Grasshopper's 8d System - Conversion tables for 1940 Global units

@Baron:

@Young:

I’ve changed Destroyers and Subs… Attacking unit dice odds

1/8 - White (12.5%)
Infantry (-16.7%)
Mech Infantry (-16.7%)

2/8 - Green (25%)
Infantry w/Artillery (-33%)
Mech Infantry w/Artillery (-33%)
Destroyer (-33%)

3/8 - Blue (37.5%)
Artillery (+33%)
Submarine (+33%)

4/8 - Purple (50%)
Tank (0%)
Fighter (0%)
Tactical Bomber (0%)

5/8 - Red (62.5%)
Strategic Bomber (-66.7%)
Cruiser (+50%)
Tactical Bomber w/Fighter or w/Tank (-66.7%)

6/8 - Black (75%)
Battleship (+66.7%)

Defending unit dice odds

1/8 - White (12.5%)
AA Guns (-16.7%)
Strategic Bomber (-16.7%)
Submarine (-16.7%)

2/8 - Green (25%)
Infantry (-33%)
Mech Infantry (-33%)
Artillery (-33%)
Aircraft Carrier (-33%)

3/8 - Blue (37.5%)
Destroyer (+33%)

4/8 - Purple (50%)
Tank (0%)
Tactical Bomber (0%)

5/8 - Red (62.5%)
Fighter (- 66.7%)
Cruiser (+50%)

6/8 - Black (75%)
Battleship (+66.7%)

Strategic bombing air combat values

1/8 - White (12.5%)
Tac Bomber (-16.7%)
Strategic Bomber (-16.7)

2/8 - Green (25%)
Fighter interceptor (+16.7%)
Fighter escort (+16.7%)

Kamikaze Token defense values

2/8 - Green (25%)
Kamikaze Token (-33%)

*Special thanks to Big Al “Mike Tyson” for contributing in this idea & Baron Munchhausen for his calculations.

@Baron:

@Young:

I really like the new chart.

It is an elegant way to not make them weaker or stronger compare to OOB.
Average strength is just below OOB strength.
A bit of iconic in a way that Artillery and Subs seems to be for attack while Destroyer is first intended as an escorting vessel.
Cruiser issue is partly resolved by making it a better attacking unit per IPC ratio compared to DD, but not for defense.

| Artillery
| attack 3
| defense 2
| move 2
| Cost 4
| +1A to Inf or Mech Inf
| Strength: 3.375 / 2.25 | Avg 2.813
| OOB: 3.00

| Destroyer
| attack 2
| defense 3
| move 2
| Cost 8
| Anti-Submarine Weapon
| Strength: 0.563 / 0.844
| Avg 0.704
| OOB: 0.750

| Submarine
| attack 3
| defense 1
| move 2
| Cost 6
| Submarine’s capacities

| Strength: 1.50 / 0.50
| OOB: 1.33 / 0.667
|
| Strength for surprise strike: 2.25 / 0.667
| OOB: 2.00 / 0.89
Compared to:

| Submarine
| attack 3
| defense 2
| move 2
| Cost 6
| Submarine’s capacities

| Strength: 1.50 / 1.00
| OOB: 1.33 / 0.67 |
| Strength for surprise strike: 2.25 / 1.50
| OOB: 2.00 / 0.89

@Young:

I really like the new chart.

Can you tell us why and what you like about these 3 units and values:
Submarine Att 3 Def 1,
Destroyer Att 2 Def 3 and
Artillery Att 3 def 2, please?

I like it too, but probably for different reasons.

Such as Submarines surprise strike being 3.6 times stronger than Cruiser instead of 4 times OOB.
But, now Cruiser in defense is same strength than Sub Surprise strike in defense (0.625 near 0.667) while OOB Cruiser was dominated 0.89 vs 0.50, 178%.

Okay Baron, show me all the unit stats for eight sided dice.

| Cruiser | attack 5 | defense 5 | move 2 | Cost 12 | Shore bombard @5 | Strength: 0.625 | OOB: 0.50
|
| Battleship | attack 6 | defense 6 | move 2 | Cost 20 | Shore bombard @6  Hits 2 | Strength: 0.707 | OOB: 0.628 |

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Here it is:

@Baron:

YG’s complete D8s roster

Here is the comparative with Enigma formula (still using a 12 IPCs unit reference) based on odds, not dice number.
144*Odds/cost^2 = strength of unit on same odds by hit per IPCs basis.

