Tell Me A Tale, Great Or Small...: Is Heroic Strike Good? (Part IV: The Probabilities (Plural) of Heroic Strike)

We come to it at last...

After a lengthy build-up breaking down all the components of a successful duel role, we're finally read to talk about Heroic Strike itself. When does it make sense? And when is it *gasp* a waste?

There's actually a lot to unpack here (far more than I thought there'd be when I originally started writing on it). We'll start by looking at the overall mechanics of Heroic Strike, mostly to understand exactly what benefits (and risks) it carries with it. It's that balancing of benefit and risk that makes it game-winning or worthless (and a lot in-between).

Because Blogger doesn't like long write-ups, this one is split into two parts. This section (part IV) goes over the three basic probability mechanics in Heroic Strike (the D6 roll to increase Fight Value, the "10" cap on Fight Value, and expected distributions of probable outcomes as dice scale upwards). Part V (which you can read here) applies those principles to a case study between two combatants (Faramir vs. Eomer), and also discusses an alternative use of Might ("boosting") and how that might--or might not--make Heroic Strike more attractive (sorry--those "might" puns were purely unintentional).

(Potential math overload ahead)

The Anatomy of Heroic Strike

Let’s start with a refresher on Heroic Strike (p. 72 of the Rules Manual):

Heroic Strike (Fight Phase)

Outnumbered and fighting for their life, it is times like these when a hero must dig down deep in order to fight off their foes with all the skill they can muster.

A Hero model who declares they are using Heroic Strike adds D6 to their Fight value for the duration of the Fight phase. This is rolled for at the start of the model’s Duel roll and lasts until the end of the turn. This cannot increase a Hero model’s Fight value above 10. Note that this bonus is applied before other effects are taken into account. Thus, a Hero who is Engaged in a Fight with a Bat Swarm would add D6 to their Fight value (to a maximum of 10) and then halve the total due to the Bat Swarm’s Blinding Swarm special rule.

There’s an intuitive understanding that this is a powerful ability. And it is. But it isn’t. It all depends on your point of view.

As we discussed when we began this series, a duel consists of eight components:

1. Gather and designate dice;

2. Declare special strikes or two-handed weapons;

3. Roll all dice;

4. Apply modifiers to dice as designated in #1 and #2;

5. Apply rerolls;

6. Apply Might to appropriate designated dice;

7. Determine the winner of the duel (breaking drawn combats by Fight Value or a roll-off); and

8. The winner makes strikes (if the winner did not shield or is otherwise prevented from striking).

Heroic Strike has no impact on most of this. It doesn’t alter the number of dice we’re rolling. It doesn’t add any modifiers, positive or negative. It doesn’t let us reroll any dice. If anything, it reduces our ability to modify dice with Might (by depleting our Might store). Its clearest application is only at step #7, because if we get the higher Fight Value via a successful Strike, we’ll win a drawn combat outright. So we’ll start with that interaction.

Note: It’s worth pointing out that while Strike only directly impacts #7, it will indirectly impact #8 as well, because the higher our chance of winning the combat, the higher our chance of wounding because our to-wound chance if we lose a duel is always 0% (unless you’re Haldir). It may also indirectly impact #6: theoretically, our opponent may not spend as much Might if he knows we can match his spending and will win ties (which would be good for us). Having said that, my (purely anecdotal and concededly limited) experience is that most opponents will go ahead and spend the Might anyway, to prey upon our inclination to throw even more of our hero’s valuable Might into winning the combat … more on that in our "Strike in Miniature" discussion, here.

The Value of Contesting “Drawn Combats”

Here’s the key observation: a successful Heroic Strike is going to confer the same benefit as higher fight value confers generally: we win drawn combats, when both our highest duel roll and our opponent’s highest duel roll have the same value (see Rule #2: Having higher fight value (of any value) than our opponent helps us win the fight, but only if we have a drawn combat). Because each opponent in a duel uses only their highest duel roll, this always comes down to pitting one six-sided die against another six-sided die, for a total of thirty-six possible roll combinations, and three possible outcomes: “Win,” “Lose,” or “Draw” (if both sides have the same fight value). Having the higher fight value than our opponent (either natively or because of a successful Strike) converts those six “draw” outcomes to “wins”:

Possible Duel Roll Outcomes

Challenger has Same Fight Value (X)

