scouting), maybe Forcemaster Solo would be a more fair and stylish way of beating Jokhtari BM (I believe the Americans call it a "chick fight"?). Especially if you have 4 Force Holds (reveal when target is activated) to keep the threats at bay (they will be dispelled) and to catch up with the swift Jokhtari BM. Mind Control the Grizzly into the Obelisk then get down to a "fair" girl-on-girl contest (you will need 2x Cheetah Speed 2x Mongoose Agility yourself to catch her though).

I am so jealous you have a local league. We just play it casually and the game has died down recently too. The new set will hopefully get them back playing....

[Zuberi]

I edited my original post. Using Tarkin's formula, I attempted to more accurately solve for the Average Damage of Burn depending on the length of a match.

[tarkin84]

This would be outrageously unfair but... Play Earth Wizard with 4 Iron Golems and 4 Hurl Boulders, ideally via 2 Wands and 2 Towers. His Wounded Prey is no benefit. Your Hurl Boulders have faster tempo and range 2. Once you Dissolve the Bow a few times, Sleep the Grizzly. Earth Wizard can be played aggro-control with your Golems guarding your Idol of Pestilence, Deathlock and Suppression Orb. Deathlock may be key as Nature mages rely on passive healing to outlast counter-aggro.

This is why I never play the same book twice in a row. The game is lovely as there will always be some counter to every build (thankfully). Even Golem Pit should lose to Priestess turtle, as Koz highlighted.

However, if Earth Wizard seems cheesy (outrageous scouting), maybe Forcemaster Solo would be a more fair and stylish way of beating Jokhtari BM (I believe the Americans call it a "chick fight"?). Especially if you have 4 Force Holds (reveal when target is activated) to keep the threats at bay (they will be dispelled) and to catch up with the swift Jokhtari BM. Mind Control the Grizzly into the Obelisk then get down to a "fair" girl-on-girl contest (you will need 2x Cheetah Speed 2x Mongoose Agility yourself to catch her though).

I am so jealous you have a local league. We just play it casually and the game has died down recently too. The new set will hopefully get them back playing....

Sorry for the offtopic that derived from my original post (I wanted to point out how time is a crucial factor when talking about fire).

Regarding our league game, my opponent has time to change also his book so building any sort of counter-book could be futile. That's why I do these mental match-ups just as a hobby and I'll bring whatever book I feel more confortable with. Anyways, thank you for your advice, which will be taken into account (all those Golems together are very evil). Your posts and decklists are always very enlightening

Don't give up hope man! Here in Madrid we were just four of us MW players just before summer and now we are holding a league of 12 people. I'd really like to hear from you that your community has also grown in this fashion!

[sIKE]

I played a game recently with a Priest, got a Burn on him round 3. Against all probability's it stayed on the entire game. I got another one on him 3rds before the end and it stayed. And finally on the last round I played one more on him.

The round he cast a Heal upon himself and was at 25 Life. I damaged him for 5 damage (and that last burn). During upkeep he rolled (2 + 2 crit +1 crit)  and dies (Air Wizard). We were both stunned at how deadly the burns were this game. We estimated that he took about 20 points of damage from burns alone. On the hand, I have had games where the burns I placed came off each time with a blank die roll.

My thinking that Burns are nice, burns are direct damage and by pass armor/ageis etc. Players can remove them (if they have the ability), cast Geyser or spell with the extinguish trait, or just live with it. Most importantly if the dice are going your way they can truly affect the game and whoever has them.

The moral is that Burns many times will or will not exceed the sum of there averages. I have started including Renewal Spring in my Priest build to help with conditions and provide a touch of healing. As a result you can say that Burns have changed the way that I play

[Wiz-Pig]

I played a game recently with a Priest, got a Burn on him round 3. Against all probability's it stayed on the entire game. I got another one on him 3rds before the end and it stayed. And finally on the last round I played one more on him.

The round he cast a Heal upon himself and was at 25 Life. I damaged him for 5 damage (and that last burn). During upkeep he rolled (2 + 2 crit +1 crit)  and dies (Air Wizard). We were both stunned at how deadly the burns were this game. We estimated that he took about 20 points of damage from burns alone. On the hand, I have had games where the burns I placed came off each time with a blank die roll.

My thinking that Burns are nice, burns are direct damage and by pass armor/ageis etc. Players can remove them (if they have the ability), cast Geyser or spell with the extinguish trait, or just live with it. Most importantly if the dice are going your way they can truly affect the game and whoever has them.

The moral is that Burns many times will or will not exceed the sum of there averages. I have started including Renewal Spring in my Priest build to help with conditions and provide a touch of healing. As a result you can say that Burns have changed the way that I play

I will agree that burns are great on an opponent mage.