## 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

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.

Ending State | Probability |
---|---|

1-1-7 Missed | 22.458% |

1-1-7 d1 | 3.867% |

1-1-7 Disabled | 26.271% |

1-1-7 d1 + Disabled | 1.645% |

1-1-7 Sunk | 45.759% |

1+2-7 Missed | 47.222% |

1+2-7 d1 | 2.898% |

1+2-7 d2 | 2.970% |

1+2-7 Disabled | 26.389% |

1+2-7 d1 + Disabled | 0.750% |

1+2-7 d2 + Disabled | 0.777% |

1+2-7 Sunk | 18.994% |

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

Result | Probability | Notes |
---|---|---|

Both Removed | 20.584% | Tie: No Control |

1+2-7 Removed Only | 26.325% | Allies Win and Control |

1+2-7 Removed | 46.909% | Allies Breaks Control |

1-1-7 Removed Only | 53.090% | IJN Win and Control |

1-1-7 Removed | 73.674% | IJN Breaks Control |

Both Sunk | 5.139% | - |

1+2-7 Sunk Only | 13.855% | - |

1-1-7 Sunk Only | 40.620% | - |

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

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

First we'll start with the

Result | Probability | Notes |
---|---|---|

Both Removed | 16.386% | Tie: No Control |

6-9-5 Removed Only | 2.012% | Allies Win and Control |

6-9-5 Removed | 18.399% | Allies Breaks Control |

1-1-7 Removed Only | 81.601% | IJN Win and Control |

1-1-7 Removed | 97.988% | IJN Breaks Control |

Both Sunk | 0.038% | - |

6-9-5 Sunk Only | 0.025% | - |

1-1-7 Sunk Only | 65.906% | - |

Next we'll look at the

Result | Probability | Notes |
---|---|---|

All Removed | 11.637% | Tie: No Control |

6-9-5 Removed Only | 35.340% | Allies Win and Control |

6-9-5 Removed | 46.977% | Allies Breaks Control |

Both 1-1-7s Removed Only | 53.023% | IJN Win and Control |

Both 1-1-7s Removed | 64.660% | IJN Breaks Control |

All Sunk | 0.337% | - |

6-9-5 Sunk Only | 1.777% | - |

Both 1-1-7s Sunk Only | 26.022% | - |

__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

Result | Probability | Notes |
---|---|---|

All Removed | 25.751% | Tie: No Control |

LBA Removed Only | 17.748% | Allies Win and Control |

LBA Removed | 43.499% | Allies Breaks Control |

CV Removed Only | 56.501% | IJN Win and Control |

CV Removed | 78.671% | IJN Breaks Control |

Both Sunk/Elim. | 10.793% | - |

LBA Elim. Only | 17.748% | Same as Removed |

CV Sunk Only | 22.114% | - |

__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!