Is Heroic Strike Good? (Part V: A Case Study in Miniature)

Photo Credit: Dork Lords

This is the second-part of our mathematical write-up on Heroic Strike itself (cut for length because Blogger didn't like the full feature in a single post--boo!). If you missed the first part, you can find it here. To summarize, we looked at the basic components of Heroic Strike. Here are the high points:

• Heroic Strike, like fight value boosts in general, only improve our odds to win drawn combats, and only drawn combats;
• The Strike itself has a lot of variance, since it hinges on a single, unmodifiable D6;
• That variance can be reduced somewhat by the overall Fight Value cap of "10," and can also be reduced by having higher Fight Value to start with; and
• Even though Heroic Strike only impacts the six drawn combat outcomes (of thirty-six total combat outcomes), those scenarios are among the most probable outcomes once we start rolling two or more duel dice, and only become more probable the more dice we add to the duel rolls.

Now, to synthesize all of this, let's take a look at a practical case study between two opponents who share almost identical stat profiles, except for their Attacks. How might calling Heroic Strike (or not calling it) impact their odds to win the fight? And are there other considerations, beyond the mere probabilities of winning and losing, that caution for (or against) spending Might to Strike?

(Maths ahead, as the rubber hits the road.)

Heroic Strike in (Miniature) Action: Faramir vs. Eomer

Suppose a hero with F5, two attacks (Faramir, let’s say) is up against a hero with F5, three attacks (Eomer), in one-on-one combat. Faramir is thinking about trying to Strike up to get the fight advantage. He knows from the chart above that he has a 30.56% chance of rolling a 6 high and thus winning the fight, so long as he gets to F6 or above (which he will regardless of what he rolls). The risk is that Eomer still has Might, and can strike himself, in which case he will have a 42.13% chance of rolling a 6 high, to Faramir’s 30.56% (because Eomer has +1 Attack). Faramir also knows that since both he and Eomer will start their Strike-Off at Fight 5, the winner of that Strike-Off is (literally) a toss-up. Should Faramir force the issue by Striking, or not?

Let’s start by figuring out what the current probable outcomes are for highest duel die if nothing changes: 

		Duel Dice Pool
		2d6	3d6
Highest Duel                  Die Result	1 High	2.78% (1/36)	0.46% 1/216
	2 High	8.33% (3/36)	3.24% 7/216
	3 High	13.89% (5/36)	8.80% 19/216
	4 High	19.44% (7/36)	17.13% 37/216
	5 High	25.00% (9/36)	28.24% 61/216
	6 High	30.56% (11/36)	42.13% 91/216

Faramir, with only two attack dice, has a 75% chance of getting at least a 4 high, which sounds good. Until we compare it to Eomer’s odds, with that extra attack die, which is almost 90% (87.50%), with a 42% chance to roll at least one six compared to Faramir’s 30% chance. With Eomer’s three attack dice pushing the odds even more towards higher duel rolls, we see even more of a concentration towards the top rolls, but with more of the weight shifted to Eomer’s side of the field:

Highest Duel Roll: Faramir vs. Eomer
		Eomer (FV: 5) Duel Roll (3d6)
		"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 5) Duel Roll (2d6)	"1"	0.01%	0.09%	0.24%	0.48%	0.78%	1.17%
	"2"	0.04%	0.27%	0.73%	1.43%	2.35%	3.51%
	"3"	0.06%	0.45%	1.22%	2.38%	3.92%	5.85%
	"4"	0.09%	0.63%	1.71%	3.33%	5.49%	8.19%
	"5"	0.12%	0.81%	2.20%	4.28%	7.06%	10.53%
	"6"	0.14%	0.99%	2.69%	5.23%	8.63%	12.87%

Even though the total number of “win” scenarios is equally split between Eomer and Faramir (15/36 wins, 15/36 losses, and 6/36 draws), the odds of one of Eomer’s fifteen win scenarios happening (a combined 47.16%) is almost twice the odds of Faramir’s fifteen win scenarios (28.07%). In fact, Faramir’s fifteen win scenarios are only slightly more probable on the whole than the six draw scenarios (which combine for 24.77% of all expected outcomes). The most common scenario of all thirty-six scenarios is that both Faramir and Eomer will each roll a “6” (12.87%). And the odds of at least one side having a 6 high is massive (a combined 59.81%, essentially six of every ten fights).

