A Further Role of the Dice
(More odds for VitP)
By Bill Burch
I like to start by acknowledging Brad Solberg's article "The Dice Got Me Again!" Without his article it would be unlikely that this amateur mathematician, in a pique of curiosity, would have calculated and generated the tables presented later in this article.
Brad's article presents many of the numbers needed by the player to do some of the elementary odds calculations required to make game decisions based on the odds of an event occurring (game theory). Unfortunately, I feel that by analyzing, reworking, and changing the presentation of this data the player may better discern what are the "best" decisions possible.
Several things attracted my attention while I was reading Brad's article for the first time. First were the missing armor factors of 7 and 8. Being a QA professional, completeness of a data set is always a high priority. Second was the lack of digits (e.g. significant figures) in his tables. Again the QA in me wanted as many digits in the answer as manageable. Since the numbers in the tables are intermediates to the ends desired by players, rounding off the numbers should be delayed until the final answer is calculated. Third, and final, was the lack of direction of what to do with the data (please no suggestions Brad!) or how to practically apply it to game situations. Practical uses of the data are required to help the gamer make decisions during game play. This is especially useful in PBEM games where there is less of a time constraint than in a FTF game or a Tournament game.
At times different viewpoints will help a gamer make a decision within the game. Hence, one could look at the game only from one's own point-of-view and seeing where he can best sink (kill) or remove his opponent's units. Or he can also look at it in another way. From your opponent's point-of-view, he will also be trying to sink or remove your units. This suggests that you should also be looking at how your own units are going to resist being sunk (survival) or resist being removed (remaining engaged).
This brings us to the most important point. What do you, as the gamer, want to accomplish? Stated in another way, what do you tactically want to achieve? Do you want to stop your opponent from accomplishing an objective? Maximize damage on your opponent's fleet? Sink as many of your opponent's ships as possible? (Note: These last two are not the same.) Not lose many ships to your opponent? Remain in, and control the area? Accomplish your own objective?
Choosing what you want to accomplish is only the first step. Applying the data from the tables to anything but a trivial situation would try many a mathematician. Therefore, I have attempted to summarize the trends in the tables in guidelines and I have also come up with a method to help the player get a feel for who will win a large fleet battle.
Tables Old and New
The first tables to examine are the redo of the six tables from Brad's article. Here again are the tables for Day/Night Odds, Probability of Disable in X shots, Probability of Sinking (normal shots), Probability of Sinking (bonus shots), Probability of Removal (normal shots), and Probability of Removal (bonus shots).[Updated Odds Table]
The numbers in these tables are different from Brad's numbers (but only by small amounts) and were calculated using a cascade-event matrix to generate the distribution of damage for a given number of hits, with the results linked to the sums of a series of binomial distributions. This takes advantage of the fact that we are only interested in the sum of results from the hits generated (combinations only) and using a sum of binomials that can be linked to the damage results while exactly duplicating the trinomial distribution of results (hit, disabled, miss). All this was done to generate the additional digits, results for Armor Factors 7 & 8, higher confidence in the odds stated, and making the table a little more readable.
The rearranging of how the odds were calculated had the added advantage of allowing overall odds of different categories of results to be calculated. Hence, given the same 1 to 12 shots from Brad's tables I could calculate the odds for each of the following categories: (1) All miss, (2) Disabled only, (3) Hit only, and (4) Hit and Disabled. This certainly adds some new ways of looking at the results of our actions versus the Sunk, Disabled, and Removed of the first set of tables.[New Catagories Table]
Fortunately this wasn't enough to satisfy my curiosity. I continued to play with the category tables until I came up with a way to precisely calculate the odds of categories taking into effect both number of shots and armor factors. From this I generated a family of eleven tables that delineates results for each of the possible offensive shots (1-6 normal or 1-5 bonus shots). The new categories calculating the odds in this manner are: (1) All miss, (2) Disabled, (3) Damaged only, (4) Damaged and Disabled, and (5) Sunk. Again adding more information for the gamer to use. The sum of categories 2, 4, and 5 is the same as the original removal tables. Since two different methods were used to calculate removal odds (original and categories) with identical results, this new way of categorizing results verify my earlier odds calculations.
