Thursday, April 12, 2012

Declaration of Nerd-dom: Statistics in the Lord of the Rings SBG

I think in numbers. When I'm having problems staying awake, I perform mathematics. When I'm trying to fall asleep...I try not to think of numbers. If I'm lying in bed trying to decide whether I want to wake up, I think of numbers. Yes, I'm a nerd. So to show a bit of the math I began one day in bed and finished here at the computer, we're going to do some statistical looks at common situations that have faced my Dwarf army in the Strategy Battle Game.

Scenario #1: Entanglement against an army of Haradhrim

Haradhrim, up until the new rules, were the best bargain for cost that an Evil player could get. With the new rules, the army costs roughly the same as other armies but still poses a strong threat to an army of Dwarves. In large part, this is due to the large amount of bows that the army can field, but another element to the threat is the vast proliferation of spears in an army from Harad. Since the Dwarf units have (by and large) no spears in their army, confrontations with Harad means that there will be more foes standing against your Dwarves in combat. We'll look at the following army as an example, facing off against an army of Dwarves which is nearly like one that I intend to showcase in a near-future article:

Haradhrim Army: The Serpent's Sting

Golden King of Abrakhan - 115 points
Suladan with bow - 95 points
2 Haradhrim Chieftains - 90 points
12 Serpent Guard - 160 points
20 Haradhrim Warriors with bows - 140 points
8 Abrakhan Merchant Guards - 64 points

600 points, 44 units, 21 bows, 4 heroes

Dwarf Army: The Folk of Durin

Gimli, Son of Gloin - 90 points
Balin, Son of Fundin - 75 points
2 Shield-Bearers - 120 points
6 Khazad Guards - 66 points
8 Dwarf Warriors with shields - 72 points
10 Dwarf Warriors with Dwarf bows - 90 points
6 Dwarf Rangers with throwing axes - 60 points
2 Dwarf Rangers with two-handed axes - 14 points
1 Dwarf Ranger with Dwarf longbow - 10 points

597 points, 37 units, 10 Dwarf bows + 1 Dwarf longbow + 8 throwing axes, 4 heroes

As you can see in this sample army (most cheaply built if one converts Haradhrim Warriors with spears into Serpent Guard), the Haradhrim with the new army rules can sport an army with a lot of bows and a lot of spears, including 20 Fight 4 warriors with poisoned blades or the Chop! special rule (not to mention 21 S2 bows with poisoned arrows). As a Dwarf commander, you're faced with the need to kill the Serpent Guards and the Abrakhan Guards for sure, but the heroes and two volley teams will provide some difficult before the melee is engaged. What do you do? Statistically, let's look at what you might do...

Chances of winning a fight: quick calculation
When two dice are rolled, there are only 36 combinations that could arise. 15 cases will result in one dice being higher than a second, while another 15 cases will result in the one dice being lower than the second. The remaining 6 rolls will be ties. This means that, without ties in consideration, a player has a 42% chance of rolling a winning roll when he and his opponent are rolling a single dice.

The ties, of course, are won by the player with the highest Fight value. Against Serpent Guard and Abrakhan Guards, however, the ties "statistically should" split evenly between the Dwarf and the Haradhrim unit he faces. This means the chance of winning the fight is a 50% chance. If we consider the roll to wound, then, we will have a 33-50% chance to wound, depending on whether the Dwarf fighting against the foe has Strength 3 or 4. With the match-ups below, here's what the battle looks like:

-2x6 Serpent Guard
-1x8 Abrakhan Guard
-TOTAL: 14 attackers

-1x6 Khazad Guard
-1x8 Dwarf Warrior with shield
-TOTAL: 14 defenders

If we compare the 8 fights between the Abrakhan Guards and the Dwarf Warriors with shields, the break-down goes as follows:
-4 Dwarves win their fights, 1.33 Wounds dealt
-4 Abrakhan Guards win, 1.33 Wounds dealt
Reasoning: the D7 of your front line means that both you and the Abrakhan Guards win the fight and wound the foe evenly. This is unmatched in the rest of Middle-Earth, but it's still risky.

In the 6 fights between the Serpent Guards and the Khazad Guards, we need a different statistic to determine who wins the fight. In this case, the Dwarf is the "attacker" and the Haradhrim are the "defenders". The result, as you can see, is that the Dwarves win 33% of the fights and the Serpent Guards win 66% of the fights:
-2 Dwarves win their fights, 0.67 Wounds dealt
-4+4 Serpent Guards win, 1.6 Wounds dealt
Reasoning: the D7 of the Khazad Guards require a roll of a 6 to wound. The Poisoned Blades of the Serpent Guard, allows them to reroll 6 of 36 combinations, one of which would succeed in wounding. With 7/36 of the rolls resulting in a wound, 20% of their attacks succeed in killing their opponent. Unlike the fight before, this fight is a bit less fair, with the Haradhrim scoring twice as many kills as the Dwarves.