Photo Credit: The Fandomentals

What this means is that while Faramir looks to be about-even with Eomer stat-wise (he has multiple dice and the same Fight Value as Eomer to start with), he is actually at a significant disadvantage before any dice are rolled. His odds of winning the fight outright are actually not that great (28%), and his odds of losing the fight outright are quite a bit higher (47%). That leaves a roughly one-in-four chance for a drawn combat, which would result in a random roll-off that Faramir is likely to win 50% of the time, and lose 50% of the time. All told, Faramir is looking at something like a 40% chance of winning the fight to Eomer’s roughly 60% win odds, if nothing changes.

Now let’s compare that to what happens if Faramir Strikes. If Eomer has already called a Heroic Action, or is out of Might, this is a really great option for Faramir, as any Strike result (“1” to “6”) will give Faramir the edge in Fight Value over Eomer—and with it, transform the roughly 25% of scenarios tied up in “drawn” combats into outright “wins” for Faramir: 

Highest Duel Roll: Faramir (F6) vs. Eomer (F5)
		Eomer (FV: 5) Duel Roll (3d6)
		"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 6) Duel Roll (2d6)	"1"	0.01%	0.09%	0.24%	0.48%	0.78%	1.17%
	"2"	0.04%	0.27%	0.73%	1.43%	2.35%	3.51%
	"3"	0.06%	0.45%	1.22%	2.38%	3.92%	5.85%
	"4"	0.09%	0.63%	1.71%	3.33%	5.49%	8.19%
	"5"	0.12%	0.81%	2.20%	4.28%	7.06%	10.53%
	"6"	0.14%	0.99%	2.69%	5.23%	8.63%	12.87%

The result would still be a close match, but one that slightly favors Faramir: 52.84% odds to win (up from 28.07%), compared to Eomer’s unchanged 47.16% (as Eomer still wins outright every scenario where he rolls a higher duel die than Faramir).

What if Eomer has Might and can Strike if he wishes? Given how much the original odds slant in Eomer’s favor, it’s probably still in Faramir’s best interest to Strike if he can. If Faramir wins the Strike-off, the jump from 28% win odds to 52% win odds is a significant increase, and would wrest control over the probabilities from Eomer. While there’s no guarantee that Faramir would win a Strike-Off against Eomer, Fight 5 is right in the sweet spot where the Strike-Off is at least likely to be decisive one way or another. Of the thirty-six potential Strike-Off combinations, only eight end in draws; the other twenty-eight outcomes will all be decisive. And unlike the duel roll chart, Faramir actually has the same odds of winning the Strike-Off as Eomer does (instead of starting way behind): 14-to-14.

Both sides Start with Equal Fight Value (F5 vs. F5)
		Target (FV: 5) Duel Roll
		"1"	"2"	"3"	"4"	"5"	"6"
Challenger (FV: 5) 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	Draw
	"6"	Win	Win	Win	Win	Draw	Draw

All of this may actually make Eomer reluctant to Strike at all; after all, the odds favor him significantly as-is, so why would he spend Might on a risky gambit that may end up with Faramir having the edge?

The answer comes down to percentages: while none of the thirty-six outcomes is remarkably likely to occur, there are four scenarios that, collectively, are far more likely to occur than the rest: “5 vs. 5,” “6 vs. 5,” “6 vs. 6,” and “5 vs. 6.” Together, there is almost a forty-percent chance (39.09%) that at least one of those four scenarios will be our final outcome; Faramir wins three of them outright. And if we expand those scenarios to include the five others where both sides roll at least one “4 highest” or better (4 vs. 4, 5 vs. 4, 6 vs. 4, 4 vs. 5, and 4 vs. 6), there is a nearly two-thirds chance that one of those nine outcomes will be our final result (65.63%), with Faramir winning six of them (41.41% combined odds, out of 100%) to Eomer’s three (just 24.22% combined odds, out of 100%). In other words, Faramir’s edge in fight value increases the percentage of possible outcomes where he’ll win outright—and increases them drastically.