Challenger has Higher Fight Value (X+1)

Target (FV: X) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X) Duel Roll

"1"

Draw

Loss

Challenger (FV: X + 1) Duel Roll

"1"

Win

Loss

"2"

Win

Draw

Loss

"2"

Win

Loss

"3"

Win

Draw

Loss

"3"

Win

Loss

"4"

Win

Draw

Loss

"4"

Win

Loss

"5"

Win

Draw

Loss

"5"

Win

Loss

"6"

Win

Draw

"6"

Win

With tied Fight Value, each side wins 15 of 36 scenarios (41.67% of the time), and the remaining six scenarios (16.67%) are drawn combats which, if neither side has an elven blade, they are likely to split 50/50 (meaning each side is likely to win 18 of 36 combats, or 50%). The higher fight value would give that player the six drawn combat scenarios, too, for total odds of 21/36 (58.33%) compared to his opponent’s 15/36 (41.67%). In other words, Heroic Strike improves your odds by 8.33% (50% to 58.33%), and reduces your opponent’s odds by 8.33% (50% to 41.67%), or from equal one-of-two odds to roughly six-of-ten odds in your favor.

That 16-percentage point swing in our favor is what we’re paying for (or gambling on) if we call a Heroic Strike, assuming that we start the combat with the same fight value as our opponent. But if we start with a lower fight value than our opponent, we’re already starting at that 60-40 disadvantage, which means that while the number of scenarios in which we win doesn’t change (we still top out at 21/36 scenarios, or 58.33%), we essentially flip the chart on our opponent: instead of winning 15 scenarios outright and losing 21, we now win 21 scenarios and lose only 15. The magnitude of the swing in our favor doubles (41.67% to 58.33%, or +16.66%), as does the negative swing for our opponent (58.33% to 41.67%, or -16.66%):

Challenger has Lower Fight Value (X-1)

Challenger has Higher Fight Value (X+1)

Target (FV: X) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X-1) Duel Roll

"1"

Loss

Challenger (FV: X+1) Duel Roll

"1"

Win

Loss

"2"

Win

Loss

"2"

Win

Loss

"3"

Win

Loss

"3"

Win

Loss

"4"

Win

Loss

"4"

Win

Loss

"5"

Win

Loss

"5"

Win

Loss

"6"

Win

Loss

"6"

Win

In both scenarios, the net result is the same: we win a maximum of only six more scenarios (the ones where the highest duel roll for both side ties), but psychologically, going from a “draw” to a “win” in those six scenarios (or even better, from a “loss” to a “win”) can be enormous. As anyone who’s played a high-fight army knows, rolling a “6” that stays a “6” and wining the duel automatically is absolutely amazing. And as anyone who’s played against a high-fight army also knows, losing those combats automatically can be completely deflating.

Which brings us back to the central question: are either of these swings worth a Might point? The second scenario (going from lower fight to higher fight) is by far the easier sell: if we can not only improve our odds of winning from four-in-ten to six-in-ten, that’s valuable right there; and to be able to simultaneously flip our opponent’s odds of winning (from six-in-ten to four-in-ten) is the icing on the cake. Going from a fifty-fifty chance to win to a sixty-forty chance to win is still an improvement, but since most heroes who can Strike can only call it 2-3 times a game (because they have a limited Might store), I think it’s fair to ask if that added increase is worth it. Against an enemy army leader with a single Fate point that you have trapped against a wall? Very tempted. If you only have a single hero with two attacks against a single troll? Maybe not… especially if your Hero has help coming next turn, and has Heroic Defense in his arsenal.

Our troll scenario also raises another important factor in the value of Heroic Strike. Getting higher fight than your opponent is great and all, but how likely are we to actually get a higher fight value when we Strike? And what happens if we tie or (heaven forbid) still end up with a lower fight value than our opponent?

The curse of the D6: Variations in Final Strike Values

Of all the probabilities that factor into Heroic Strike (and there are a lot), this one—which is arguably the most important—is also the most inflexible. At the start of the Fight Phase, any hero who wants to Strike has to declare that they’re doing so (usually spending a point of Might then and there, although sometimes none and sometimes two…). Then, when we get to that hero’s combat, we roll a single D6 before any duel dice are rolled (more on them shortly).

