Axis and Allies Revised: Standard Tactics: Planning, Risk Assessment, and Calculation
Written by Bunnies_P_Wrath 20-06-2007 at Axis and Allies.org forums
I. Steps for Assessing Risk
II. Approximation Methods for Large Battles
III. Assessing Utility Gain (IPC Loss and Gain)
IV. Calculation Methods for Small Battles
V. Other Utility Gain Calculations
VI. Assessing the Board Situation
VII. Long-Term Goals
I. Steps for Assessing Risk
I thought I?d write a few words regarding proper risk assessment and calculation. The proper steps for a turn are:
1. Determine long-term goals.
2. Determine short-term goals.
3. Determine minimum acceptable required to maximum available forces to carry out those short-term goals
4. Determine differing probable battle outcomes and the utility gained from each likely scenario, and opponent responses to minimum acceptable forces to maximum available forces allocated to those goals, along with the opponent?s utility gained from each opponent?s countermove. Determine what contingency plans to undertake in case unfavorable odds occur during a battle. Determine what battles are unacceptably risky in case unfavorable odds occur.
5. Determine whether or not, by factoring in probable opponent response, possible unfavorable odds, and unacceptable risk, whether or not a given short-term goal is viable, in context of long-term goals and available units.
6. Depending on calculated net utility gain, including both your own immediate utility gain, and anticipated opponent utility gain in response to your moves, allocate forces as necessary to carry out goals of greatest importance. As final allocation of forces for the most important goal is made, the available forces for other less important goals will change, changing the maximum available forces to carry out other short-term goals. So repeat steps 3 through 6 until a list of acceptable short-term goals is made (in context of the long-term goals).
7. Purchase units and/or research tech based on what is required for the long-term goal, and make appropriate combat moves.
8. Carry out the combat whose result should be known first to determine the desired outcome of other combats.
9. During combat, press the attack or retreat as is determined to be best, based on other combats still to be carried out, the anticipated non-combat move, and the anticipated unit placement at the end of the round..
10. Repeat steps 9-10 until combat is over.
11. Make a final reassessment of the board situation, and make the appropriate non-combat movement and placement.
That is, in a few words ? figure out what you want to do in the long run, figure out what you want to do right now, figure out the what benefit your moves will give you, figure out the possible benefits your opponents can get from your moves, make sure your plans don?t conflict with one another, do what you want to do, but reassess the situation constantly, and act appropriately to your reassessments.
How can you figure out what a probable battle outcome is? How do you calculate utility gain? How can you determine what unacceptable risk is? How do you determine long-term goals, and how do you reassess the board situation?
A full answer of the last question does not fall into the scope of basic risk assessment and calculation. However, all of these questions will be addressed below.
II. Approximation Methods for Large Battles
An approximation of the most probable battle outcome can be determined by adding up the attack values and defensive values of the attackers and defenders, and dividing by 6 in each case to determine the first round losses on each side, then repeating that process to determine the second round losses, and so forth. In the case of fractions, round down or up as appropriate.
That is, suppose you have a bomber and two infantry attacking one infantry. The attackers are a bomber (which attacks at 4 or less) and two infantry (each of which attack at 1 or less). If you add up the attack values of the attackers, they add up to 6; similarly, if you add up the defense value of the defender (a single infantry, which defends at 2 or less), that adds up to 2. Dividing by 6 for both indicates the most likely first round result is the attacker inflicting (6 (total attack value) / 6), or 1 casualty, and the defender inflicting (2 (total defense value) / 6), or 0 casualties, making the most likely outcome will be a bomber and two infantry surviving, against no infantry surviving.
III. Assessing Utility Gain (IPC Loss and Gain)
Sometimes, a player will want to make a more precise estimate of gained utility, though. In such cases, the probabilities should be calculated, and the immediate utilities factored in. An example of such an estimate, and the way to estimate utilities, follows.
Example: Your opponent has 11 infantry and 4 tanks, and 1 fighter in West Russia. You have 5 infantry, 2 tanks, and 1 fighter in Eastern Europe. Your opponent controls Belorussia with 1 infantry. In a REAL game situation, the rest of the board would be relevant, but we will ignore the rest of the board for most of this example.
A veteran player would immediately discard the though of attacking Belorussia with all available forces. This is because even if Belorussia were taken with no losses, Russia could follow up with their own all-out attack, crushing the Germans. Even assuming the best case scenario of 5 infantry 2 tanks in Belorussia (having taken no casualties on the attack), you must consider the Russian attack on the Russian turn.