This seems to clinch the deal, doesn’t it? If Faramir Strikes and Eomer doesn’t contest, Eomer is giving up a major swing in Faramir’s favor. But there’s one final thing to consider.

(Okay, I lied… one final final thing to think about… and it’s a big one).

Shifting Probabilities (again) by Boosting Dice

What if instead of engaging in a Strike-Off (which Eomer may win, or may lose), Eomer decides to save that Might point (he only has three of them after all, at least most of the time) to modify his highest duel die (in case neither side rolls a “6”). Doing so should, in theory at least, increase the number of scenarios where he has at least one “6” to win the duel (a natural six or a five that is Mighted to a six)—or a “5” or a “4”—which (again, in theory) should improve the probability of a scenario where he wins outright. Does it work out that way?

Yes. Yes it does.

Highest Duel Roll: Faramir (F6) vs. Eomer (F5 + Might)
		Eomer (FV: 5) Duel Roll (3d6 + Might)
		"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 6) Duel Roll (2d6)	"1"		0.01%	0.09%	0.24%	0.48%	1.95%
	"2"		0.04%	0.27%	0.73%	1.43%	5.86%
	"3"		0.06%	0.45%	1.22%	2.38%	9.77%
	"4"		0.09%	0.63%	1.71%	3.33%	13.68%
	"5"		0.12%	0.81%	2.20%	4.28%	17.59%
	"6"		0.14%	0.99%	2.69%	5.23%	21.50%

Note that Faramir still has the higher Fight Value (6+ vs. Eomer’s Fight 5), and thus is still winning our drawn scenarios (a final duel value of “6” vs. “6,” “5” vs. “5,” and so on). What’s changed is that because Eomer is spending Might to improve his natural rolls (“boosting”), his odds of rolling a “6 high” can be combined with his odds of rolling a “5 high” (which would then become a “6,” thanks to a point of Might) to give him a final odd of ending with a “6 high” at the end of the duel. By the same token, his odds of rolling a “4 high” now become his odds of getting a “5 high” (improved again by Might), his odds of rolling a “3 high” become his odds of getting a “4 high,” and so on. This simple action (“boosting”) happens a ton in our game, often at the most critical junctures.

What this does is push Eomer’s dice—which were already inclined to have higher results than Faramir’s, since Eomer had more dice to begin with—even higher than they were before (because his 6s stay 6s, 5s become 6s, 4s become 5s). As such, Eomer is stealing outcomes out of his no-win categories (say, a 3-3 tie) and pushing them into a category when he comes out on-top (a 3-3 loss becomes a 3-4 victory). In other words, he’s draining victories from Faramir, and reappropriating them as victories for himself (which become losses for Faramir).

Photo Credit: fanpop.com

Just to illustrate this, notice that while Faramir technically still wins drawn combats, six of our original thirty-six possible outcomes (Eomer rolls a “1” vs. Faramir rolling a “1” to “6”) are completely gone now. That leaves Faramir with fifteen scenarios where he’ll score an outright victory—including that all-important “6 vs. 6” auto-win scenario, which will now happen about one-fifth of the time (21.50%). But the loss of those “Eomer rolls 1s” scenarios—all of which were Faramir victories, by the way) gets Eomer back in the game.

Now to be fair, those outcomes weren’t all that common before we started boosting (they accounted for less than half a percent (0.46%, technically) of Faramir’s 52.84% odds to win the fight as a whole. Faramir’s actually seen a small increase in the odds of a “drawn” high roll, as well (where his Fight Value advantage would earn him an outright victory): that now happens 27.98% of the time (up slightly, from 24.77%). And there’s been absolutely no change to the odds that Faramir will roll a “6” (30.56% of the time), and he still wins every time he does.