|1 Normal Shot||1 Bonus Shot|
|2 Normal Shots||2 Bonus Shots|
|3 Normal Shots||3 Bonus Shots|
|4 Normal Shots||4 Bonus Shots|
|5 Normal Shots||5 Bonus Shots|
|6 Normal Shots|
When I was analyzing the above tables, something else became obvious. My old assumption of how much damage was expected from a salvo was wrong. I used to think that the expected damage of a salvo was based on the probability of obtaining a hit and the expected damage per hit (3.5 damage/hit). Unfortunately, I overlooked that the armor factor was also a part of this function. Let me demonstrate. Let's say that Hiryu is firing at the San Francisco during a day action. Hiryu has three bonus shots and thus would expect exactly one hit. This would translate to an expected damage of 3.5 if we fail to take into account the San Francisco's armor factor. Its very hard to believe that the expected damage could be 3.5 when the maximum damage the San Francisco may take is 2.0 (sunk). Clearly a new set of tables was required to show the expected damage per salvo. The following tables show the expected damage per salvo (1 - 12 normal or bonus shots) versus armor factor from 0 to 9. Please note that it is only possible for cruisers (and other 1 shot wonders) to reach the theoretical maximum expected damage.[Expected Damage from Normal Shots]
As short sidebar articles to the use of the tables above there are several set (or common) situations in Victory in the Pacific which can lead to a more in depth analysis of the situation. One of these set situations is the Pearl Harbor air raid and you can select the link below to open this analysis. As an final addition to this article three other sidebar articles have been added and the links can be found below.[Pearl Harbor Air Raid]
The final thing I did was to try to simplify the shear mass of numbers and tables I just presented. Here I was at a quandary on how to simplify. What I chose to do was to do a mathematical process called normalization. Simply, I chose the probabilities of a 4 normal factor shot to remove or sink a 4 armor factor target, set that value equal to a value of 10, which became the basis of judging the value of all other armor factors and attack factors in the game. In other words, this became my "yardstick" which I used to measure how different attack factors and armor factors perform within the game.[Normalized Set of Tables]
The normal combat tables above are not four separate tables. Rather, they are two pairs of tables. Each pair must be used separately to evaluate the attrition and tactical (holding the area) situations. The top pair of tables (Survival/Kill) is used to evaluate which of the players will have the advantage in attrition combat (who will likely sink the most ships). The bottom pair of tables (Engaged/Removal) is used to evaluate which of the players will have the advantage in tactical combat and will hold the area for control or denial of control.
Here I put forward that a fleet's ability to survive, to kill the opponent's ships, to remain engaged with the enemy's fleet, and to remove the enemy's fleet is related to the sum of attack and defense factors stated above. Also in both of the defensive tables there is a 'NF@' armor factor. When an unequal number of ships are present in a battle, ships that are not fired at will be counted as the NF@ defense factor instead of their usual defense factor. Special circumstances are also noted in the tables to account for the effect of LBA upon a fleet combat. Stated simply, LBA's presence devalues the opposing fleets air attack factors. Some easy examples are in order to clarify what I mean.
First Example: Evaluate for tactical combat the situation where an IJN 5-5-4 faces two USN 4-5-3s. The 5-5-4 has a surface attack factor of 11.3 and a defense factor of 11.2. The two 4-5-3s have a surface attack factor of 20.0 (10.0 + 10.0) and a defense factor of 40.5 (11.2 + 29.3). Note that the second allied battleship has a much larger defense factor since it cannot be fired at because the IJN player does not have a second ship.