I bolded the greatest number between D8 or D6 for same unit.
When there is no difference, I quoted it.
So, you can see at glance which is boosted or nerfed compared to OOB.

Unit type  Combat value  (D8 system) (D6 OOB)
Infantry      A1-2 D2    (2.00-** / 4.00)        (2.67-3.92 / 5.33)
Inf A2+Arty A2 D2        2.94**  /  2.94        (3.92 / 3.92)
Inf A2+Arty A3 D2        3.67**  /  2.94        Same as above
Inf A2+Arty A3 D3        3.67**  /  3.67        Same as above

MechInfantry A1-2 D2  (1.125-** / 2.25)      (1.50-3.00 / 3.00)
MechInf A2+Art A2 D2  (2.25** / 2.25) vs      (3.00 / 3.00)
MechInf A2+Art A3 D2  (2.81** / 2.25) vs        Same as above
MechInf A2+Art A3 D3  (2.81** / 2.81) vs        Same as above

Artillery  A2 D2      (2.25 / 2.25)                  (3.00 / 3.00)
Artillery    A3 D2    (3.375 / 2.25) avg: 2.813  Same as above
Artillery    A3 D3      (3.375 / 3.375)            Same as above

Tank        A4 D4            (2.00 / 2.00)            (2.00 / 2.00)

Anti-Air A  A0 D1* (0.00 / 0.72 per plane)      (0.00 / 0.96 per plane)

---

Fighter      A4 D5    (0.72 / 0.90 )              (0.72 / 0.96)

TcBomber  A4-5 D4 (0.595-** / 0.595)      (0.595 / 0.595)

TcB A5+Tank C17    (1.12** / 1.00)        (1.16 / 1.00)
TcB A5+Fighter C21 (0.735** / 0.653)    (0.761 / 0.653)
StBomber  A5 D1    (0.625 / 0.125)        (0.667 / 0.167)

---

Submarine  A2 D1 (1.00, fs: 1.50 / 0.50 fs: 0.67)  (1.33, fs: 2.00 / 0.67 fs: 0.89)
Submarine  A3 D1 (1.50, fs: 2.25 / 0.50 fs: 0.67)  (1.33, fs: 2.00 / 0.67 fs: 0.89)
Submarine  A3 D2 (1.50, fs: 2.25 / 1.00 fs: 1.33)  (1.33, fs: 2.00 / 0.67 fs: 0.89)
Destroyer    A2 D2 (0.563 / 0.563)                        (0.75 / 0.75)
Destroyer    A2 D3 (0.563 / 0.844) avg:  0.704      (0.75 / 0.75)
Destroyer    A3 D3 (0.844 / 0.844)                        (0.75 / 0.75)
Cruiser        A5 D5  (0.625 / 0.625)                      (0.50 / 0.50)
Carrier, 2 hits A0 D2 (0.0 / 0.368)                            (0.00 / 0.491)
Carrier, 2 hits A0 D3 (0.0 / 0.552)                           As above
Battleship    A6 D6  (0.707 / 0.707)                      (0.628 / 0.628)

The real difference is between OOB ground units and naval units decreasing with higher cost compared to ground units strength decrease with higher cost compared to naval units strength increase with higher cost, if totally following Enigma formula with D8.

OOB: Inf 5.33, > MI 3.00, Art 3.00, MI+Art 3.00, > Tank 2.00
DD 0.75, > Cruiser 0.50, < BB 0.628

D8 system:
Inf 4.00, > MI D2 2.25 , Art D2  2.25, MI+Art D2 2.25, > Tank 2.00
DD A2 D2 0.563, < Cruiser A5 D5 0.625,  < BB A6 D6 0.707.

With DD A2 D3, defense is rising to 0.844 and this make DD above > Cruiser (0.625) and BB (0.707).
This is the same as OOB strength distribution: DD 0.75,> Cruiser 0.50 & BB 0.628.

Again, Cruiser on defense is still unoptimized but it is not different than OOB.
So, this can be a reason to change that way to not affect too much balance.

@Baron:

The real difference is between OOB ground units and naval units decreasing with higher cost compared to ground units strength decrease with higher cost compared to naval units strength increase with higher cost, if totally following Enigma formula with D8.

OOB: Inf 5.33, > MI 3.00, Art 3.00, MI+Art 3.00, > Tank 2.00
DD C8 0.75, > Cruiser C12 0.50, < BB C20 0.628

D8 system:
Inf 4.00, > MI D2 2.25 , Art D2  2.25, MI+Art D2 2.25, > Tank 2.00
DD C8 A2 D2 0.563, < Cruiser C12 A5 D5 0.625,  < BB C20 A6 D6 0.707.