And yet, if you were to add up the combined probability of Faramir’s fifteen victory scenarios, and compare them with the combined probability of one of Eomer’s fifteen victory scenarios, it is Eomer who has the clear advantage: 59.05% to Faramir’s 40.95%. Even though Faramir is still winning ties, and every scenario where he rolls a “6,” and the most likely outcome of them all: 6 vs. 6. So where is he losing out?

Where Faramir is suffering losses are in those scenarios where neither side ends up with a that “6” (the ones marked in yellow): 

Highest Duel Roll: Faramir vs. Eomer									Highest Duel Roll: Faramir (F6) vs. Eomer (F5)
		Eomer (FV: 5) Duel Roll (3d6)									Eomer (FV: 5) Duel Roll (3d6 + Might)
		"1"	"2"	"3"	"4"	"5"	"6"				"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 5) Duel Roll (2d6)	"1"	0.01%	0.09%	0.24%	0.48%	0.78%	1.17%		Faramir (FV: 6) Duel Roll (2d6)	"1"		0.01%	0.09%	0.24%	0.48%	1.95%
	"2"	0.04%	0.27%	0.73%	1.43%	2.35%	3.51%			"2"		0.04%	0.27%	0.73%	1.43%	5.86%
	"3"	0.06%	0.45%	1.22%	2.38%	3.92%	5.85%			"3"		0.06%	0.45%	1.22%	2.38%	9.77%
	"4"	0.09%	0.63%	1.71%	3.33%	5.49%	8.19%			"4"		0.09%	0.63%	1.71%	3.33%	13.68%
	"5"	0.12%	0.81%	2.20%	4.28%	7.06%	10.53%			"5"		0.12%	0.81%	2.20%	4.28%	17.59%
	"6"	0.14%	0.99%	2.69%	5.23%	8.63%	12.87%			"6"		0.14%	0.99%	2.69%	5.23%	21.50%

There are five scenarios where Faramir has seen pretty significant drops: 3 v. 3 (from 1.71% to 0.63%), 4 vs. 3 (2.2% to 0.81%), 4 vs. 4 (3.33% to 1.71%), 5 vs. 4 (4.28% to 2.2%), and 5 vs. 5 (7.06% to 4.28%). Each of those scenarios have gone down by at least one percentage point. The net result is that the combined probability of those scenarios drops from 20.23% before Eomer began boosting (about 1-in-5) to just 9.84% (about 1-in-10), for a loss of 10.39 percentage points overall.

Ironically, Eomer is seeing a similar siphoning in his victories when he has a “4” or a “5” highest. He’s still winning when he has a 5 and Faramir has a 4, but this now happens a lot less often (3.33% of the time, compared to 5.49% before boosting). As a result, the scenarios where he wins on a high roll of 5 or less have also gone down, from 17.9% to 10.19% (and a loss of 7.71 percentage points). When you add these siphoned results to the 10.39 percentage points covered above, and then the miscellaneous losses when we have a “2” or “3” highest result (which are pretty rare), we end up with net losses of 19.61 percentage points.

Which begs the question: where have they got to?

Well, the number that stands out the most is that enormous increase in the “6 vs. 6” box, which has ballooned from 12.87% to 21.5% (an increase of 8.63 percentage points). That increase happens to be exactly the percentage point loss that Faramir has seen in his “6 vs. whatever Eomer rolls” results (the white boxes), which are down 8.63% across the board. So that’s where at least some of them are going (i.e., all the ones where Faramir rolled a “6,” and Eomer rolled any amount). And, as expected, these all stay victories.

What this shift doesn’t account for is the siphoning that appears in all the results where Faramir and Eomer both fail to score a “6.” Where are all these results going? They’re being shifted one column to the right, which means the bulk of them are ending up in columns where Eomer scores a “6” and Faramir does not (marked in purple). These are scenarios where Eomer wins outright, because he has a higher final duel roll than Faramir. Before Eomer started boosting, these scenarios accounted for just 29.26% of our total results. Now? Those same outright wins for Eomer account for a whopping 48.87% of our total results.