Second Example: Evaluate for attrition combat the situation where two IJN 3-4-* LBA faces an USN 0-2-7(4+) CV and a British 0-2-7(2) CV. The LBA units have an air attack factor of 14.8 (7.4 + 7.4) and a defense factor of 18.0 (9.0 + 9.0). The Allied player's forces have an air attack factor of 14.8 (10.0 + 4.8) and a defense factor of 16.2 (8.1 + 8.1). Note that the air attack factor of the USN's CV was devalued from 20.2 to 10.0 since it is fighting a LBA unit. Also, if you were the Allied player evaluating this situation and knew you were going to use both CVs versus one LBA before fighting the other (doubling up) the defense factor of the IJN forces would be 22.2 (9.0 + 13.2) instead of 18.0.
Third Example: Evaluate for tactical combat the situation where three IJN 3-4-* LBA faces an Allied force composed of two British 0-2-7(2) CVs, a British 1-2-4(1), and a 2-4-* LBA. The three IJN LBA have an air attack factor of 19.3 (8.3 + 8.3 + 2.7) and a defense factor of 26.7 (8.9 + 8.9 + 8.9). Note that one of the IJN's LBA attack factor was devalued since at least one of the LBA units will have to engage the Allied LBA unit. The Allied player's forces have an attack factor of 6.2 (1.8 + 1.8 + 0.8 + 1.8) and a defense factor of 52.7 (7.8 + 7.8 + 7.8 + 29.3). Here the highest defense factor (the LBA's 8.9) was replaced with the NF@ defense factor since the Allied player has an extra unit engaged in the battle.
Notice in the three examples above that there is no mixture of air attack factors and surface attack factors. Each example was either a pure day battle or pure night battle. A slightly more complex situation would involve the analysis of fleets with both air and surface attack factors present. Air attack factors are simply summed for the entire fleet (of course devalued if engaging LBA). Surface attack factors are summed for surface ships only, unless a carrier(s) will likely be firing during a night action; in this case the surface attack factors for the firing carriers is added also. Defense factors are the sum of all the fleet's defense factors (including SNLFs and Marines if they are likely targets). There is one exception to summing all of the fleet's defense factors. When both players will agree to either a day or a night action, then only the defense factors of the part of the fleet engaged (carriers and LBA for day and surface ships for night) are summed.
After air attack factors and surface attack factors have been calculated, the air and surface attack factors must be devalued if day/night will not be agreed upon. Depending on the day/night odds, apply the correction factors from the table below.
|Case||Air Attack Factor Correction||Surface Attack Factor Correction|
|+0 (Split Bonus)||0.58||0.50|
|+1 (Day only)||0.72||0.35|
|+2 (Day and Flag)||0.83||0.22|
The sum of the above corrected attack factors, now becomes the overall attack factor for the fleet.
Fourth Example: Evaluate for attrition combat the situation where an IJN fleet composed of a 1-1-8(3+), a 1+2-7, and a 4-3-6 faces an Allied fleet composed of two 1-1-7s and a 5+9-7 in an area where the IJN player wants day and has the flag. First doing the calculations for the IJN fleet, the air attack factor is 15.3 and the surface attack factor is 16.9 (2.3 + 4.6 + 10.0). Note that the carrier's surface attack factor is included since the carrier is likely to be fired at (and return fire) during a night battle. The overall attack factor for the IJN is 16.4 ((15.3 * 0.83) + (16.9 * 0.22)), while the defense factor is 24.3 (7.2 + 8.1 + 9.0). Second doing the calculations for the Allied fleet, the air attack factor is zero and the surface attack factor is 29.1 (2.3 + 2.3 + 24.5). The overall attack factor for the Allies is 6.4 ((0.0 * 0.83) + (29.1 * 0.22)), while the defense factors is 27.2 (7.2 + 7.2 + 12.8).
Fifth Example: Redo the fourth example in an area where the IJN player wants day versus the flag. All calculations are the same except calculating the overall attack factors. For the IJN fleet, the overall attack factor is 17.3 ((15.3 * 0.58) + (16.9 * 0.50)), while for the Allied fleet the overall attack factor is 14.6 ((0.0 * 0.58) + (29.1 * 0.50)).