With DD A2 D3, defense is rising to 0.844 and this make DD above > Cruiser (0.625) and BB (0.707).
This is the same as OOB strength distribution: DD 0.75,> Cruiser 0.50 & BB 0.628.

Again, Cruiser on defense is still unoptimized but it is not different than OOB.
So, this can be a reason to change that way to not affect too much balance.

Compared to D10, D8 sided dice brings a more interesting scale of strength for warships.
Simply because 6/8 get 75% for BB, which is the highest acceptable combat values vs 4/6 (67%) or 7/10 (70%)
And Destroyer can be lowest but acceptable: 2/8 get 25% compared to 2/6 (33%) or 3/10 (30%)
Cruiser can reach the most interesting middle range: 5/8 (62.5%) compared to 3/6 (50%) or 6/10 (60%).

DD C8 A2 D2 0.563, < Cruiser C12 A5 D5 0.625,  < BB C20 A6 D6 0.707

DD C8 A2 D3 avg A/D 0.704, > Cruiser C12 A5 D5 0.625,  < BB C20 A6 D6 0.707

And this different scale makes DD much stronger than Cruiser on defense, but Destroyer on average it is the same as BB.

@Baron:

Scale on D10:

| **Unit
type  ** |     | **D10 com.
values  ** | **OOB odds
offense  ** | **OOB odds
defense  ** | OOB values |

| Infantry | A2-3 D3 | 17-33% | 33% |             | A1-2 D2 |
| Mechanized
Infantry | A2-3 D3 | 17-33% | 33% |             | A1-2 D2 |
| Artillery | A3 D3 | 33% | 33% |             | A2 D2 |
| Anti-Aircraft
Artillery | A0 D1 | 0% | 17% |             | A0 D1 |
| Tank | A5 D5 | 50% | 50% |             | A3 D3 |
| Fighter | A5 D7 | 50% | 67% |             | A3 D4 |
| Tactical
Bomber | A6-7 D5 | 50-67% | 50% |             | A3-4 D3 |
| Strategic
Bomber | A6 D2 | 67% | 17% |             | A4 D1 |
| Submarine | A3 D2 | 33% | 17% |             | A2 D1 |
| Destroyer | A3 D3 | 33% | 33% |             | A2 D2 |
| Cruiser | A6 D6 | 50% | 50% |             | A3 D3 |
| Carrier | A0 D3 | 0% | 33% |             | A0 D2 |
| Battleship | A7 D7 | 67% | 67% |             | A4 D4 |

Destroyer strength: .30144/8^2  = 0.675
Cruiser     strength: .60144/12^2= 0.600
Battleship strength: .701442.618/20^2 = 0.660

Even such cost structure would not solve the warships DD vs Cruiser vs BB issue.
D8 allows it, because it goes from DD@2 25%, CA@5 62.5%, BB@6 75%:
Destroyer    A2 D2 (0.563 / 0.563)
Destroyer    A3 D3 (0.844 / 0.844)
Cruiser        A5 D5  (0.625 / 0.625)
Battleship    A6 D6   (0.707 / 0.707)

So, buying cheap you get weaker unit but costlier it becomes stronger, as it is suppose to be.

Is it what you will use?

I would probably ponder about AAA, StB and Sub defense values.
Maybe the weak odds are more realistic @1 out of 10 instead of rising them to 20%.

I might go this way:
AAA A0 D1 vs up to 3 planes but lower cost to 3 IPCs each.
OOB you get near 50% when 3 planes targeted for 5 IPCs: 10% per IPC.
Here you keep same ratio: 30% for 3 IPCs: 10% per IPC.

StB A7 D1
Bombers were made for offense and already very good at it.

Sub A3 D2
Because Subs on defense are usually trapped by planes and 1 DD.
The game mechanic make Subs too much vulnerable. Defense @2 is a small compensation.

I would prefer TcB A6-7 D5, that way combined arms simply gives +1A to Inf, MI and TcB.
Fg A5 D7 vs TcB A6 D5 seems a nice way to make both planes different.

@TripleThreat:

Baron, do believe the d8 system would also work well with your modified A&A unit roster? or would you have to change the prices of units.

I have a question about the formulas that you are using which I find very interesting by the way. I get that to find the relative strength you are dividing the probability by the cost squared, but why are you multiplying the probability by 100 sometimes and 144 other times?