Melee comparison: If you take the wound ratings shown above and compare that to each individual unit, the Abrakhan Guards have a 17% chance of killing a Dwarf, while the Serpent Guard have a 7% chance of killing a Dwarf. Pretty clear that we should be killing the Abrakhan Guards, right? We'll let's look quickly at the archers...

21 bows (including Suladan) means a lot of arrows all the time. Calculating a statistical chance of wounding with bows is much easier, as you can see with the explanation below:

Volley Fire: 21 bows x (16% chance to hit) x (8% chance to wound) = 0.29 Wounds/round
Direct Fire: 21 bows x (50% chance to hit) x (8% chance to wound) = 0.88 Wounds/round

The shots calculated here consider how easy it is to wound a Dwarf Warrior with shield or a Khazad Guard. The ability to wound a Dwarf Warrior with Dwarf bow or a Dwarf Ranger will be easier and double the chances of wounding. If this is the case, 21 archers are supposed to kill nearly 2 Dwarves at close range and 1 Dwarf every 2 turns at volley. This means that each archer has an 8% chance of killing a Dwarf with D5-D6, making the Abrakhan Guards the most likely to kill any Dwarf at a given time. Statistically speaking, it's best to kill these guys. Since we've selected the Abrakhan Guards as our targets, we'll see how many hits and wounds we get with our Strength 3 Dwarf bows:

Volley Fire: 10 Dwarf bows x (16% chance to hit) x (33% chance to wound) = 0.56 Wounds/round 
Direct Fire: 10 Dwarf bows x (50% chance to hit) x (33% chance to wound) = 1.67 Wounds/round

If your volley team is left alive by the time it is able to fire directly, they are supposed to kill 1-2 Haradhrim each turn. For having half of the archer force of your opponent, that's not bad. At the same time, you might kill 1 unit every 2 turns while volleying, so you're not going to kill many units before battle begins. A good Dwarf general would be careful then, before rushing into battle, as the overwhelming numbers of the enemy (outnumber by 7 in this case) could really hurt your army.

Scenario #2:  Entanglement against an army of Uruk-Hai

The comparison between Dwarves and Uruk-Hai is similar to the comparison against Haradhrim with a few exceptions: first, Uruk-Hai have Defense 6 instead of Defense 4, and second, Uruk-Hai have crossbows and smaller numbers. Here's a sample army we'll look at:

Uruk-Hai Army: The Slaughter Lines

2 Uruk Captains with shields - 110 points
*Vrasku - 60 points
Uruk-Hai Shaman - 50 points
17 Uruk-Hai Warriors with shields - 170 points
10 Uruk-Hai Warriors with pikes - 100 points
10 Uruk-Hai Warriors with crossbows - 110 points

600 points, 41 units, 11 crossbows*, 4 heroes

On the whole, the Uruks barely outnumber the Dwarves (by 4), so this game should be close. In melee combat, there are 27 Uruks to fight 14 heavily-defended Dwarf Warriors and Khazad Guards. Here's the break-down of the fights as estimated:

-2x10 Shields + Pikes
-2x3 Shields + Shields
-1x1 Shields
-TOTAL: 14 attackers

-1x6 Khazad Guard
-1x8 Dwarf Warrior with shield
-TOTAL: 14 defenders

We're going to assume that the double-up fights of the Uruks are occurring on the flanks and so we'll have those three fights against shield-bearing Dwarf Warriors. If we compare 5 fights between the Uruk Shields + Pikes and the Dwarf Warriors with shields (and 5 fights between similar units and Khazad Guards), the break-down goes as follows:
-2 Dwarf Warriors and 2 Khazad guards win their fights, 0.33 + 0.67 Wounds dealt (1 dead Uruk)
-6+6 Uruks win, 2.00 Wounds dealt
Reasoning: the D7 of the Dwarves make it harder for the Uruks to win (and completely negates the Strength 4 stat of the Uruk-Hai). The D6 of the Uruks, however, negates most of the chance of the Dwarf Warriors of scoring wounds, though the Khazad guards have it much, MUCH easier. Still, the Uruks have the advantage because of their pike support.

The remaining fights break down as follows:
-1 Dwarf vs. 1 Uruk: each has a 50% chance of winning the fight and a 16% chance of killing their foe...
-2 Dwarves vs. 4 Uruks: the Dwarves have a 33% chance of winning their fights and a 16% chance of killing their foe, while the 4 Uruks have a 66% chance of winning their fights and a 16% chance of killing their foes; final tally: the Dwarves have an 11% chance of killing their foes, while the Uruks have a 44% chance of killing their foes. Having the extra guys really, really helps.

The key to stopping the melee fights, therefore, is to remove the ability to get more than 1 Uruk in each fight. This, of course, means, we're shooting at the pikemen:

Volley Fire: 10 Dwarf bows x (16% chance to hit) x (33% chance to wound) = 0.56 Wounds/round 
Direct Fire: 10 Dwarf bows x (50% chance to hit) x (33% chance to wound) = 1.67 Wounds/round

I'll mention now that the chance of wounding the pikemen here is just as easy as wounding Abrakhan Guards in the Haradhrim army. The ability to kill some of these pikemen before the fight is possible but shouldn't be counted on as a way to whittle down the foe. We must consider, however, if we should instead shoot at the crossbows instead (same likelihood of wounding, so we're focusing on damage caused).