Though you now have a lot of new information to help estimate the odds in a battle, one more step is needed. Using just four numbers (the two attack factors and defense factors from the two fleets), two ratios can be calculated for either the attrition or removal situation. These ratios are: (1) IJN attack factor / Allied defense factor and (2) Allied attack factor / IJN defense factor. Whoever has the highest attack factor to defense factor ratio will most likely win the battle. In addition, the ratio gives you an idea how much wild swings of luck are likely to effect the battle. Ratios less than 0.5 will be prone to wild swings of luck, while ratios over 0.8 will be fairly stable and less prone to swings of luck.
Sixth Example: We will continue with the fifth example. The IJN fleet has an attack factor of 17.3 and a defense factor of 24.3, and the Allied fleet has an attack factor of 14.6 and a defense factor of 27.2. The IJN attack/Allied defense ratio is 0.64 versus the Allied attack/IJN defense ratio of 0.60. Since the ratios are almost equal and both are fairly low (<0.8 but >0.5), luck will likely be the deciding factor in determining who will win the attrition battle. Most players would recognize this as little more than a 50/50 gamble for the IJN player.
Seventh Example: Using the same forces and situation as used in the sixth example, lets shift our focus to the ability to control (or deny control) the area. Without the math being shown, the IJN attack factor/Allied defense factor ratio is 0.46 and the Allied attack factor/IJN defense factor is 0.19. In the tactical battle the IJN players will most likely win due to the larger ratio, but since both ratios are very low (<0.5), there will likely be large swings of luck.
Though the examples above were kept simple, the normalized combat tables were intended to be applied to large fleet battles (10+ ship per side). Applying these tables to smaller battles (as in the examples above) will still yield useful numbers, but luck will play even a larger part. One further caution, normalized functions are usually based on well-behaved functions. In VitP the presence of a large number of cruisers (>50% of the fleet) or many large armor factor ships (6+) may warp the results.
General Trends and Guidelines
This section is for the people who shy away from math intensive operations. I have spent a lot of time looking over the data presented in the tables above and have been able to come up with some general guidelines from the trends I found. Remember in the introduction when I asked, "What do you, as the gamer, want to accomplish?" Here's what the odds/numbers suggest:
A Complex Example
It is Turn 5 and the IJN Player is ahead by 25 PoC and controls the Hawaiian Islands. The IJN perimeter consists of the Hawaiian Islands, Marshall Islands, the Marianas, and Indonesia. The IJN Player lost control of the South Pacific Ocean and Lae fell to the Allied Player on Turn 4 in a major fleet action that caused significant carrier losses to both sides.
Turn 5 opens with the IJN Player placing patrol ships in each of his rear areas. He also sends three 1+2-7 cruisers, two 1+1-7 cruisers, and two 5-5-4 battleships (one failing its SR) to patrol Indonesia. The Allied Player responds with scant surface patrols, but sends two British 4-4-3 battleships (both fail their SRs), a USN 1-1-7 cruiser, and a USN 5-5-3[d1] battleship to patrol and contest Indonesia.
During LBA Placement the IJN uses his five LBA to secure the rest of his perimeter against the Allied Player's Yorktown and Victorious by placing them in the Hawaiian Islands, Marshall Islands, and the Marianas. The Allied Player uses his ten LBA to secure his areas devoid of patrols and places the remaining six LBA in Indonesia.
The IJN Player responds to the Allied Player's challenge by placing his sole SNLF in Indonesia. The Allied Player promptly ups the ante by sending one marine to both Indonesia and the South Pacific Ocean.
The IJN then sends a cruiser to the Coral Sea to challenge a lone patrolling allied cruiser, a damaged 4-4-4 battleship to the SPO to attempt to sink the marines, and the rest of his ships go to Indonesia including seven CV/CVLs. The Allied Player responds by sending all his raiders to Indonesia. A 1-1-7[d1] damaged cruiser makes its SR to Indonesia, while a 4-5-3 battleship and a 5+6-5 battleship fail their SRs and return to Samoa.
