Is Heroic Strike Good? (Part I: Fight Value and the Anatomy of a Duel)

Having finished a lengthy discussion of an oft-sneered-at heroic action that nobody uses (Heroic Strength), I figured it'd be fun to examine a heroic action at the other end of the spectrum that everyone values: Heroic Strike.

One of the first things new players learn in our game is the importance of fight value. If your fight value is higher than your opponent's, you can auto-win a duel if you roll a six (or can obtain a six through other shenanigans, usually involving banners and/or character-specific special rules and/or Might). You also win if you happen to roll higher on any die than your opponent, or if you roll the same as your opponent. If your fight value is tied with your opponent's, you only win outright if you roll higher than your opponent, and if you tie (or both roll a "6"), then you go to the dreaded roll-off where your fate is (literally) decided by (even more) random chance (than usual). And if your fight value is lower than your opponent's ... well, let's just say it can get very frustrating very quickly.

Most heroes boast a fight value of 5 or 6 (there are notable exceptions on both ends, of course). But as any seasoned player knows, there's a massive difference between FV5 and FV6. FV5 is usually good enough for dealing with regular troops, but not all elites, most heroes (even at low hero tiers), and almost all monsters. FV6 is good enough for dealing with most elite troops (but not all) and most low-end heroes, but not mid-to-high tier heroes, and again, most monsters will still pose a problem.

Enter Heroic Strike: the pinnacle of rule-breaking, at the cost of a Might point it allows our lowly hero to become as adept at fighting at the Balrog himself (although usually "as good as Gil-Galad" is good enough). Game-winning when it works, game-breaking when it doesn't, and usually game-slogging all the time, it is largely considered the most important heroic action to have in a hero's arsenal (now that not all heroes can call all heroic actions).

But are we really sure it's good?

(Controversial maths ahead...)

The (Rules) Anatomy of a Duel

Before we delve into Heroic Strike, we need to understand bit about how fights work, and the probabilities involved. Our principal guide is pp. 43-44 and 47 of the Middle Earth Strategy Battle Game Rules Manual, though there are other pages that we'll refer to as well. Page 43 provides us with a handy flow chart for resolving a basic fight. The first six constitute the "duel," which determines who wins the overall fight:

1. First, we gather the number of six-sided dice we need for the Duel roll, making sure we use different colors and types of six-sided dice for the different types of models we have with different modifiers (more on that in a moment).
2. Once the dice are gathered (but before they are rolled--this is important) each player declares if they are using any special strikes or two-handed attacks. There are loads of special rules about special strikes on page 87 of the Rules Manual, which we'll discuss later, but the basic gist for winning a duel is that one-hand-a-half special strikes will reduce our model's fight value; one special strike may reduce our opponent's future fight value; and using a two-handed weapon may also reduce the results we roll on our dice.
3. After we declare all this, we roll all of our dice together. Our opponent usually does the same with his dice at the same time, although not always.
4. Once all our dice are rolled, we apply any modifiers we have to our dice rolls to any dice we appropriately designated as needing modifiers in step #2. There are only ever bad modifiers to a duel roll, so if we're doing anything at this stage, it's making our dice results worse, not better. Our opponent does this as well (again, usually at the same time), in which case his bad modifiers are good for us (thereby breaking the aforementioned absolute statement that there are only ever bad modifiers to a duel roll).
5. We then use any re-rolls or special rules that our models might have. This is where banners come in (YAY!). Our opponent uses his banners here, too (BOO!). 
6. Finally, everyone can use might to modify any dice that are of the proper color/ size/ etc. designated in step #2 as dice that could be modified with Might (see, step #2 is important). Page 66 has important additional instructions for this step if both players have might in the same duel. While either player can declare they're using might at any time, the impetus is on the player whose hero is currently losing: that player has the first opportunity to use Might. If he does, the opposing player may use Might to win again (or force a tie), and the players go back and forth until someone runs out of might (or someone can no longer win or force a tie). Page 66 also clarifies that heroes can only use Might to modify their own dice (not the dice of their friends), so again: designation in step #2 is important. And, for what it's worth, page 67 explicitly allows heroes to use might to improve their score (dice rolls) in a duel. In case you wanted to make absolutely sure you're playing by the book.