Direct Fire: 10 crossbows x (50% chance to hit) x (16% chance to wound) = 0.83 Wounds/round
Direct Fire (Vrasku): 2 crossbow shots x (66% chance to hit) x (16% chance to wound) = 0.22 Wounds/round

Each crossbowman has an 8% chance of killing his foe, while the Uruks in close combat have a 16% chance when fighting two-to-one. This means that the targets should be the pikemen, as we estimated before.

Scenario #3:  Entanglement against an army of Goblins

Goblins provide a different dynamic when running the statistics: no Fight 4 units. The Bat Swarm can change this quite a bit, but on the whole, the Dwarves will win ties. Here's a sample Goblin army to show what the statistics are:

Goblin Army: The Host of the Mountains

Durburz, Goblin King of Moria - 60 points
2 Goblin Captains with shields - 80 points
1 Goblin Shaman - 45 points
16 Goblin Warriors with shields - 80 points
16 Goblin Warriors with spears - 80 points
12 Goblin Warriors with Orc bows - 60 points
2 Cave Trolls - 160 points
Bat Swarm - 35 points

600 points, 51 units, 12 Orc bows, 4 heroes

The fact that Goblins have so many units is, in fact, a bother to most Dwarves (outnumbering the Dwarves by 14 in this set-up). We all know we need to shoot Trolls to death before they get to us, but we just won't have the units to fight off over 40 Green Skins by the time they arrive, ready for battle. Let's see why:

-2x12 Goblin Warriors with shields + spears
-4x2 Goblin Warriors shields + spears
-TOTAL: 14 attackers

-1x6 Khazad Guard
-1x8 Dwarf Warrior with shield
-TOTAL: 14 defenders

Here's how the 2-on-1 fights go:
-3 Dwarf Warriors + 2 Khazad Guards win their fights, 1.67 Wounds dealt
-7+7 Goblin Warriors win their fights, 1.20 Wounds dealt
Reasoning: despite winning many more fights, the Goblins have the statistical likelihood of wounding fewer Dwarves than the Dwarves wound Goblins. One of twelve dead Dwarves, of course, is larger than 1.5 dead Goblins out of twenty-four, but it's fairly close.

The 4-on-1 fights, we will assume, involve Dwarves with shields. By using the shielding, rule, the calculation is fairly similar to rolling 1 dice against 2 (a few percentage points in favor of the Goblins, but it's a small difference). These uneven fights have a similar trend, then, as shown above. In this case, of course, the Goblins who win their fights double their likelihood of wounding solely because they have more dice, upping the likelihood of wounding from those shown above. The Dwarves deal 0.25 Wounds, while the Goblins deal 0.8 Wounds...not very fair, no?

Shooting at Goblin Warriors is a simple choice from the perspective of archery: they're all D4 or D5, so you wound them on 5s. Your archers do as they have in the last few times (we'll review again for good measure):

Volley Fire: 10 Dwarf bows x (16% chance to hit) x (33% chance to wound) = 0.56 Wounds/round 
Direct Fire: 10 Dwarf bows x (50% chance to hit) x (33% chance to wound) = 1.67 Wounds/round

The Goblin archery, though is less impressive than previous cases have been:

Volley Fire: 12 bows x (16% chance to hit) x (8% chance to wound) = 0.17 Wounds/round
Direct Fire: 12 bows x (33% chance to hit) x (8% chance to wound) = 0.33 Wounds/round

Even with direct fire, the archers are only set to kill one Dwarf (if he is D7) every three rounds. Volleying against the archers may be useful to keep fire from your troops, but on the while, you shouldn't lose many units to these shots. Targeting the spearmen will probably do more of an impact on the game (at least until the archers join in the melee).

So...there was some interesting math from the perspective of my doughty Dwarves. I didn't include the likelihood of the throwing axes or Dwarf longbow dealing damage, but the choice of who you shoot with these weapons tend to be less tactical since the choices are much more limited. If you've thought through this as well, I'd love to hear from you in comments.

1 comment:

  1. too many numbers after too much spanish to try to check your math, but that was an interesting breakdown. Not that there was ever any doubt that you would be targeting my Abrakhans - if for no other reason than they give you the heeblee jeeblees ;)

    this is why they are always at the _back_ of the formation - where they can't easily be targeted by archery.

    I'm also grateful that the game itself involves more subtleties than this - like the fact that I would volley at your archers - not your annoying D7 troops. But that gets into far more complicated statistics.

    I am also intrigued that your scenario had you facing off against Suladan - who I have never fielded against you (far preferring the stealthy Dalamyr).

    We totally need to get another game sometime soon. I just got some palms to start building my desert terrain collection.