And for the Battleship  why do you multiply be 2.618 instead of 2 for the hit points?

The 100 factor is used mostly by Vann Formula.
I rather prefer to use 6 IPCs Tank or 12 IPCs Cruiser/StB as the benchmark.

So, I use 36* or 144* in Enigma (inspired by Vann) Formula.

It is only a matter of convenience when comparing various strength values. There is more opportunity to get well rounded numbers.

Probably if using D8, the benchmark would be better by using an 8 IPCs units reference like a Destroyer, so 8^2= 64 and D10 a Fighter as benchmark at 10^2= 100.

On 2 hits unit, you have to remember that this formula is not totally arbitrary and is linked somehow to Lanchester Laws which describe fairly well the strength of 2 packs of units in battle when input to a BattleCalculator.

This 1.618 factor result from an empirical analysis of many combats on AACalc to find the 50%-50% odds of survival between a 1 hit unit and a 2 hit unit.
It might be 1.6 or 1.618, the AACalc left rooms for such.

The main research are in these threads:
https://www.axisandallies.org/forums/index.php?topic=40346.msg1684954#msg1684954

https://www.axisandallies.org/forums/index.php?topic=40347.msg1684962#msg1684962

At that time, there was no need to directly use odds, so we only consider attack/defense value based on D6.
Because we needed to compare between D6 and D8 system, a “2” on D6 (0.33) is not the same as “2” on D8 (0.25), that’s why relying directly on odds of making a hit was better.

About my modified A&A unit roster, you have to show me which roster you are talking about.
My favourite roster was an evolutionary process going along as other revamp project were discussed (mainly G40 Enhanced and G40 Redesign).
So, I need to understand which one you saw.

Refence unit cost^2actual unit Power1 HitPoint/(actual unit Cost^2){1+[(nb hit -1)/11.618034] }= offense or defense power*hit factor

I have always wonders how to exactly figure out how effective a 2 hit point unit is. I wonder what number you would use for a 3 hit point unit.

Refence unit cost^2actual unit Power1 Hit Point/(actual unit Cost^2){1+[([b]3 -1)/11.618034] }= offense or defense powerhit factor
Let’s suppose a Battleship A6 D6 C20:
3675%1/400 (1+21.618) = 0.28593
36.75/400 *4.236

So, instead of *2.618 for 2 hits unit, you use *4.236 for 3 hits unit

If using C6 Tank as benchmark:
I changed the actual unit Power with odds of unit.

Tank A 0.50% D 0.50% C6 HP1: Efficiency will be: 0.50
It means 36*(4/8)/(6^2) = 0.50
How can you get 0.8889?

And if only using “4” as combat value, it translates into 4.00.
Tank A 0.625% D 0.625% C10 HP2: Efficiency will be:
It means (36*(5/8)/(10^2) )* 2.618 = 0.58905

So, 0.589 is slightly above 0.50.
That way, such 10 IPCs 2 HPs Tank is an optimized purchase.

However, is it correct that it works like a Battleship?
So, if you have an army core of 10 such Tanks, it can absorb 10 hits before losing fodder units.
How do you repair such Tank?

I’m not sure if this can be a correct way of creating an Heavy Tank.

About TcB able to target Tank, the formula cannot consider such special capacity.
It considers an abstract situation of 2 stacks made of a single unit type, and loosing all the additional hits before the first unit is taken as casualty.
(It is called the skew effect.)
With TcB targeting Tank, it is a kind of reverse skew effect.
It needs to rely on a mixed stack of Infantry fodder and Tanks.
There is so many variations that it is impossible to get an average situation.

It is more realistic to move warships from one SZ to another.
But, would you retreat more than 1 TT to repair your Tank?
(Land movement are much slower and falling back damaged Tanks while staying in position with Art and Inf make such delay an issue. You also have to compare between planes for the cost. Aircraft mobility and versatility is way more interesting than 2 hits units which have to return to IC for repair.)

IMO, it is easier to consider that such unit are repaired right on place with spare parts.

A different procedure I suggested a while ago is that you can only repair one and have to pick a damaged Tank as casualty before allocating a hit to damage another Tank.

So, instead of 10 Tanks staying on battle until the 11th hits.
A second hits eliminates a first Tank.
This becomes far less effective, but it is easier to follow a single damaged Tank per stack.

However, I did not search on how to calculate such way of allocating damage.