Hard Numbers Are Hard To Come By (The Trouble With Hard Numbers) By Bill Burch

Introduction

While my article was being reviewed by John Pack for inclusion on his web site, he made several suggestions. One of those suggestions was to include more applications of the numbers and statistics to common situations that occur in almost every game of VitP. Given that my article was very large already, I disagreed with John. But John's idea was very worthwhile so I proposed addressing these applications by adding another set of sidebar articles. John agreed and here is one of the results.

Why Use Fuzzy Numbers?

If you have read either the main article or any of the other sidebar articles, then you would have seen that I rarely state the exact probabilities of any outcome. You might have noticed the condensed categories of 'Sunk' or 'Removed' or the normalized tables from the main article. Or maybe the critical ratio from the cruiser sidebar. How about the maximized kills, bottomings, or damage from the 'Pearl Harbor' sidebar? There's a reason for why I don't take the numbers to the nth degree.

Lets take a little example. Say a Allied 1-1-7 cruiser fires at a target. You roll one die. So how many different outcomes are there to this one die roll? Six? Three? No, the answer is eight without knowing what the target is. If you thought it was six, you were probably thinking about the six different sides of the die {1,2,3,4,5,6}. If you thought it was three, you might have been thinking about the three categories of outcomes {miss, disabled, and hit}. But the hit gets rerolled to show the amount of damaged caused. So the eight outcomes possible are miss, disabled, 1 damage, 2 damage, 3 damage, 4 damage, 5 damage, and 6 damage.

Now lets say the Allied 1-1-7 cruiser was firing at an IJN 1+2-7 cruiser. By naming the target the number of possible outcomes changes from eight to five {miss, disabled, 1 damage, 2 damage, and sunk}. Starting to get the idea how VitP's combat system generates unique challenges to the mathematician?

Lets go look at a multiple-shot salvo. Say a IJN 4-3-6 battlecruiser is firing at a target. There are 44 possible outcomes to the battlecruiser firing {miss, disabled, 1 damage ... 24 damage, 1 damage + disabled ... 18 damage + disabled}. If the battlecruiser is firing at a USN 4-5-3 battleship the number of possible outcomes drops from 44 to 13 {miss, disabled, 1 damage ... 5 damage, 1 damage + disabled ... 5 damage + disabled, sunk}.

Example #1: Take the above one-on-one match-up {4-3-6 vs. 4-5-3}. The 4-3-6 battlecruiser has 13 possible results versus the battleship, while the 4-5-3 battleship has 9 possible results versus the battlecruiser. Multiplying the number of results possible gives us 117 possible outcomes from the first round exchange of the battle.

Example #2: Take the worst case one-on-one match-up {6-9-5 vs. 5+9-7}. The 6-9-5 has 21 possible results versus the 5+9-7, while the 5+9-7 also has 21 possible results versus the 6-9-5. Multiplying the number of results possible gives us 441 possible outcomes from the first round exchange of the battle.

Take a look at the examples above and it becomes crystal clear that some way of simplifying the calculations of the outcome odds must be found. Unfortunately there are many obstacles to finding these simplifications. The first obstacle comes with the realization that the outcomes form an infinite series. This infinite series at first appears to be a multinomial distribution. But there are problems. Multinomial distributions require that the events of the series be independent of each other. But in the combat system used in VitP, the results of the die rolls are not independent of each other. Disabled results apply only once and any further disabled result has no effect on the result. Damage results add together until they exceed the armor factor of the target. All further damage beyond the armor factor plus one also has no effect on the result. These two facts make it clear that the results of combat die rolls are partially dependent on one another. This and other factors such as LBA deleting disable results and cancelling the USN carrier's airstrike bonus leaves the mathematician with little recourse but to resort to exhaustive and brute-force methods to calculate the 'hard numbers'. To this end, I have created three spreadsheets to look at three specific combat situations and show the exact results (with less than .001% error).

Easy One First?

Easy is a relative term for this first one and required a 277 KB spreadsheet to solve. This example pits a IJN 1+2-7 versus an Allied 1-1-7. The IJN 1+2-7 has a possibility of ending each round in one of seven states {Missed, 1 Damage, 2 Damage, Disabled, Disabled +1 Damage, Disabled +2 Damage, and Sunk}. The Allied 1-1-7 has a possibility of ending each round in one of five states {Missed, 1 Damage, Disabled, Disabled +1 Damage, and Sunk}. The two cruisers continue to fight round after round until one or the other (or both) are removed. The two cruisers have six (2x3) possible continuing states. Fifteen rounds were required to lower the sum of all continuing states to <.0005%. Therefore the results form a 4D (5x7x6x15) matrix of 3,150 cells. This is for a very simple battle.

Before giving the results most players would expect the IJN cruiser to have about a 3-to-2 advantage in removing the Allied cruiser and about a 2-to-1 advantage sinking the Allied cruiser due to the likelyhoods of hits and disables being rolled on their respective combat dice. Let us see if this holds true.

