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Monday, 17th November, 2014, 08:48 PM #1
Cost/Benefit Analysis of True Strike
Warning: the following post has math, but you don't have to read that far if you don't want to. I summarize my findings first.
In the "lowlevel wizards suck" thread I didn't think True Strike was ever worth it until someone pointed out that it is only a cantrip (which I had forgotten) and that it can cancel disadvantage. So I decided to do some algebra and work out exactly when casting True Strike before an attack is worth it.
SUMMARY: I analyzed the probabilities of hitting and expected damage of two scenarios: casting true strike then attacking once, or attacking twice on subsequent rounds without true strike. I conclude that when the attack has no cost (e.g. a melee attack or cantrip), casting true strike is only worthwhile when you have disadvantage AND you need an 11 or better to hit the target. On the other hand, when the attack has a cost (like a spell slot), then casting True Strike beforehand gives a better damage/slot ratio than casting the spell twice.
NUANCE: The spell description for TS says "You point your finger at a target...", perhaps implying that you can't use TS on a target you can't see or pinpoint somehow. However, the most common source of disadvantage on attacks is not being able to see the target. Since the expected damage advantage of TS+Attack over Attack+Attack is small, if your DM rules that you can't TS someone you can't see, then I think the cost of learning this cantrip is not worth the benefit if you intend to use it mainly with free attacks.

All the following compare two scenarios:
S1 = Cast True Strike in round 1, then make an attack in round 2.
S2 = Make an attack in round 1, make the same attack in round 2.
MATH: Free attacks, no disadvantage
Using True Strike with a melee attack or cantrip is never worth it in this case. It's easy to see why: both cases give you two rolls to hit, but in S1 you can never hit more than once. To put it another way, there are 400 possible outcomes of the 2 d20 rolls, which you can visualize on a 20x20 grid. The horizontal axis represents the first roll, and the vertical axis represents the second roll. Let's call the number you need to roll to hit x. In S1, you miss if the two rolls land on a point in the (x1) x (x1) square in the lower left corner of the grid; otherwise you hit and do the expected damage from one attack. In S1, the regions of hits and misses look the same. However, there is a (21x) x (21x) square of outcomes in the upper right corner of the grid that means you landed two hits, not one, so the expected damage of those outcomes is higher. Therefore, S2 is always better than S1.
MATH: Free attacks, disadvantage
In this situation the probability of landing a hit in S1 is easy to calculate: it's just (21x)/20 where x is the number needed to hit. The expected damage is proportional to the probability. The expected damage in S2 can be calculated by again visualizing the possible outcomes on a grid. Except this time, we're going to look at a 400x400 grid. In effect, I model each roll with disadvantage (2d20) as if it were a d400, since there are 400 possible outcomes of that roll.
If you need x to hit, then on your disadvantaged attack roll there are (21x)^2 outcomes where you hit and 400(21x)^2 outcomes where you miss. If the xaxis is your first attack and the yaxis is your second attack, imagine drawing a cross on the grid with horizontal and vertical lines that cross the axes at position 400(21x)^2. This cross divides the grid into four regions: the square in the lower left where you miss both times, the square in the upper right where you hit both times, and the two rectangles left over where you hit once.
The two rectangles each have area [400(21x)^2]*(21x)^2, and the square in the upper right corner has area (21x)^4.
The expected damage is therefore [2*[400(21x)^2]*(21x)^2 + 2*(21x)^4]/400^2, equal to the probability of getting one hit plus the probability of getting 2 hits weighted double. Conveniently, the terms proportional to (21x)^4 in the numerator cancel each other out and we are left with:
S2 exp. damage = 2(21x)^2/400
S1 exp. damage = (21x)/20
Equating the two and solving for x gives x=11. So if the number you need to hit the target is 11 or more, the expected damage from using true strike to cancel out your disadvantage, then attacking, is greater. However, the difference is not great. For low values of x, your expected damage in S2 is vastly greater since you have a good chance of hitting twice. For high values of x, the difference is only a few percent. It's worth casting True Strike, but you shouldn't expect a huge difference in the outcome. For example, if x=12 then your E.D. is 0.45 in S1 or 0.405 in S2. If x=20 then your E.D. is 0.05 in S1 or 0.0025 in S2.
MATH: Costly attacks, no disadvantage
Let's say you have two choices: you can buy a coupon for 10$ that gives you 10 free pies at your local bakery, or a coupon for 20$ that gives you 15 free pies. Which is better? Obviously the one with the higher pietodollar ratio, the first coupon.
Similarly, if each attack costs a spell slot, then your decision of whether or not to cast True Strike depends on the cost, and you should prefer the strategy with the higher damagetoslot ratio. I used the same sort of analysis described above, but on a 20x20 grid, and divided the expected damage of S2 by two (because it costs 2 spell slots, or superiority dice, or whatever other limited resource you want to model). We get:
S1 exp. damage / cost = 1  (x1)^2/400 (the whole grid minus the lower left square of misses)
S2 exp. damage / cost = (21x)(x1)/400 + (21x)^2/400 (rectangles/2 + 2*upper right square/2)
It turns out that in the range 120, S1 always has a higher expected damage than S2. So it is nearly always worthwhile for a wizard to set up their spell requiring an attack roll with True Strike.
MATH: Costly attacks, disadvantage
The probabilities are the same as in "Free attacks, disadvantage", but we divide the term for S2 by 2 to get the exp. damage / cost ratio. It turns out that the two ratios are pretty similar over the whole range x=[1,20]. The ratio for S1 is a diagonal line from 1 at x=1 to 0.05 at x=20. The ratio for S2 is a shallow parabola with similar endpoints. The expected damage of S1 is always greater. For example, if you need an 11 to hit, then the damage/cost ratio for S1 is 0.5 and the damage/cost ratio for S2 is 0.25. That's about where the difference is greatest; if you need a high or a low number to hit then the difference is only a few percent.
Note that I am comparing the damagetocost ratios, not the expected damage. If you need a high roll to hit, then the expected damage of S1 is slightly greater but the damagetocost ratio is also greater, reinforcing the advantage of this scenario. If you need a low roll to hit, then the expected damage of S2 is greater but pursuing this strategy also costs two spell slots instead of one.