Generally speaking, the fight is won by whoever has the highest dice score (p. 43). But if there's a tie ("drawn combat"), the first tie-breaker is fight value, which means the model with the higher fight value will win on a tied dice score (p. 44). If neither model has the higher fight value, we go to the dreaded 50-50 roll-off (p. 44), which may not in fact be a 50-50 roll-off depending on the wargear on our own models and/or our opponent's models (p. 85).

The final step in a fight is that the winner of the duel (whether by dice roll, fight value, or roll-off) gets to make strikes at the loser (provided the winner wasn't prone, cursed, or otherwise incapable of making strikes). But we're not interested in that here (check out Appendix A and Appendix B on Heroic Strength if you want those maths).

Now, if you've made it through all the above without your eyes glossing over, you will have noticed that most of this is dependent on random luck: we have set modifiers (always bad for us, unless they're our opponent's modifiers), set fight values (which may be good or bad), and a set number of dice (again, could be good or bad). Everything else is either a random roll or a random re-roll or a random roll-off.

But don't let that discourage you! Even though all of this is random, there are some rules that govern our randomness. They aren't guarantees (mostly), and they won't save us from the proverbial "bad roll," but they can help us plan and strategize--both in list-building and on the table-top--in ways that will improve our odds of winning duels. This helps us both on offense (to kill enemy models, or at least to kill more enemy models, we usually have to win duels) and on defense (if our enemy can't win duels, they usually can't kill us, or at least can't kill us more).

The (Mathematical) Anatomy of a Duel

We'll be looking at the probabilities of duels from multiple angles, in both this write-up and the next. We're starting here with the three basic probability rules for duels:

RULE #1: 

In an even duel, both sides have an even chance of winning.

By "even duel," we mean a fight when everyone's duel conditions are the same. Think a single warrior of Minas Tirith vs. a single Easterling warrior: both have equal attacks (one) and fight value (three), which means whoever rolls higher on their attack die is going to win, whoever rolls lower on their attack die is going to lose, and if they draw we go to a 50-50 roll-off. They might as well be fighting a mirror match. Across the board, everyone's odds are exactly the same. See?

 Even Duel Outcomes: x1 Warrior of Minas Tirith vs. x1 Easterling Warrior

	Easterling “1”	Easterling “2”	Easterling “3”	Easterling “4”	Easterling “5”	Easterling “6”
WOMT “1”	DRAWN	Easterling	Easterling	Easterling	Easterling	Easterling
WOMT “2”	WOMT	DRAWN	Easterling	Easterling	Easterling	Easterling
WOMT “3”	WOMT	WOMT	DRAWN	Easterling	Easterling	Easterling
WOMT “4”	WOMT	WOMT	WOMT	DRAWN	Easterling	Easterling
WOMT “5”	WOMT	WOMT	WOMT	WOMT	DRAWN	Easterling
WOMT “6”	WOMT	WOMT	WOMT	WOMT	WOMT	DRAWN

This is a simple chart showing the thirty-six possible outcomes if each player rolls one six-sided dice. Each die can roll a single value between "1" and "6." If the Easterling's value is higher, he will win the duel and the Warrior of Minas Tirith will lose. This happens 15 times out of 36, or approximately 41.67% of the time. If the Warrior of Minas Tirith's value is higher, he will win the duel and the Easterling will lose. This also happens 15 times out of 36, or approximately 41.67% of the time. The remaining six times, both combatants roll the same value on their dice, resulting in a drawn (tied) combat. This then goes to a 50-50 roll-off, where the player with priority rolls a single six-sided die. The Easterling will win on a roll of a 1, 2, or 3 (50% of the time) and the Warrior of Minas Tirith will win on the roll of a 4, 5, or 6 (50% of the time). Smashed together, each warrior's odds of winning the duel roll is 15/36, and their odds of winning the roll-off are 3/6, resulting in an overall probability of winning 18/36 fights, or exactly 50%.

