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Why BIND Rules Don't Allow Players to Go for the Eyes

(a story about spreadsheet failure)

I've considered changing BIND's 'to-hit' system to let players 'go for the eyes' (or a headshot, or otherwise decide to attempt a vitals shot), and decided against it. My reasons sit below, but expect lots of boring numbers. You have been warned. (or just skip to the conclusions)

Quick Recap

Consider someone with a shortsword with +2 Strength - they deal 1D6 + 2 Damage, or 5.5 on average (this could also be 1D8 +1 or whatever). Let's also assume that the opponent has the same stats, making the Tie Number (TN) '7'.

If the player rolls above the TN, they deal Damage.
If the player rolls below the TN, they receive Damage.
If the player hits the TN exactly, they could both deal and receive Damage.

┌────┬─────────────┬───────┬──────────┐
│    │ Probability │ Dealt │ Received │
╞════╪═════════════╪═══════╪══════════╡
│ 2  │       2.78% │       │  0.153   │
├────┼─────────────┼───────┼──────────┤
│ 3  │       5.56% │       │  0.306   │
├────┼─────────────┼───────┼──────────┤
│ 4  │       8.33% │       │  0.458   │
├────┼─────────────┼───────┼──────────┤
│ 5  │      11.11% │       │  0.611   │
├────┼─────────────┼───────┼──────────┤
│ 6  │      13.89% │       │  0.764   │
├────┼─────────────┼───────┼──────────┤
│ 7  │      16.67% │ 0.917 │  0.917   │
├────┼─────────────┼───────┼──────────┤
│ 8  │      13.89% │ 0.764 │          │
├────┼─────────────┼───────┼──────────┤
│ 9  │      11.11% │ 0.611 │          │
├────┼─────────────┼───────┼──────────┤
│ 10 │       8.33% │ 0.458 │          │
├────┼─────────────┼───────┼──────────┤
│ 11 │       5.56% │ 0.306 │          │
├────┼─────────────┼───────┼──────────┤
│ 12 │       2.78% │ 0.153 │          │
├────┼─────────────┼───────┼──────────┤
│    │       Total │ 3.209 │  3.209   │
└────┴─────────────┴───────┴──────────┘

Armour

Now let's add in chain armour, with Damage Resistance 4. If the player rolls 1 or 2 above the TN, their Damage is reduced by 4.

┌─────────────┬──────┬────────┐
│ Probability │ Roll │ Damage │
╞═════════════╪══════╪════════╡
│    16.7%    │  1   │   0    │
├─────────────┼──────┼────────┤
│    16.7%    │  2   │   0    │
├─────────────┼──────┼────────┤
│    16.7%    │  3   │   1    │
├─────────────┼──────┼────────┤
│    16.7%    │  4   │   2    │
├─────────────┼──────┼────────┤
│    16.7%    │  5   │   3    │
├─────────────┼──────┼────────┤
│    16.7%    │  6   │   4    │
└─────────────┴──────┴────────┘

Average Damage: 1.667

So the average Damage is precisely 10 times the chance of hitting a '1' on the D6. Isn't that pleasing? But it's also poor average Damage - it's a lot lower than the old average Damage of 5.5.