Photo Credit: Flame of Udun

And remember, that’s just the percentage of times that Eomer wins with a “6 high”—if we add his odds of winning the rarer (but not impossible) scenarios where Eomer wins with a “5 high” or less, his total odds of victory are 59.05%. Which means Faramir’s total odds of wining—even with the higher Fight Value from Strike—are just 40.95%... which is still better than his odds before he struck (28%), but lower than Eomer’s odds of winning the fight outright.

Concluding Thoughts: So what should Faramir do?

So let’s go back to our initial scenario, and think about what all this mean for poor Faramir. At the start of the Fight Phase, he’s locked into combat with Eomer. They have tied Fight Value, so there are fifteen victory scenarios for each of them, and six draws. There’s a 28% chance that one of Faramir’s victory scenarios will happen; a 47% chance that one of Eomer’s fifteen victory scenarios will happen; and a 25% chance of a drawn combat. 

Highest Duel Roll: Faramir vs. Eomer
		Eomer (FV: 5) Duel Roll (3d6)
		"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 5) Duel Roll (2d6)	"1"	0.01%	0.09%	0.24%	0.48%	0.78%	1.17%
	"2"	0.04%	0.27%	0.73%	1.43%	2.35%	3.51%
	"3"	0.06%	0.45%	1.22%	2.38%	3.92%	5.85%
	"4"	0.09%	0.63%	1.71%	3.33%	5.49%	8.19%
	"5"	0.12%	0.81%	2.20%	4.28%	7.06%	10.53%
	"6"	0.14%	0.99%	2.69%	5.23%	8.63%	12.87%

If Faramir Strikes, and Eomer does not, Faramir will gain sole control of all the drawn combats, bumping his win scenarios from 15 to 21, with a combined probability of 52.84% that one of those 21 victory scenarios will occur (to Eomer’s 47.16% combined probability). If Eomer had no Might, this would be pretty low-risk, as Faramir would gain the higher Fight Value no matter what he rolls on the Strike die, and Eomer couldn’t boost his dice (which severely blunts Faramir’s gains). But Eomer does have Might, which means one of two responses are possible.

First, Eomer could respond by calling his own Strike. Since Eomer and Faramir have the same Fight, and that fight is pretty low (5), there’s a lot of variance in outcomes, but Faramir only gets the higher fight value about 40% of the time (38.89% technically). A drawn result (22.22% chance) will give Eomer a 50-50 shot at all those juicy drawn combats. And if Eomer wins the Strike-Off, Faramir’s chances of winning the combat plummet. There’s a lot of risk here for Faramir: the odds are pretty good (better than 60%) that Eomer ends up with no negative impact or a gained advantage. Not exactly the stuff dreams are made of.

Both sides Start with Equal Fight Value (F5 vs. F5)
		Target (FV: 5) Duel Roll
		"1"	"2"	"3"	"4"	"5"	"6"
Challenger (FV: 5) 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	Draw
	"6"	Win	Win	Win	Win	Draw	Draw

Second, Eomer could go more conservative, and do nothing for now, banking his valuable Might point to boost his duel roll. As we just saw in the pie chart, while Eomer will lose if Faramir rolls a “6,” Eomer still has a sneaky 60% chance (okay, okay—59.05% chance) to steal the victory because boosting allows him to get consistently higher dice rolls than Faramir.

This more conservative approach also has an underrated psychological and resource-management component to it. Unlike Faramir, who has to essentially gamble with his Might point (he has to declare a Strike without knowing if Eomer will Strike-Off, and what the result of that potential Strike-Off will be), Eomer has the luxury of waiting to spend his Might until after all duel rolls are made and modified, and any banner rerolls have occurred. If Faramir has a “6” and will win anyway, Eomer can pocket that Might for later instead of burning it unnecessarily (perhaps to modify an unfortunate Fate roll if Faramir manages some wounds, or even for a Strike of his own next turn, when Faramir’s Might store is depleted). If Faramir is winning a drawn combat on a roll of “5” or less, Eomer can boost his die to give him the victory—or worse, force Faramir to commit even more of his Might to the fight (matching Eomer’s boosting in kind), so as to not “waste” the Might he spent on the Heroic Strike. There’s a very real temptation to “double-down,” by committing even more might to ensure a Heroic Action works: otherwise, we feel that action is wasted (it happens a lot with Heroic Combats).