To make it easier to read the table below summarizes the results.

How did the assumptions of 3-to-2 for removal and 2-to-1 for sinking end? The 3-to-2 for removals holds fairly close to the assumed value of 1.50 at 1.57. The 2-to-1 for sinkings is a little further off from the assumed value of 2.00 at 2.41. So for this simple example the hard numbers seem to be reinforcing our feel for the game.

What About Extra Ships?

The second example comes from John's editorial process of the origonal article. Example #1 in that article had a 6-9-5 battleship matching up versus two 1-1-7 cruisers instead of the current example. This example existed to show how a "not fired at" ship changed the defensive value of a side by replacing the defensive value with the 'NF@' value. For attrition combat the 'NF@' value is 13.2 and not much larger than the 10.0 for a 4 AF ship, but for control combat the 'NF@' value is 29.3 and about triples the 10.0 for a 4 AF ship. How can the first cruiser engaged against the battleship be worth a normalized value of 6.7 and the second cruiser be worth a normalized of 29.3? John rightfully thought that BB vs 2xCA example was a poor choice for an example. But the question remains. How much does an unengaged ship change the battle results?

This is a complex situation to calculate the exact probabilities for and required a 4.1 MB spreadsheet to solve. This example pits a IJN 6-9-5 versus two Allied 1-1-7s. The IJN 6-9-5 has a possibility of ending each round in one of 21 states {Missed, 1 Damage ... 9 Damage, Disabled, Disabled +1 Damage ... Disabled +9 Damage, and Sunk}. The Allied 1-1-7s have a possibility of ending each round in one of 10 states {Missed, 1 Damage, Disabled, Disabled +1 Damage, and Sunk for each cruiser}. The two sides continue to fight round after round until one or the other (or both) are removed. The two sides have 40 (4x10) possible continuing states. Eleven rounds were required to lower the sum of all continuing states to <.0005%. Therefore the results form a 4D (10x21x40x11) matrix of 92,400 cells. This is for a three-ship battle. The nice thing about doing this battle is that with the altering of two numbers in the matrix the results for a 6-9-5 vs single 1-1-7 is also given.

First we'll start with the 6-9-5 battleship fighting a single 1-1-7 cruiser. With the battleship having six attack dice to the cruiser's single die and with the battleship being able to withstand nine damage before sinking to the cruiser's one we see a very one-sided fight in favor of the battleship. Indeed the lone cruiser will only win this battle a little over 2% of the time and the you should never see the battleship sunk (about 4000-to1 against it).

Next we'll look at the 6-9-5 battleship fighting two 1-1-7 cruisers. With the battleship having six attack dice to the cruisers' two dice and with the battleship being able to withstand nine damage before sinking to the each cruiser's one damage we see a very one-sided fight in favor of the battleship again. Right? No, this time the odds change dramatically. The two cruisers will win this battle over a third (>35%) of the time and the you might just see the battleship sunk (about 55-to1 against it). This large change in the odds is not due mainly to the additional firepower and defense the added cruiser brings to the battle, but it is due to the battleship's inability to fire on more than one target during a single round. Unengaged ships will quickly push the odds to the larger fleet's side during a battle.

Carriers versus LBA

No other situation in VitP sparks misconceptions as carriers fighting LBA. So for the last of the exhaustive numbers crunching, lets look at a common 1-on-1 matchup between carriers and LBA.

This is a simplier situation than the last one to calculate the exact probabilities for and required a 303 KB spreadsheet to solve. This example pits a IJN 3-4-* LBA versus an Allied 1-3-7(4+) carrier. The IJN LBA has a possibility of ending each round in one of five states {Missed, 1 Damage ... 3 Damage, and Eliminated}. The Allied 1-3-7(4+) carrier has a possibility of ending each round in one of nine states {Missed, 1 Damage, 2 Damage, 3 Damage (crippled), Disabled, Disabled + 1 Damage, Disabled +2 Damage, Disabled +3 Damage, and Sunk}. The two sides continue to fight round after round until one or the other (or both) are removed. The two sides have 12 (4x3) possible continuing states. Eight rounds were required to lower the sum of all continuing states to <.0005%. Therefore the results form a 4D (5x9x12x8) matrix of 4,320 cells. Care was taken in calculating the final odds to take into account that a crippled CV cannot continue combat but will win control of the area if the LBA is eliminated on the same round that the CV was crippled.

In Conclusion

Though I have shown that "hard numbers" are possible to calculate, hopefully noticing the size and effort required to generate the spreadsheets I have also shown that it is impractical to do such in all but the smallest battles. In addition, it is possible (but not shown) to compute the odds for mixed day/night cases where carriers face surface units. Even for die hard mathematicians (pun intended), the "fuzzy" numbers I presented in the main article and the other sidebar articles should suffice. If anyone wants to try a more complicated case, I wish him/her luck. But for me this ends this series of articles. Good Luck!

Spreadsheet Files