While 50% isn't a bad percentage (because it happens as often as it doesn't), it's also not a reliable percentage (because it doesn't happen as often as it does). If our strategy comes down to 50/50 rolls, we should expect to win only half the time, and expect to lose half the time. In other words, we should settle for 50/50 roll-offs only when that's the best we can do. And usually, it isn't.

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.

Now let's say that instead of a Warrior of Minas Tirith fighting this Easterling warrior, we now have a ranger of Gondor, who is FV4 instead of FV3. The models' attack values haven't changed (we're still rolling the same number of dice), so both models are stuck with what they roll. So the only thing that changes probability-wise is that our ranger now wins all of the six "drawn" combats:

Duel Outcomes: x1 Ranger of Gondor vs. x1 Easterling Warrior

	Easterling “1”	Easterling “2”	Easterling “3”	Easterling “4”	Easterling “5”	Easterling “6”
Ranger “1”	Ranger	Easterling	Easterling	Easterling	Easterling	Easterling
Ranger “2”	Ranger	Ranger	Easterling	Easterling	Easterling	Easterling
Ranger “3”	Ranger	Ranger	Ranger	Easterling	Easterling	Easterling
Ranger “4”	Ranger	Ranger	Ranger	Ranger	Easterling	Easterling
Ranger “5”	Ranger	Ranger	Ranger	Ranger	Ranger	Easterling
Ranger “6”	Ranger	Ranger	Ranger	Ranger	Ranger	Ranger

Our Easterling Warrior is still winning all the fights where he has a higher dice roll (15/36, 41.67%), while the Ranger now wins all fights where he has the higher roll (15/36, 41.67%) and all fights where the dice rolls are drawn (6/36, 16.67%), for a grand total odds of 21/36, or 58.33%. This nets our Ranger an improvement of +8.33 percentage points in his odds to win the duel, and a corresponding loss of -8.33 percentage points for our Easterling warrior. This is a definite improvement, although it's worth noting that our low-fight Easterling warrior still has a 15/36 (41.67%) chance of winning the fight. And while this particular Easterling warrior has FV3, he'd have the same chances of winning (41.67%) if he had FV 2 or even FV1, because improved fight value only comes into play when the highest dice rolls are tied (6/36 times). By the same token, if our opponent has FV3, and our models have FV5, we get no greater benefit against our opponent than if our models had FV4 (or FV9, for that matter). Higher Fight Value is binary: we either have more of it than our opponent, or we don't.

RULE #3:

Adding more dice (on its own) is more likely to help us win fights than improved Fight Value (on its own).

Once we start adding multiple dice, we escalate quickly from two-dimensional models to three-dimensional models (and maybe more), which don't translate well to graphs (for starters, we'd need to graph 216 potential results for just three six-sided dice--6x6x6--which is way more time consuming to make or read than it's worth).

From this point on, I'm going to be using Jeremy Hunthor's fabulous "LOTR Maths" calculator, which is available here (I encourage you to listen to his math-related episodes on the Green Dragon Podcast as well for a digger deep-dive into how it works). I'll go ahead and run our basic query above: one Warrior of Minas Tirith (FV3) vs. one Easterling Warrior (FV3):

x1 Warrior of Minas Tirith vs. x1 Easterling Warrior

	Win %	Kill %	Death %
Good	50.00%	8.33%	8.33%
Evil	50.00%	8.33%	8.33%

We have, unsurprisingly, a 50% chance for both good and evil to win this fight (a combination of winning outright through a higher dice roll and their odds of winning the roll-off). The chart also usefully tells us the chance that each model will kill its opponent (which I have labeled "Kill %) and the chance each model will itself be killed (which I have labeled "Death %"). A couple things about these percentages:

• Bear in mind that both percentages reflect the probability that our model will kill or be killed by its opponent before the fight begins, not our model's chance of killing its opponent assuming it has won the fight (which would be exactly twice 8.33%--16.67%--because S3 wounds D6 on a "6", and thus wounds on 1-roll-in-6, or 16.67%). While we can still look at individual factors if we wish, the beauty of this calculator is that it is going to allow us to look at holistic probabilities of killing/dying (factoring in everything that can happen in a fight, including who wins the duel), which I find provides much more interesting information.
• Part and parcel with the calculator factoring in our kill/death probabilities before the fight begins means our odds of killing are always going to be hard-capped by our model's chance of winning the duel (because our model can't kill its opponent's in a fight if we don't first win the duel). In this case, the maximum kill % any model could obtain is 50%, because that's each model's chance of winning the duel roll and making strikes. Here, both models have an 8.33% chance of killing their opponent, which looks bad at first until you factor in that it's an 8.33% chance out of a maximum of 50% (or a 1-in-6 chance if we win the fight). It's still not great, but it's not bad either. Conversely, while we might feel confident in our chance of surviving if our opponent has only an 8.33% chance of killing us, remember that his odds are essentially doubled as well.

Now let's see what happens if we add a second Easterling Warrior with a pike to the fight:

x1 Warrior of Minas Tirith
vs.
x1 Easterling Warrior + x1 Easterling Warrior (Pike)

	Duel Win %	Kill %	Death %
Good	33.80%	5.63%	20.23%
Evil	66.20%	20.23%	5.63%

All the models still have FV3, which means they're still splitting drawn combats 50/50. The only change is that the side with two models (Evil) is rolling two dice instead of one. As we saw in Appendix B, adding additional dice affects our expected and at-least distributions, by making it more likely we will roll a high number to win the duel outright:  

Expected distribution of Outcomes on 1 attack die

	6s	5+	4+	3+
0	83.33 %	66.67 %	50.00 %	33.33 %
1	16.67 %	33.33 %	50.00 %	66.67 %

Expected distribution of Outcomes on 2 attack dice

	6s	5+	4+	3+
0	69.44 %	44.44 %	25.00 %	11.11 %
1	27.78 %	44.44 %	50.00 %	44.44 %
2	2.78 %	11.11 %	25.00 %	44.44 %

At Least distribution of Outcomes on 2 attack dice

	6s	5+	4+	3+
0+	100.00 %	100.00 %	100.00 %	100.00 %
1+	30.56 %	55.56 %	75.00 %	88.89 %
2+	2.78 %	11.11 %	25.00 %	44.44 %

Our Warrior of Minas Tirith has a 16.67% chance of rolling a "6" on his single die; our Easterlings have a 27.78% chance of rolling a "6" on one of their two dice, and an additional 2.78% chance of rolling two sixes, for a combined (at least) total of a 30.56% chance of rolling at least one "6." If our WOMT manages to roll that six, he'll force a roll-off. But he's almost half as likely to do so as the Easterlings are. What if no one rolls a "6"? Well, our WOMT has a lower chance of rolling a 5+ (33.33%) than the Easterlings do (55.56%). And the trend continues for every number, across the board.

As a result, our WOMT's overall chance to win the duel has dropped from 50% to 33.80%, while our Easterlings' chance to win has climbed from 50% to 66.2%. Notice as well that our WOMT's chance of killing (and, conversely, our Easterling's chance of dying) has taken a slight tumble from 8.33% to 5.63% (-2.3 percentage points, out of a maximum of 33.80%, or roughly 1-in-6), but our Easterling's chance of killing (and, conversely, our WOMT's chance of dying) has spiked all the way to 20.23% (+11.9 percentage points, out of a maximum of 66.23%, or roughly 1-in-3). Not only do more dice improve our chance of winning the fight, but they also improve our chance of wounding (as we saw in Appendix A and B).

Note as well that the resulting shift (+/- 16.2 percentage points) is almost twice as large as the jump we saw from the improved fight value of a single Ranger of Gondor (+/- 8.33%), which suggests that increasing our number of dice is more valuable than improving our fight value. But what happens if we replace our WOMT with a ranger? Well... it's kind of a mixed bag:

x1 Ranger of Gondor (FV4)

vs.
x1 Easterling Warrior (FV3) + x1 Easterling Warrior (FV3, Pike)

	Duel Win %	Kill %	Death %
Good	42.13%	7.02%	32.15%
Evil	57.87%	32.15%	7.02%

Looking solely at the duel roll, our Ranger's odds have actually improved. Instead of winning just 33.8% of the time like our WOMT, he's now winning 42.13% of the time because he's back to winning drawn combats (+8.33 percentage points... funnily enough, exactly the same boost he got against a single Easterling warrior for having fight value). But the Easterlings are still winning more than half the fights (57.87%) because their odds of rolling a given high number (6, 5, etc.) is still higher than the Ranger's odds of rolling that number because the expected and at-least distributions on one die vs. two dice haven't changed (two dice have better odds of rolling at least one high number than one die has).