The importance of that single dice roll is pretty self-explanatory: whatever result that single D6 produces is the amount our Hero’s fight value goes up. The inflexibility is also pretty clear, if we stop and think about it. This die can’t be modified. It can’t be rerolled. And we don’t get any second-chances (unless we have multiple heroes who’ve called a Heroic Strike in the same fight… which I’ve done exactly once, against Sauron, just to make sure).

The wonderful (or terrible) thing about a single D6 is that we know exactly how probable each result is (assuming the die isn’t rigged, or cursed, of course): we have a 16.67% chance of getting a “1,” a 16.67% chance of getting a “6,” and a 16.67% chance of every other result between them. Yes, our “average” roll is a 3.5. But all that means is that half the time our result will be lower than that (1, 2, or 3) and half the time it will be higher (4, 5, or 6). There’s a lot of difference between increasing your fight value by “1” and by “6.”

Now there are some things that can reduce this variance. The biggest is that a hero’s maximum fight value is always capped at “10,” which means that heroes that start with higher fight values will have more “10” results, making that single D6 less of a gamble:

Heroic Strike Results Based on Starting Fight Value

		Starting Fight Value
		FV 4	FV 5	FV 6	FV 7	FV 8	FV 9
Heroic Strike Roll	*"1"*	5	6	7	8	9	10
	*"2"*	6	7	8	9	10	10
	*"3"*	7	8	9	10	10	10
	*"4"*	8	9	10	10	10	10
	*"5"*	9	10	10	10	10	10
	*"6"*	10	10	10	10	10	10

Another way to look at the same information from a different angle is through an “at least” chart. We discussed at least charts in ourseries on Heroic Strength, so if you want to read more about what they do and how to create your own, you can check out that series. This time, rather than an at least chart that shows our odds of scoring wounds, we’ll look at our chance of scoring “at least Fight Value X” when we call a Heroic Strike:

At Least Fight Value after Heroic Strike

		Starting Fight Value
		FV 4	FV 5	FV 6	FV 7	FV 8	FV 9
At Least Fight Value from Heroic Strike	*FV5+*	100.00%
	*FV6+*	83.33%	100.00%
	*FV7+*	66.67%	83.33%	100.00%
	*FV8+*	50.00%	66.67%	83.33%	100.00%
	*FV9+*	33.33%	50.00%	66.67%	83.33%	100.00%
	*FV10*	16.67%	33.33%	50.00%	66.67%	83.33%	100.00%

If our hero has a starting Fight Value of 4, they have only a small chance of reaching Fight 10 (16.67%). It can and will happen, but is probably not something we can (or should) count on. A much more realistic expectation is for our hero to end up somewhere between Fight 6 (five-of-six results on the Strike die) and Fight 8 (three-of-six results on the Strike die). By contrast, a hero who starts with Fight 8 has a five-in-six chance of reaching Fight 10—that’s what we call good odds.

The variance is also helped (usually) by the fact that we have a target Fight Value we’re trying to reach. If we have a Fight 6 hero (like Aragorn), and he’s confronted by a Fight 7 Mordor Troll, we know Aragorn needs to at least get to Fight 7—which he will 100% of the time, if he Strikes. Ideally, we’d like to get a higher Fight Value than 7, so that Aragorn will also win ties. If he Strikes, he can do this pretty reliably as well—83.33% of the time (i.e., as long as he doesn’t roll a “1” to Strike).

The problem is that this target isn’t always static. If our Mordor Troll is instead a Mordor Troll Chieftain, that Chieftain can also call a Heroic Strike—which means his Fight Value when all is said and done is going to be at least 8, and could be as high as 1, just like Aragorn’s. It’s this interplay of changing fight values where Heroic Strike tends to be the most important, and also the most unpredictable.

Double the curses! (Heroic Strike-Offs)

Now before we get into the mechanics of the Strike-Off, let’s just reiterate what we’ve said before: winning (or losing) a Strike-Off does not impact our odds of winning the fight as a whole. It impacts our odds of winning the fight only in six specific scenarios, where both sides roll exactly the same value on their highest duel roll—a “6” v. “6,” “5” v. “5,” and so on down the line:

		Target (FV: X) Duel Roll
		*"1"*	*"2"*	*"3"*	*"4"*	*"5"*	*"6"*
Challenger (FV: X) Duel Roll	*"1"*	Draw	Loss	Loss	Loss	Loss	Loss
	*"2"*	Win	Draw	Loss	Loss	Loss	Loss
	*"3"*	Win	Win	Draw	Loss	Loss	Loss
	*"4"*	Win	Win	Win	Draw	Loss	Loss
	*"5"*	Win	Win	Win	Win	Draw	Loss
	*"6"*	Win	Win	Win	Win	Win	Draw

As with our one-sided Strike, the goal is still to try to flip the chart. If our challenger wins the Strike-Off, he’ll end up with higher fight, and thus convert all those “draws” (if he’s tied fight) or “losses” (if he’s lower fight) to “wins.” If he loses the Strike-Off, the best-case scenario is that nothing changes (“draws” remain “draws” or “losses” stay “losses”). The worst-case scenario is that an opponent who was previously tied Fight now has higher Fight, and those “draws” become “losses”—disaster!

Challenger has Higher Fight Value (X+1)

Challenger has Lower Fight Value (X-1)

Target (FV: X) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X + 1) Duel Roll

"1"

Win

Loss

Challenger (FV: X-1) Duel Roll

"1"

Loss

"2"

Win

Loss

"2"

Win

Loss

"3"

Win

Loss

"3"

Win

Loss

"4"

Win

Loss

"4"

Win

Loss

"5"

Win

Loss

"5"

Win

Loss

"6"

Win

"6"

Win

Loss

The only difference between a one-sided Strike and a Strike-Off is that both sides of the duel are trying to win control of those six drawn duel rolls. They do this by each rolling a D6; those two D6 results are then added to each side’s starting Fight Value, and then the final Fight Values are compared to each other to determine who now has the highest Fight.

Because we’re comparing two D6s, we once again have thirty-six possible roll combinations. Assuming that each side starts with the same Fight Value, each of them will have exactly 15/36 outcomes in which they will “Win” (end the Strike-Off with the higher Fight Value), and 15/36 outcomes in which they “Lose” (end the Strike-Off with the lower Fight Value). This leaves 6/36 outcomes where both sides will “Draw” by rolling the same result on the Strike-Off, leaving them with the same final Fight Value as their opponent:

*Both sides Start with Equal Fight Value (X)*
		Target (FV: X) Duel Roll
		*"1"*	*"2"*	*"3"*	*"4"*	*"5"*	*"6"*
Challenger (FV: X) Duel Roll	*"1"*	Draw	Loss	Loss	Loss	Loss	Loss
	*"2"*	Win	Draw	Loss	Loss	Loss	Loss
	*"3"*	Win	Win	Draw	Loss	Loss	Loss
	*"4"*	Win	Win	Win	Draw	Loss	Loss
	*"5"*	Win	Win	Win	Win	Draw	Loss
	*"6"*	Win	Win	Win	Win	Win	Draw

If one opponent starts with a higher Fight Value than their opponent, the results become more compressed. The likelihood of “wins” goes up for that player, and the number of draws steadily decreases. The higher that opponent’s starting Fight Value, the more severe the disparity. A challenger who starts one Fight behind his opponent is only likely to roll higher in a Strike-Off in 10 of 36 scenarios (27.78%) and draw in only 5 of 36 (13.89%); his opponent, who starts with higher fight, will end up with the higher score in the remaining 21 scenarios (or 58.33% of the time). If the challenger starts off trailing his opponent by four fight, his chance of outright overcoming his opponent’s Strike score is just 1 in 36 (2.77%), and the chance of drawing is just 2 in 36 (5.56%), compared with his opponent’s 33 in 36 odds of winning outright (91.67%):

Challenger starts with -1 Fight Value (X-1 vs. X)

Challenger starts with -4 Fight Value (X-4 vs. X)

Target (FV: X) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X-1) Duel Roll

"1"

Loss

Challenger (FV: X-1) Duel Roll

"1"

Loss

"2"

Draw

Loss

"2"

Loss

"3"

Win

Draw

Loss

"3"

Loss

"4"

Win

Draw

Loss

"4"

Loss

"5"

Win

Draw

Loss

"5"

Draw

Loss

"6"

Win

Draw

Loss

"6"

Win

Draw

Loss

Fortunately for our challenger, there is a catch: because a hero’s Fight Value is capped at a maximum of 10, it becomes much easier for a challenger with lower fight value to catch and match the Strike roll of an opponent that starts with higher Fight. If Faramir Strikes against Sauron and rolls a “5” and adds that to his basic Fight of “5,” he goes to Fight “10.” If Sauron Strikes (at Fight 9), he will automatically go to Fight “10” regardless of what he rolls; which means if he also rolls a “5,” he ends up with exactly the same Fight Value as Faramir, even though he rolled a Strike Score of “14” (9+5).