Armour Adjustments and DR

┌────┬─────────────┬───────┬──────────┐
│    │ Probability │ Dealt │ Received │
╞════╪═════════════╪═══════╪══════════╡
│ 2  │       2.78% │       │  0.153   │
├────┼─────────────┼───────┼──────────┤
│ 3  │       5.56% │       │  0.306   │
├────┼─────────────┼───────┼──────────┤
│ 4  │       8.33% │       │  0.458   │
├────┼─────────────┼───────┼──────────┤
│ 5  │      11.11% │       │  0.158   │
├────┼─────────────┼───────┼──────────┤
│ 6  │      13.89% │       │  0.231   │
├────┼─────────────┼───────┼──────────┤
│ 7  │      16.67% │ 0.278 │  0.278   │
├────┼─────────────┼───────┼──────────┤
│ 8  │      13.89% │ 0.231 │          │
├────┼─────────────┼───────┼──────────┤
│ 9  │      11.11% │ 0.185 │          │
├────┼─────────────┼───────┼──────────┤
│ 10 │       8.33% │ 0.458 │          │
├────┼─────────────┼───────┼──────────┤
│ 11 │       5.56% │ 0.306 │          │
├────┼─────────────┼───────┼──────────┤
│ 12 │       2.78% │ 0.153 │          │
├────┼─────────────┼───────┼──────────┤
│    │       Total │ 1.611 │  1.611   │
└────┴─────────────┴───────┴──────────┘

Here, any roll which beats the TN by 3 produces a 'Vitals Shot', bypassing armour. Damage has reduced significantly, as the most likely numbers to come up have a serious Damage deficit.

The Alternative System

Now let's imagine players can elect to take a 'vitals shot' not by rolling high, but by taking a -1 penalty to their roll. If they hit, it's a vitals shot!

We're going to take the damage Dealt from the first chart, but miss out that sweet '7' spot, reducing the average Damage from 3.2087 to 2.29185.

The average Received damage is taken from the second chart, as the opponent may still hit the player's armour.

┌────┬──────────────┬───────┬──────────┐
│    │ Probability  │ Dealt │ Received │
╞════╪══════════════╪═══════╪══════════╡
│ 2  │        2.78% │       │  0.153   │
├────┼──────────────┼───────┼──────────┤
│ 3  │        5.56% │       │  0.306   │
├────┼──────────────┼───────┼──────────┤
│ 4  │        8.33% │       │  0.458   │
├────┼──────────────┼───────┼──────────┤
│ 5  │       11.11% │       │  0.611   │
├────┼──────────────┼───────┼──────────┤
│ 6  │       13.89% │       │  0.231   │
├────┼──────────────┼───────┼──────────┤
│ 7  │       16.67% │       │  0.278   │
├────┼──────────────┼───────┼──────────┤
│ 8  │       13.89% │ 0.764 │  0.231   │
├────┼──────────────┼───────┼──────────┤
│ 9  │       11.11% │ 0.611 │          │
├────┼──────────────┼───────┼──────────┤
│ 10 │        8.33% │ 0.458 │          │
├────┼──────────────┼───────┼──────────┤
│ 11 │        5.56% │ 0.306 │          │
├────┼──────────────┼───────┼──────────┤
│ 12 │        2.78% │ 0.153 │          │
├────┼──────────────┼───────┼──────────┤
│    │ Average Dam. │ 2.292 │  2.269   │
└────┴──────────────┴───────┴──────────┘

The average results look pretty similar.

Disappointing Conclusions

While I'm usually a fan of spreadsheets, these actually tell us nothing. Without any numbers, we can see:

If the opponent is put at a disadvantage, they can just take their own -1 penalty, and we return precisely to the chart where everyone gets no armour, with the same results. * In this system, wearing no armour would mean the opponent doesn't have to take that -1 penalty, so armour does not reduce damage but effectively adds a +1 'to-hit' bonus (madness!).
If this does not put the opponent at a disadvantage, then the player isn't put at an advantage, and they simply don't use the system, making the entire thing pointless.

However one adjusts the numbers, the same effects happen. You can increase the cost of a Vitals Shot to a '-2' penalty instead of just '-1', but it doesn't change the result. The system can serve to give the illusion of choice to someone who doesn't have time for spreadsheets, but the best case scenario here is a few players who don't know which number should prompt them (and the opponent) to attempt a Vitals Shot.

Once the mask of Maths has been lifted, the system offers a bunch of faff to arrive at a single, best result.

So all in all, I'll be sticking with the original system: armour reduces Damage, unless one gets a Vitals Shot by hitting a high number. It may not feel engaging, but at least the system's honest.