Since there are so many units involved, the approximation system already described will be used. The Russian attack of 11 infantry 4 tanks 1 fighter has an attack value of 26 against the Germans? defense value of 16 for 5 infantry and 2 tanks. Even were the odds skewed in favor of the Germans again, giving the attacker only 4 inflicted casualties on the first round and the defender 3 inflicted casualties, that still leaves the Russians with 8 infantry 4 tanks 1 fighter (attack value 23) against the Germans with 1 infantry 2 tanks (defense value Cool, with the next round likely ending with the Russians with 6 infantry 4 tanks 1 fighter (at the worst), and the Germans with nothing. In the end, then, the Russians lost 1 infantry in Belorussia, and 5 more infantry on the attack, which is 18 IPC worth of units. The Germans lost 5 infantry and 3 tanks, which is 30 IPC worth of units, and gained Belorussia for a turn, gaining 2 IPC of territory for one turn. All in all, though, 18 + 2 = 20, and 20 is much less than 30. So if the Germans stand to gain 20 IPC worth of units and IPCs, and lose 30 IPC worth of units and IPCs, then the Germans should not carry out the attack.
It may be, though, that a lesser force could attack, and thereby increase Germany?s expected utility. Consider what happens if a German infantry and a German fighter attack the Russian infantry in Belorussia.
IV. Calculation methods for small battles
For this small battle of three units, although the probability calculation is a bit cumbersome, it is not awful. So, I will list it, and also show how simple utility gain (or IPC gain) is calculated.
The German fighter has a 3/6 chance of inflicting a casualty, and a 3/6 chance of missing. The German infantry has a 1/6 chance of inflicting a casualty, and a 5/6 chance of missing. The Russian infantry has a 2/6 chance of inflicting a casualty, and a 4/6 chance of missing.
The probability the Germans will get two hits, that is, both the fighter hitting and the infantry hitting, is 3/6 * 1/6, or 3/36. The probability the Germans will get one hit is the probability of only the fighter hitting (3/6 * 5/6) added to the probability of only the infantry hitting (3/6 * 1/6), or 15/36 + 3/36, or 18/36. The probability the Germans will get no hits is (3/6 * 5/6), or 15/36. To error check, we add up the probabilities; 3/36 + 18/36 + 15/36 = 1. The probabilities are correctly estimated. Since the Germans can only get one casualty (even if they hit twice), the probability of wiping out the Russians is thus 3/36 + 18/36, or 21/36. Note that this is NOT the same as you would get if you simply added the probability of the German fighter hitting to the probability of the German infantry hitting. (That is, if you add 3/6 to 1/6, you will get 4/6, or 24/36. You will not get 21/36). The probability of missing is still 15/36.
The probability the Russians will get one hit is 2/6, and the probability the Russians will miss is 4/6.
Now we combine the two probability sets. The probability the Germans will get one hit and the Russians will get one hit is 21/36 * 2/6, or 42/216. The probability the Germans will get one hit and the Russians will get no hits is 21/36 * 4/6, or 84/216. The probability the Germans will get no hits and the Russians will get a hit is 15/36 * 2/6, or 30/216. The probability the Germans will get no hits and the Russians will get no hits is 15/36 * 4/6, or 60/216. Again, we add up the probabilities to error check; 42/216 + 84/216 + 30/216 + 60/216 = 216/216 = 1. The probabilities are correctly calculated.
If the Germans inflict a casualty, and the Russians inflict a casualty, the combat is over. If the Germans inflict a casualty, and the Russians do not inflict a casualty, the combat is over. If the Germans do not inflict a casualty, and the Russians do inflict a casualty, a new situation arises. If neither the Germans nor the Russians inflict a casualty, the exact same combat of one German fighter and one German infantry against one Russian infantry repeats.
Let us examine what happens when a new situation arises (that is, when the Germans do not inflict a casualty, and the Russians do inflict a casualty). Either the German infantry will be chosen as a casualty, or the German fighter will be chosen as a casualty, and at that point, the German player will either choose to press the attack or retreat. What should the German player do?
If a German fighter is chosen as a casualty, the Germans immediately lose 10 IPC worth of units. If the German infantry is chosen as a casualty, the Germans immediately lose 3 IPC worth of units.
If a German fighter attacks a Russian infantry, the German fighter has 3/6 a chance of hitting. The Russian infantry has a 2/6 chance of hitting. Calculating the probabilities as above, the chance of both the German fighter and Russian infantry hitting is 6/36, the chance of the German fighter hitting and the Russian infantry missing is 12/36, the chance of the German fighter missing and the Russian infantry hitting is 6/36, and the chance of the German fighter missing and the Russian infantry missing is 12/36.
Whether the Germans and/or the Russians inflict a casualty, the combat is over. If neither Germans nor Russians inflict a casualty, the combat repeats. If we assume that the German player will continue to press the attack (after all, the German player could have retreated to begin with) until the battle is decided, then we can effectively eliminate the probability of neither side inflicting casualties, as the combat will repeat until one side or the other or both is destroyed.