Before I apply the Tables what is your analysis of the situation? Forces are list below. There are 59 total units involved in the battle (28 IJN & 31 Allied).
Air: 1-4-6(4+), 1-1-8(3+), 1-2-4(3+), 1-2-4(3+)[d1], 0-1-5(2+), 2x 0-0-5(2+)
Surface: 2x 6-9-5, 2x 5-5-4, 2x 4-4-4, 4x 4-3-6, 5x 1+2-7, 3x 1+1-8, 2x 1+1-7
Other: 0-3-3 SNLF, 1-0-* I-Boat
Air: 6x 2-4-* LBA, 0-2-7(4+)
British Surface: 2x 4-4-3, 2x 1-1-7
USN Surface: 2x 5+6-5, 5-6-5, 5-6-5[d1], 5-5-3[d1], 2x 4-5-3, 11x 1-1-7, 1-1-7[d1]
Other: 0-4-3 Marine
The IJN Fleet:
Air Attack Factor (Attrition): 46.6
Air Attack Factor (Control): 17.2 (vs LBA) or 66.1 (vs ships)
Air Defense Factor (Attrition): 61.5 (CV/CVLs + SNLF)
Air Defense Factor (Control): 77.7 (CV/CVLs + SNLF)
Surface Attack Factor (Control): 160.7 (includes 3 CVs) or 153.8 (without 3 CVs)
Surface Defense Factor (Control): 177.5 (without 3 CVs) or 202.0 (with 3 CVs)
Fleet Defense Factor (Control): 229.2 (includes SNLF) or 220.3 (without SNLF)
The Allied Fleet:
Air Attack Factor (Attrition): 49.0
Air Attack Factor (Control): 48.7 (with LBA) or 12.1 (without)
Air Defense Factor (Attrition): 62.1 (CV + LBA) or 72.1 (CV + LBA + marines)
Air Defense Factor (Control): 61.2 (CV + LBA)
Surface Attack Factor (Control): 148.1
Surface Defense Factor (Control): 194.0
Fleet Defense Factor (Control): 255.2 (if 3 IJN CV engage) or 305.6 (if not)
In the overall situation (a Control fight), the IJN Player has an overall attack factor of 86.9 ((17.2 * 0.58) + (153.8 * 0.50)), and a fleet defense factor of 220.3. The lower surface attack factor was used to calculate the overall attack factor since the IJN Player had no intention of allowing the Allied Player to 'clump' salvos versus his patrolling ships and 3 firing CVs during a night action. The Allied Player has an overall attack factor of 102.3 ((48.7 * 0.58) + (148.1 * 0.50)), and a fleet defense factor of 305.6.
These numbers yield an IJN Attack/Allied Defense ratio of 0.28 and an Allied Attack/IJN Defense ratio of 0.46. Meaning that the Allies have a fair-to-heavy advantage in the fight but wide swings of luck will likely determine the result.
Don't agree? Let's break the battle down into three cases based on the Day/Night dice roll results.
Case 1: (Day) = 41.67%
A quick look the fleets show each side has 7 air units and 1 amphibious unit. The Allied Player must engage the SNLF to keep his LBA in the area and the IJN Player will likely engage the marines since he will not see if his SNLF remains until after his shots and he hopes to save the Philippines. This day battle can be analyzed as either an attrition situation or a control situation. If the IJN Player want to 'shoot and scoot' in case his SNLF fails to take Lae, then it is an attrition situation and he concedes control of Indonesia to the Allies. For attrition the IJN/Allied ratio is 0.65 (46.6/72.1) and the Allied/IJN ratio is 0.80 (49.0/61.5). The Allies have a fair advantage in an attrition battle and will likely cause more damage (sinkings) than the IJN Player. But the Allied units are almost all LBA, so the IJN fleet can safely disengage and fight again next turn at the cost of losing control of Indonesia. If the IJN Player intends to stay and contest control then the situation can be analyzed as a control situation. For control the IJN/Allied ratio is 0.28 (17.2/61.2) and the Allied/IJN ratio is 0.63 (48.7/77.7). The IJN Player should not engage the USN amphibious unit since it is a moot point if the IJN controls Indonesia along with the Marianas. Disregarding extreme luck (fairly possible given the very low IJN/Allied ratio) or the SNLF taking Lae, a control air fight is an Allied win with the Allies controlling Indonesia, holding Lae, and possibly capturing the Philippines. [Note: For argument sakes say the 0-2-7(4+) will engage the SNLF with a 83.99% chance of removal.]