This artificial cap on fight value means that our final charts (for Fight 5 and higher, at least) end up with far more draws than if we were just looking at the Strike rolls themselves:

Challenger starts with -1 Fight Value (X vs. X+1)

Challenger starts with -1 Fight Value (F5 vs. F6)

Target (FV: X+1) Strike Roll

Target (FV: 6) Strike Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X) Strike Roll

"1"

Loss

Challenger (FV: 5)

Strike Roll

"1"

Loss

"2"

Draw

Loss

"2"

Draw

Loss

"3"

Win

Draw

Loss

"3"

Win

Draw

Loss

"4"

Win

Draw

Loss

"4"

Win

Draw

Loss

"5"

Win

Draw

Loss

"5"

Win

Draw

"6"

Win

Draw

Loss

"6"

Win

Draw

The charts on the left show what the results would be if there were no max cap on Fight; the charts on the right show the same results, assuming our challenger is Fight 5, the target starts at Fight 6, and there’s a maximum Fight cap of 10. Our challenger goes from winning 10/36 (27.78%) to winning 9/36 (25%), losing 21/36 (58.33%) to losing 18/36 (50%), and drawing 5/36 (13.89%) to drawing 9/36 (25%). In other words, our lower-fight hero has gone from not winning any drawn combats to have a 50/50 chance of either winning those drawn combats outright (25%) or forcing a roll-off (25%). Not bad.

Now unfortunately for our challenger, the greater the difference between his starting Fight Value and his target’s starting Fight Value, the less likely he is to notch outright “wins” (meaning he’ll end with the undisputed highest Fight Value). This intuitively makes sense: the lower your starting fight value is than your opponent’s, the higher you’ll have to roll (and the lower he’ll have to roll) to beat him; conversely, the larger the lead your opponent starts out with, the lower he’ll have to roll (and the higher you’ll have to roll) to beat you. Fortunately, the cap at Fight 10 helps keep our Fight 10 hero in the running, replacing what would be outright losses with contested roll-offs for drawn combats:

Challenger starts with -2 Fight Value (X vs. X+2)

Challenger starts with -2 Fight Value (F5 vs. F7)

Target (FV: X+2) Strike Roll

Target (FV: 7) Strike Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X) Strike Roll

"1"

Loss

Challenger (FV: 5)

Strike Roll

"1"

Loss

"2"

Loss

"2"

Loss

"3"

Draw

Loss

"3"

Draw

Loss

"4"

Win

Draw

Loss

"4"

Win

Draw

Loss

"5"

Win

Draw

Loss

"5"

Win

Draw

"6"

Win

Draw

Loss

"6"

Win

Draw

Challenger starts with -3 Fight Value (X-3 vs. X)

Challenger starts with -3 Fight Value (F5 vs. F8)

Target (FV: X+3) Strike Roll

Target (FV: 8) Strike Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X) Strike Roll

"1"

Loss

Challenger (FV: 5)

Strike Roll

"1"

Loss

"2"

Loss

"2"

Loss

"3"

Loss

"3"

Loss

"4"

Draw

Loss

"4"

Draw

Loss

"5"

Win

Draw

Loss

"5"

Win

Draw

"6"

Win

Draw

Loss

"6"

Win

Draw

Challenger starts with -4 Fight Value (X-4 vs. X)

Challenger starts with -4 Fight Value (F5 vs. F9)

Target (FV: X+4) Strike Roll

Target (FV: 9) Strike Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: X) Strike Roll

"1"

Loss

Challenger (FV: 5)

Strike Roll

"1"

Loss

"2"

Loss

"2"

Loss

"3"

Loss

"3"

Loss

"4"

Loss

"4"

Loss

"5"

Draw

Loss

"5"

Draw

"6"

Win

Draw

Loss

"6"