There’s nothing wrong with doubling down if it’s likely to accomplish an important objective. In fact, if Faramir doubles-down and boosts his duel roll, his odds of winning go up significantly—he outright wins 63.37% of all scenarios compared to Eomer’s paltry 36.63%, including the absurd 39.09% of scenarios where both sides have a “6”: 

		Eomer (FV: 5) Duel Roll (3d6 + Might)
		"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 6) Duel Roll (2d6 + Might)	"1"
	"2"		0.01%	0.09%	0.24%	0.48%	1.95%
	"3"		0.04%	0.27%	0.73%	1.43%	5.86%
	"4"		0.06%	0.45%	1.22%	2.38%	9.77%
	"5"		0.09%	0.63%	1.71%	3.33%	13.68%
	"6"		0.26%	1.80%	4.89%	9.52%	39.09%

(By the way, those eleven scenarios where at least one side has a “6 high” when both sides are boosting their dice? They account for 86.83% of our likely scenarios. We thought anecdotally that this happens a lot, and this is why.)

Now all of this is anecdotal, dealing only with one very limited scenario. Next time, we'll take a look at some of the more common scenarios involving heroes and troops, and identify some general scenarios where Strike greatly increases our chances to win, scenarios where it marginally increases our chances to win, and scenarios where it's unlikely to impact who wins one way or another.

But before we get there, let me leave you with some food for thought on the topic of boosting:

Assume that, knowing all we've talked about, we go back to the very beginning, when Faramir is trying to decide if he’s going to Strike at all. Now let’s suppose that instead of opting to Strike, Faramir opts to do something else: he passes, opting to hold his Might, like Eomer could, to boost his duel roll. What happens then? Well, there are three possibilities. Option 1: Eomer Strikes. At that point, Faramir would still have the option to counter, if he wanted, or to hold his Might in reserve. Option 2: Eomer doesn’t Strike, and calls some other Heroic Action (maybe Heroic Strength for more damage output). Faramir would then have a second chance to call a Heroic Strike if he wanted, which would now carry far less risk—Eomer can’t match his Strike. Or he could still hold on to his Might, to win that duel (or at least force a roll-off). Option 3: Eomer doesn’t call anything, and calls the start of the Fight Phase. Both players now opt to boost, which sets us up for (probably) a fifty-fifty roll-off where both sides have a 6 high:

		Eomer (FV: 5) Duel Roll (3d6 + Might)
		"1"	"2"	"3"	"4"	"5"	"6"
Faramir (FV: 5) Duel Roll (2d6 + Might)	"1"
	"2"		0.01%	0.09%	0.24%	0.48%	1.95%
	"3"		0.04%	0.27%	0.73%	1.43%	5.86%
	"4"		0.06%	0.45%	1.22%	2.38%	9.77%
	"5"		0.09%	0.63%	1.71%	3.33%	13.68%
	"6"		0.26%	1.80%	4.89%	9.52%	39.09%

Eomer’s odds of winning outright? 36.63%, which is higher than Faramir’s outright win chance of 19.44%, but way lower than the 40-60% odds he had in the other scenarios we had above. The chance of a draw? A whopping 43.93%. Which leads to the absurd scenario where both combatants have more than a 50% chance not to lose the fight outright: Eomer will win or draw 80.56% of the time, and Faramir will win or draw 63.37% of the time. Which means if Faramir sits still and does nothing with his Might until dice are actually rolled, he actually has a shockingly good chance of getting a 50% chance to win the fight… on a fifty-fifty roll-off.


Photo Credit: jc_special

So now that you know all the options available, if you were controlling Faramir, what would you do?

Something to think about.