Case 2: (Day followed by Night) = 16.67%
This starts the same as Case 1 above without the IJN Player being able to choose to make it an attrition situation. It's an air control battle with the added benefit/curse of a surface control battle to follow. As above the air control battle follows the same reasoning as above and will not likely impact (with the exception of lost IJN CVs) the surface control battle. For the surface control battle the IJN/Allied ratio is 0.83 (160.7/194.0) and the Allied/IJN ratio is 0.73 (148.1/202.0). Disregarding extreme luck (possible but unlikely given a day and a night battle) or the SNLF taking Lae, a control fight is an Allied win with the Allies controlling Indonesia, holding Lae, and likely capturing the Philippines, with the IJN fleet forced to disengage having suffered attrition to their surface fleet in addition to the losses from the air battle.
Case 3: (Night) = 41.67%
In this case the SNLF will land at Lae and remove the 6 Allied LBAs from Indonesia after the surface battle. This night battle should be analyzed as a control battle since the IJN Player will have both day and flag bonuses to the Day/Night rolls and at least a 4 to 1 carrier ratio following this exchange. The Allied player has a simple choice of bunching his shots on the 6 IJN patrolling ships, or lapping the IJN's surface ships to engage 3 carriers. If the Allied Player bunches his shots he may be able to remove the flag from Indonesia and may chance landing his marines in the Philippines. However even with the large number of shots there is no certainty of removing six patrolling vessels (see Brad's article for this argument). Furthermore, when the Allied Player does disengage the IJN Player will have 7 carriers and a larger number of surface vessels than normal to pursue him. If the Allied Player laps the IJN surface ships and engages three carriers he will likely remove 2 or 3 carriers from the following pursuit, but its almost impossible to remove the IJN patrollers and there is no purpose in landing the marines in the Philippines. Assuming the IJN Player will not engage with 3 carriers, the IJN/Allied ratio for the surface battle is 0.79 (153.8/194.0) and the Allied/IJN ratio is 0.73 (148.1/202.0). After this near even exchange of ship removals we need to recalculate the overall factors based on the loss of the LBA units. Without showing the calculations, the new overall IJN/Allied ratio is 0.28 (86.7/305.6) and the Allied/IJN ratio 0.19 (42.3/220.3). These are calculated without knowing the actual losses from both sides after the night battle, but they do show that the IJN Player would have a heavy advantage in any further rounds or pursuit in Indonesia. Indeed the most likely result of a night battle for the Allies would be the temporary loss of Lae, heavy attrition of his surface forces and the possible losses of a carrier and a marine with Indonesia remaining IJN controlled.
When I started this trip through the world of probability and statistics, my main intent was to find some way to better describe the tactical situations we found through playing VitP. This evolved into the "yardstick" of the normalized combat tables. But be warned the analogy of a yardstick is apt here. We are using that yardstick to measure a curved surface. The numbers generated using the normalized tables are not exact and can never be taken to be exact measurements. Unfortunately, exact measurements lie outside the scope of an article.
Victory in the Pacific is a strategic game that generates many tactical battles. Hopefully, with what I have covered you can better understand some of the tactical choices you may use to pursue your strategic goals. Whatever your choices (attrition, survival, maximum damage, etc.), remember that tactics are a way to obtain your goals, but in most cases good strategy will win out over good tactics.- wcmb