Draw

It’s also worth noting that the Fight cap of 10 also alters the results of Strike-Offs where the heroes have the same starting fight value. Their odds of winning the Strike-Off doesn’t change (each fight has the same chance as their opponent does to “win” and to “lose” the Strike-Off), but the number of “draw” outcomes increases significantly the higher that starting Fight Value is. A Strike-Off between two Fight 4 heroes (which would be rare, but can happen) results in fifteen wins for each side, and six draws. That same Strike-Off between two Fight 6 heroes (which happens a ton) results in only 12 outright wins for our challenger, 12 outright wins for the target, and 12 draws (i.e., inconclusive results). And, of course, a Strike-Off between two Fight 9 combatants (Sauron vs. Gil-Galad) results in thirty-six inconclusive outcomes:

Both sides Start with Equal Fight Value (F4 vs. F4)

Both sides Start with Equal Fight Value (F5 vs. F5)

Target (FV: 5) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: 5) Duel Roll

"1"

Draw

Loss

Challenger (FV: 5) Duel Roll

"1"

Draw

Loss

"2"

Win

Draw

Loss

"2"

Win

Draw

Loss

"3"

Win

Draw

Loss

"3"

Win

Draw

Loss

"4"

Win

Draw

Loss

"4"

Win

Draw

Loss

"5"

Win

Draw

Loss

"5"

Win

Draw

"6"

Win

Draw

"6"

Win

Draw

Both sides Start with Equal Fight Value (F6 vs. F6)

Both sides Start with Equal Fight Value (F7 vs. F7)

Target (FV: 5) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: 5) Duel Roll

"1"

Draw

Loss

Challenger (FV: 5) Duel Roll

"1"

Draw

Loss

"2"

Win

Draw

Loss

"2"

Win

Draw

Loss

"3"

Win

Draw

Loss

"3"

Win

Draw

"4"

Win

Draw

"4"

Win

Draw

"5"

Win

Draw

"5"

Win

Draw

"6"

Win

Draw

"6"

Win

Draw

Both sides Start with Equal Fight Value (F8 vs. F8)

Both sides Start with Equal Fight Value (F9 vs. F9)

Target (FV: 5) Duel Roll

"1"

"2"

"3"

"4"

"5"

"6"

"1"

"2"

"3"

"4"

"5"

"6"

Challenger (FV: 5) Duel Roll

"1"

Draw

Loss

Challenger (FV: 5) Duel Roll

"1"

Draw

"2"

Win

Draw

"2"

Draw

"3"

Win

Draw

"3"

Draw

"4"

Win

Draw

"4"

Draw

"5"

Win

Draw

"5"

Draw

"6"

Win

Draw

"6"

Draw

So here’s the basic pattern:

· Strike-offs between heroes with low, tied starting fight (F4 or F5) tend to be crap shoots. It is literally a random roll-off, and the results are incredibly variable, but mostly decisive. With tied F6 or F7 (most heroes in the game who can Strike), there’s still a lot of variance, and the variance tends to result in more results that are indecisive (i.e., both sides reach Fight 10, and the winner of the duel hinges on a random roll-off). And at tied F8 or F9, a “Strike-Off” has very little variance that pretty much guarantees (or does guarantee, if you’re F9) an indecisive result.

· Strike-offs between low-fight (F4-5) and mid-fight (F6-7) heroes are heavily swung in favor of the mid-fight heroes (they will win or draw 2/3 of the time, or better). Forcing a Strike-Off between a low-fight (F4-5) and high-fight (F8-9) hero is of much greater benefit for the low-fight hero, who is unlikely to make their current situation much worse, and a decent chance at forcing a draw; there’s a lot less to be gained by the high-fight hero.

Still following this? Good. One last thing to talk about…

So after we have our final fight values… what next?

We started off this discussion by emphasizing that all a Strike-Off impacts is six of thirty-six possible results in a duel outcome: when our highest duel roll ties our opponent’s highest duel roll. The goal is to turn those six results into “wins” (or at least into “draws,” if we start the fight behind our opponent in Fight Value). But even if we’re successful, we still need to roll those duel dice, and we still need a highest duel roll that will at least match (if not exceed) our opponent’s highest duel roll.