To make matters worse, our Ranger is now likely to die 32.15% of the time (out of a maximum of 57.87%, or more than half the time) and only has a 7% chance of killing an Easterling out of 42% (or 1-in-6). In other words, this is not a winning position for our ranger, even with the higher fight value. Now to be fair, there are other factors at play here than just fight value (most notably the Ranger's defense of 4, which means the Easterlings are wounding him on "5s" instead of "6s"). But we can adjust for that by replacing our Ranger with an Osgiliath Veteran within 3" of Boromir or Faramir (who is FV4, D6):

x1 Osgiliath Veteran (FV4)

vs.

x1 Easterling Warrior (FV3) + x1 Easterling Warrior (FV3, Pike)

	Duel Win %	Kill %	Death %
Good	42.13%	7.02%	17.68%
Evil	57.87%	17.68%	7.02%

While the kill odds go down significantly, the Easterlings are still significantly more likely to wound the Osgiliath Vet before the fight even begins than he is to wound them (17.68% vs. 7.02%), because their additional die makes more of a difference in winning the duel roll than the Osgiliath vet's higher fight value does. Conversely, if the Osgiliath Veteran is supported by a WOMT with spear, the odds dramatically reverse yet again:

x1 Osgiliath Veteran (FV4) + x1 WOMT (FV3, Spear)

vs.

x1 Easterling Warrior (FV3) + x1 Easterling Warrior (FV3, Pike)

	Duel Win %	Kill %	Death %
Good	61.03%	18.65%	11.91%
Evil	38.97%	11.91%	18.65%

Make no mistake: fight value contributes to our odds of wining the fight. Both sides are now more likely to kill or be killed than when just one model was involved on each side (8.33%), but good has a decided advantage thanks to their higher fight value (FV4 vs. FV3), because (a) they now have the same expected and at least probabilities as evil, and (b) they are now winning drawn combats. If we replace our Easterling pikeman with a Black Dragon pikeman (FV4), we end up essentially with a drawn combat again, albeit a more deadly one:

x1 Osgiliath Veteran (FV4) + x1 WOMT (FV3, Spear)

vs.

x1 Easterling Warrior (FV3) + x1 Black Dragon Warrior (FV4, Pike)  

	Duel Win %	Kill %	Death %
Good	50.00%	15.28%	15.28%
Evil	50.00%	15.28%	15.28%

Conversely, if the Black Dragon drops out, giving Gondor the advantage both on dice and fight value, the odds tip dramatically in Gondor's favor:

x1 Osgiliath Veteran (FV4) + x1 WOMT (FV3, Spear)

vs.

x1 Easterling Warrior (FV3) 

	Duel Win %	Kill %	Death %
Good	74.54%	22.78%	4.24%
Evil	25.46%	4.24%	22.78%

To summarize: fight value is important for winning fights, but it's not as important as adding additional dice (at least not when we're dealing with small numbers of dice to begin with). This is why multiple attack models of any fight value (F3 Riders of Rohan/Orc Captains, S2 Goblin Captains) can be tough for normal troops to deal with, why shield walls backed by spears and pike formations are difficult to move, even for elves, and why multiple attack models with high fight value (Uruk-Hai Ferals and Berserkers, Elf Cavalry, and most heroes, especially when mounted) are so dangerous.

This wraps up the three core rules for winning duels: even duels are 50/50 duels, having a higher fight value than our opponent improves our odds of winning the duel, and having more dice than our opponent gives us an even greater improvement in our odds of winning the duel (and usually wounding as well).

But what if you have no multiple-dice models or can't take spears/pikes? Are you doomed to losing? What about negative modifiers? And is there any point where adding more dice suffer diminishing returns (and/or end up making less of a difference than improved fight value)?

More to come.