Monday, 17th November, 2014, 08:53 PM #2
Pit Fiend (Lvl 26)
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Another thing that gets overlooked is that an Eldritch Knight can use truestrike to grant him or her advantage on their attack every round using their 7th level class ability.

Monday, 17th November, 2014, 10:03 PM #3
Lama (Lvl 13)
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Monday, 17th November, 2014, 10:10 PM #4
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One thing missing from your analysis is the chance that in round n+1, your attack is meaningless because the target is dead. Equally, there is a chance that maximum damage from attack cantrip + costly spell ends an attack in round two, where as true strike + costly spell does not because though the expected damage is higher the maximum damage is not.
I'd expect that there is a fairly tight grouping between single target damage and spell level, that suggests that the greater the spell level of your costly spell, the more desirable it is to set up with true strike relative to setting up with an attack cantrip. I'm not sure that I buy that cast true strike + 1st level spell, always has a higher expected damage than attack cantrip + 1st level spell. Particularly in the case of like a 5th level caster, True Strike + 3rd level spell seems likely to be higher expected damage, but I'd bet on attack cantrip + 1st level spell over true strike + 1st level spell in terms of expected damage because of the fact that at 5th level, cantrip damage just about doubles IIRC.

Monday, 17th November, 2014, 10:11 PM #5
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A key point to bear in mind: Time also has a cost. 50 points of damage dealt on round #1 is worth more than the same 50 points of damage dealt on round #2, because that damage might kill an enemy which would otherwise hit back on its turn. 50 points of damage dealt on round #10 is practically worthless, since the fight is almost certain to be over by then.
It's very hard to calculate the value of time, but as a general rule, the faster combat goes (in rounds, not time at the table), the more valuable it is to have your damage frontloaded. 5E combat tends toward the short and brutal.

Monday, 17th November, 2014, 10:31 PM #6
Hydra (Lvl 25)
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Yep, only casting True Strike on round one (and doing nothing else) doesn't feel constructive to me.
Even if the math tells me I have set up a better damage probability on round two, that doesn't take into account the risk that attack has become irrelevant by then:
The monster could be dead. Worse, the monster could have hidden or otherwise defended itself from your attack. Even worse, a fellow party member could be in trouble, requiring you to abandon your plan and do something else.
I believe you are much better served to cast a spell that makes a difference here and now, in turn one. Then, in turn two, you can cast the spell that's best for that scenario. Things change pretty quick in D&D magical combat. (Which, of course, is the source of fun)
However, if you can pull off both the cantrip and payload spell in your turn, casting both before anyone can react, then we're talking...
So, yeah, this is just a technical note. The math is still sound (I presume!)

Monday, 17th November, 2014, 10:40 PM #7
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Monday, 17th November, 2014, 10:41 PM #8
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I liked True Strike better when I thought you could cast it on your allies. (I had skimmed the spell and just looked at the range.)

Tuesday, 18th November, 2014, 12:17 AM #9
Lama (Lvl 13)
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Would it be a good idea to 'fix' the spell by upgrading it to level 1 and making it a reaction? Say the offensive counterpart to Shield? You can cast it as a reaction when you miss with an attack roll and add 5 to the roll you just missed. This would make it useful, but prevent spamming it by requiring a spell slot.

Tuesday, 18th November, 2014, 01:51 AM #10
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There are plenty of formulas used to calculate the value of time. The hard part will be getting a reasonable approximation of the deflation rate for damage in a general case. But I think as a logical exercise, you could pick a fairly reasonable number  say 10%. That is, doing 6 damage on 4 successive rounds (24 damage total) is probably more valuable than doing 35 on the fourth only. There are of course counter examples  it's a single PC versus a single monster with 34 hit points. But for every such extreme case, we could come up with examples on the opposite extreme  a single PC versus 4 6 hit point foes.
In any event, while I love the analysis, the point is that if it is a close thing, you should never cast true strike. Take the case casting attack cantrips at a disadvantage. fuindordm concludes in this case: "It's worth casting True Strike, but you shouldn't expect a huge difference in the outcome." But because you don't expect a huge difference in outcome, you shouldn't cast True Strike. Not just because the value of X damage now is greater than the value of X damage in round n + 1, but because we can't necessarily be sure that the situation that grants us disadvantage will persist. If we expect no hugely favorable improvement in the outcome, we certainly shouldn't delay.
The only time I can see it being useful (if it isn't free) is if we are preparing to cast a very costly spell and its early in the fight.
For example, suppose we are prepared to spend Chromatic Orb as a third level spell doing 5d8 damage, and we hit on an 11+. If we cast firebolt now, we may do 2d10 damage with, hitting on an 11+, for an average of 5.5 damage, then the orb next round for 11.25 more for a total of 16.75 expected. Or we can cast True Strike now, then the orb next round for 16.875 expected damage. This is a slight improvement but hardly enough to justify delaying, and I'd guess the delay strategy would be less effectual in the long run over the course of many encounters. And note that if we were only setting up a 1st level Orb, it wouldn't be worth it. But if we perhaps had prepared some 3rd level single target spell that did 8d6 damage on a hit, then we could probably justify the delay to give us the best chance of hitting.
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