Now at some level, our overall results chart isn’t going to change, because we always only ever compare two dice (our highest vs. our opponent’s highest), and those dice can only ever produce a result that is a whole number equal to or between “1” (no lower) and “6” (no higher). Which means, at the end of the day, there are only thirty-six duel roll outcomes, no matter how many dice we roll or how many modifiers we apply:

		Target (FV: X) Duel Roll
		*"1"*	*"2"*	*"3"*	*"4"*	*"5"*	*"6"*
Challenger (FV: X) Duel Roll	*"1"*	Draw	Loss	Loss	Loss	Loss	Loss
	*"2"*	Win	Draw	Loss	Loss	Loss	Loss
	*"3"*	Win	Win	Draw	Loss	Loss	Loss
	*"4"*	Win	Win	Win	Draw	Loss	Loss
	*"5"*	Win	Win	Win	Win	Draw	Loss
	*"6"*	Win	Win	Win	Win	Win	Draw

Unfortunately, this chart creates the impression that all of these outcomes are equally viable. And if we were simply fighting a duel that consists of one duel die on one duel die, they would be. But that only tends to happen when one warrior is fighting another (and to be honest, even single warrior-on-warrior combats are pretty rare, unless the battle lines have completely broken down). And it almost never happens for heroes, and almost never never when there’s a Strike-Off on the line because Heroic Strike heroes almost all have two or more attacks (Gandalf the Grey Striking against a Witch-King without the Crown of Morgul is perhaps the only scenario where you’d see that… at which point, someone’s doing it wrong). Outside of those very rare corner-case scenarios, we have at least two dice on each side, and more often three or four if not more.

Here’s why this matters: Heroic Strike only changes the outcome of six of the thirty-six possible scenarios. On its face that may sound insignificant (only 1-in-6? Why bother?). It may also sound drastically important (I can win a whole 16.67% more fights? Sign me up!). But unless each side is only rolling one duel die (which, as we said, will almost never happen in a Strike-Off), those 1-in-6 figures and 16.67% percentages aren’t actually the odds for those scenarios. Why? Because the more dice you roll, the higher your odds of rolling a high duel roll—and the less your odds of rolling a poor one.

Let’s start with the simplest example: If each side rolls a single d6 for its duel roll, and has no modifiers for that duel roll, each of our six outcomes (“1” through “6”) is equally likely, for each side:

		Player 1's Duel Pool			Player 2's Duel Pool
		1d6			1d6
Highest Duel Die Result	1 High	16.67%	Highest Duel Die Result	1 High	16.67%
	2 High	16.67%		2 High	16.67%
	3 High	16.67%		3 High	16.67%
	4 High	16.67%		4 High	16.67%
	5 High	16.67%		5 High	16.67%
	6 High	16.67%		6 High	16.67%

By the same token, because each participant’s duel outcomes are all of equal probability, there’s an equal probability that each of the thirty-six possible outcomes (1 v. 1, 2 v. 1, 2 v. 2, and so on) has an equal chance of coming about: 1-in-36, or 2.78%:

*Tied Fight, each side rolls one (1) Duel Die*
		Target (FV: X) Duel Roll (1d6)
		*"1"*	*"2"*	*"3"*	*"4"*	*"5"*	*"6"*
Challenger (FV: X) Duel Roll (1d6)	*"1"*	2.78%	2.78%	2.78%	2.78%	2.78%	2.78%
	*"2"*	2.78%	2.78%	2.78%	2.78%	2.78%	2.78%
	*"3"*	2.78%	2.78%	2.78%	2.78%	2.78%	2.78%
	*"4"*	2.78%	2.78%	2.78%	2.78%	2.78%	2.78%
	*"5"*	2.78%	2.78%	2.78%	2.78%	2.78%	2.78%
	*"6"*	2.78%	2.78%	2.78%	2.78%	2.78%	2.78%

Coincidentally (or perhaps not so coincidentally), this gives the Challenger a combined chance of 2.78% * 15 (41.7%) to win the fight, a 2.78% * 15 chance to lose the duel (41.7%), and a 2.78% * 6 chance (16.68%) chance to draw the duel. Which happen to be numbers we’ve seen before… a lot.

