Victory in the Pacific

This is my attempt to devise an easily used and remembered system to compare the widely varied forces used in VitP, to assist in combat planning and withdrawal decisions.  It uses additive scores for each ship and LBA intended to 'predict' what side will be remaining at end of battle (equivalent to denying control of the area to the other side).  It does not predict a side's control (if patrols are a small proportion of own force) nor relative losses - ships sunk.  I used the Lanchester model (deterministic) to obtain
Equivalent Values that are close to the centre-points of the random distributions. From my experience, the approximation is very good - much better than adding firepower!  Modelling errors are much smaller than the ample randomness present in the game of VitP. It's still prudent to use enough superiority (20-50%).

Airpower Comparison

This is most often the decisive component of a combined air/surface battle, being the fastest-deciding part (compared to gun combat).  Land-based-air (LBA) units rule the skies, and carriers (CV) rule over everything else:

Note:  If 2-4-* LBA are an important part of the force, raise all LBA values (only) to correct for 'intransitivity':

Very important may be the presence of Marines to threaten and/or take critical bases, attracting massive enemy fire in the first round to prevent that!  Subtract about 5p from enemy air value for each own Marine present (The damage not done on own carriers while they attack the Marines).  Long-term strategic value of bases may be much more.

Surface (Gun-firing) Ship Comparison

This is a separate comparison, 'surface value' points are less valuable than the air value above. The 44* battleship (BB) owned by all powers is taken as a reference of 10p.  Roughly a cruiser (CA) is worth *half* a BB, not 1/4!

Planes vs. Guns Comparison

If one side has carriers and the other has surface ships in excess of the opponent's number, the outcome will depend heavily on the day/night dice roll(s). Averages are still useful for light carriers (CVL) that usually do not decide in one round. Removal probabilities were weighted by the average number of Day rounds before the first Night round (I consider a Day/Night as two sub-rounds).

Average D/N ratio = (P(D)+P(DN))/(P(DN)+P(N))

The comparison is again to the 444 BB, these are Surface-equivalent points. One Air-point is worth 1.4-2.8 Surface-points.  The control value of LBA against surface ships alone should be infinite, but its firepower may be not enough to prevent the enemy fleet from sinking most of the own fleet before they retreat.

Carrier Value in Surface Points

This value depends largely on the Day/Night roll modifier advantage for the carriers -- such that three values are shown for each type of carrier depending on the flag situation.  Hosho is worth a 444 BB in a flag defense!  The number under the drm is the number of day actions expected for each night action.

Shadow Forces

This refers to the Allied advantage of Ultra intelligence - deciding moves *after* knowing enemy dispositions. So the prudent IJN player should count the maximum value of US forces that could appear in every area, even if this means counting the same unit several times!  If arriving to fight depends on 'speed rolls', multiply the value by the probability (ex. 5+65 raiding 3 areas deep: 12p x 4/6 = 8p).  Even the US can think this way (IJN shadow carrier-based air force), because IJN raids are decided after Allied LBA.

But the IJN can benefit from 'partial deterrence' because each US ship actually appears only once. ex. IJN defence is so arranged that A cannot be taken (easily) simultaneously with B.  US chooses to fight for A, and not B or C.  "He who defends everything defends nothing."

The Lanchester Model

Lanchester's model (1916) shows each side 'attritioning' the other in fixed proportion to its own strength, using deterministic differential equations:

dX/dt = -ky * Y(t)
dY/dt = -kx * X(t)

In this application, the 'attrition coefficients' kx and ky are the probabilities of Unit Removal (ship sunk, disabled, or carrier losing airpower).

The Equivalence Value (EV) is the force ratio X/Y at which both sides attrit each other exactly to zero (in the deterministic model):

EV = sqrt (ky/kx)

The Scores proposed are these EV, multiplied by 10 (for a 'reference' enemy unit) and rounded to the nearest integer.

Example 1: 137/4+ CV vs 34* LBA

LBA is immune to Disable, use Sinking probability without Bonus and against Armor=3 because equal dice (4) removes it. CV Armor is considered 1 less because equal dice (3) remove its air attack making it useless.

kx=32% Sinking: CV 4 on LBA 3 Armor
ky=63% Removal: LBA 3 on CV 2 Armor

Equivalent Value EV=sqrt(.32/0.63)=0.713
x10 and rounded = 7p for the carrier (LBA=10p)

Example 2: 2*(1+17) CA vs 444 BB

I do not compare 1 CA+ vs 1 BB, being too far from equivalence, but a 2:1 match that is much closer. Here kx is the contribution of each CA+ in removing the BB.  Generally, 'ganging up' the own forces on the enemy decreases own efficiency in removing them (but prolongs the battle and increases total ships sunk for both sides).

2+ fire on 444 P(remove)=2*kx=51%
(for one ship) kx=26.5%
4 fire on 1+17 ky=77%

Equivalent Value EV=sqrt(26.5/77)=0.587
x10 and rounded = 6p for one cruiser

Example 3: 12*/3+ CV vs 444 BB

CV side controls area (DRM+2, or 3 Days/Night)

Day: 3+ fire on 444 P(remove)=68%
Night: 1 fire on 444 P(remove)=22%
Average: kx=(3*0.68 + 0.22)/4 =56.5%

Only at night: 444 fires on 12*. P(remove)=77%
Average: kx=(3*0 + 0.77)/4 = 19.25%

Equivalent Value EV = sqrt (.565/.1925) =1.713
x10 and rounded = 17p for one 1-2-*/3+ carrier

Comments

* Randomness: I don't know yet the exact relationship between the superiority ratio (using these scores) and the probability to win, but guess it's a S-shaped curve. If anyone knows better, please let me know !

• More is always better, but a very large superiority gives little more insurance against extremes.
• Big is safer: the relative effect of luck decreases as both forces are increased in same ratio. I guess 50:40 points may be 70% to win but 100:80 may be 85%.
• LBA (as targets) have the largest variance, compared to their small probability to be knocked out. Use more superiority (even 2:1 or 3:1) to be really sure.
• Big surface-ship battles are easier to predict.
• There are no guarantees - anything may happen!

* Intransitivity: With this model, if 1 A = 2 B and 1 B = 1.5 C, it does NOT result that 1 A = 3 C!  The errors are smaller if the 'common reference' enemy unit is in the middle of 'combat speeds' and 'vulnerability'. That's why the 4-4-4 was proposed.

I hope and dream these will help VitP players as much as the 4-3-2-1 point system (by Charles Goren) aided Bridge players!

Comments, criticism, improvements are welcome!  Happy gaming!
...but don't forget the real victims of war

Mircea Pauca, Bucuresti, Romania