Now watch what happens when we add a single die to each side of the fight (two heroes with two attacks each). Because we’re now using only the highest result of those two dice, our odds of getting lower numbers change. Here’s each side’s chance of rolling a “1” to “6” highest (calculated by our good friends at anydice.com):

		Duel Dice Pool
		1d6	2d6
Highest Duel Die Result	1 High	16.67% (1/6)	2.78% (1/36)
	2 High	16.67% (1/6)	8.33% (3/36)
	3 High	16.67% (1/6)	13.89% (5/36)
	4 High	16.67% (1/6)	19.44% (7/36)
	5 High	16.67% (1/6)	25.00% (9/36)
	6 High	16.67% (1/6)	30.56% (11/36)

Notice how our odds of rolling only a “1 High” has plummeted, because now it only happens if we roll “double 1s” (which we have only a 1-in-36 chance of doing). If we roll a “1” plus any other outcome, we won’t roll a “1 High” (assuming no negative modifiers from banners, etc.). Our odds of rolling a “2 High” and a “3 High” have also dropped, although by less, and our odds of a “4 High,” “5 High,” and that all-important “6 High” have risen—in the case of the “6 High,” significantly. This is because of our thirty-six scenarios, eleven of them include at least one six, giving us a “6 high” (6:1, 6: 2, 6:3, 6:4, 6:5, 6:6, 5:6, 4:6, 3:6, 2:6, and 1:6). 11-of-36 translates into (you guessed it) 30.56%.

How do these changing odds impact our win-lose-draw odds? Like this:

*Tied Fight, each side rolls two (2) Duel Dice*
		Target (FV: X) Duel Roll (2d6)
		*"1"*	*"2"*	*"3"*	*"4"*	*"5"*	*"6"*
Challenger (FV: X) Duel Roll (2d6)	*"1"*	0.08%	0.23%	0.39%	0.54%	0.69%	0.85%
	*"2"*	0.23%	0.69%	1.16%	1.62%	2.08%	2.55%
	*"3"*	0.39%	1.16%	1.93%	2.70%	3.47%	4.24%
	*"4"*	0.54%	1.62%	2.70%	3.78%	4.86%	5.94%
	*"5"*	0.69%	2.08%	3.47%	4.86%	6.25%	7.64%
	*"6"*	0.85%	2.55%	4.24%	5.94%	7.64%	9.34%

Instead of an even distribution of likely outcomes, we now have outcomes that are heavily skewed towards higher duel rolls. Just 6.25% of our outcomes involve a “high” duel roll for both sides of three or less (compared to 25% of our outcomes with just one die). By contrast, more than half the likely outcomes involve both sides having a “high” of four or more (56.25%), and almost one-third of the time, each side will have at least a “5 High” on their duel roll (30.86%). Most importantly, those “drawn combats” only accounted for 16.67% of all results when each side was rolling just one die. But with each side rolling two? 22.07%, or almost one-in-four combats, end in a “draw” where fight value—and Heroic Strike—could provide a decisive edge. And of that percentage, nearly half (9.34%) arise when both sides roll a 6 High, when only Fight Value (or a roll-off) will determine the winner.

Those percentages only go up as we add more and more dice to the pool, each of which pushes up our chance of that “6 High,” to the point when not rolling a six high becomes the exception instead of the expected outcome:

		Duel Dice Pool
		1d6	2d6	3d6	4d6	5d6	6d6	7d6	8d6
Highest Duel Die Result	1 High	16.67%	2.78%	0.46%	0.08%	0.01%	<0.01%	<0.01%	<0.01%
	2 High	16.67%	8.33%	3.24%	1.16%	0.40%	0.14%	0.05%	0.02%
	3 High	16.67%	13.89%	8.80%	5.02%	2.71%	1.43%	0.74%	0.38%
	4 High	16.67%	19.44%	17.13%	13.50%	10.04%	7.22%	5.07%	3.51%
	5 High	16.67%	25.00%	28.24%	28.47%	27.02%	24.71%	22.06%	19.35%
	6 High	16.67%	30.56%	42.13%	51.77%	59.81%	66.51%	72.09%	76.74%

Now I won’t do the full duel roll probability charts for all those outcomes (mostly because by the time we get even to four dice, we’re dealing with an absurdly large denominator of more than a million). Suffice it to say, once we start rolling three attack dice or more against three attack dice or more, the odds of both sides getting at least one “6” high are pretty good (about 20% and higher). As that percentage increases, so does the benefit of having the higher Fight Value (and thus, Heroic Strike… probably).

Due to an arbitrary Blogger cut-off, this is the abrubt end to this discussion. Click here for Part V (our Faramir vs. Eomer case study).

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Monday, June 3, 2019

Is Heroic Strike Good? (Part IV: The Probabilities (Plural) of Heroic Strike)

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