So, instead of having the probability of both German fighter and Russian infantry hitting as 6/36, or 6 out of 36 times, the probability of the German fighter hitting and the Russian infantry missing at 12/36, or 12 out of 36 times, the probability of the German fighter missing and the Russian infantry hitting at 6/36, or 6 out of 36 times, and the probability of both missing at 12/36, or 12 out of 36 times, we are going to calculate as if the last case did not exist. That is, the 12 out of 36 times that both Germans and Russians miss will be eliminated.
After that happens, there will only be 24 different instances instead of 36. And of those 24, both Germans and Russians will hit 6 times, German hit and Russian miss 12 times, and German miss and Russian hit 6 times. (Effectively, you eliminate the probability of all parties missing, and you create a new denominator based on the sum of the remaining numerators, then you slap that denominator on all the remaining fractions).
That is, 6 out of 24 times (1/4) of the time, the Germans will hit and the Russians will hit, 12 out of 24 times (1/2) of the time, the Germans will hit and the Russians will miss, and 6 out of 24 times (1/4) of the time, the Germans will miss and the Russians will hit.
At this point, we will calculate a simple version of utility gain and loss. In a real game, actual utility gain and loss depends on the placement of units on the board as well as their raw IPC value; ten infantry in Western Europe would probably be infinitely more valuable in the Balkans on the Germans? second turn, for example. But I digress; that is a topic that is beyond the scope of this article.
If the Germans and the Russians both hit, the Germans lose a 10 IPC fighter and the Russians lose a 3 IPC infantry. If the Germans hit and the Russians miss, the Germans lose nothing, and the Russians lose a 3 IPC infantry. If the Germans miss and the Russians hit, the Germans lose a 10 IPC fighter and the Russians lose nothing. We will sum this up as the Germans losing 7 IPC, the Germans gaining 3 IPC, and the Germans losing 10 IPC, respectively. (That is, the amount of IPC damage that the Germans inflict, minus the amount of IPC damage in units that the Germans take.)
Figuring this out, that?s ((1/4) * -7) + ((1/2) * 3) + ((1/4 * -10) = -7/4 + 3/2 + -10/4, or -11/4. The Germans should expect to gain -11/4 IPC worth of units if they send one German fighter against one Russian infantry. That is, the Germans should expect to LOSE unit value overall. So unless there are mitigating factors, the Germans should not carry out the attack of a single German fighter attacking a single Russian infantry.
What are mitigating factors? What may be lost or gained as an indirect consequence of the battle, what the overall board situation is, and (closely related to the overall board situation) the perceived value of a piece. That is, suppose that Moscow is only defended by four infantry, and IF Germany succeeds in its one German fighter on one Russian infantry attack, twelve Japanese tanks will be free to blitz into Moscow. (Assume there is no UK unit that can move into that emptied territory on the UK turn.) Or, say that the Germans are capturing Ukraine, and want to leave enough units there so that Russia can?t take Ukraine back on its turn; in that case, German fighters may be chosen as casualties before German tanks, as German fighters can?t land in the Ukraine to strengthen the German defense of that territory, but German tanks can stay in the Ukraine, strengthening the German defense there. And so forth, including values of mobilized units that are on the front as opposed to IPCs in the bank, and so on. The subject of mitigating factors does not fall within the scope of an article addressed to beginners, though, so there will be no further explanation of that topic here.
Getting back to the subject ? notice that in the calculations for a German fighter going against a Russian infantry, the fact that a German infantry is already lost at that point is not taken into account, as at this point, the German infantry is lost. The fact that the German infantry could be lost is figured into the calculations at the time when deciding what forces to allocate to the initial attack.
Probability of both German infantry and Russian infantry hitting is (1/6) * (2/6), or 2/36. Probability of German infantry hitting and Russian infantry missing is 4/36. Probability of German infantry missing and Russian infantry hitting is 10/36. Probability of both missing is 20/36. Again, we eliminate the last possibility, arriving at a final probability of 2/16 chance of both Russians and Germans hitting, a 4/16 chance of the Germans hitting and the Russians missing, and a 10/16 chance of the Germans missing and the Russians hitting.
This time, if the Germans and Russians both hit, both sides lose 3 IPC worth of units. If the Germans hit and the Russians miss, the Germans lose nothing, gain a 2 IPC territory (the value of Belorussia), and the Russians lose a 3 IPC infantry. If the Germans miss and the Russians hit, the Germans lose a 3 IPC infantry. We will sum this up as the Germans losing 0 IPC, gaining 5 IPC, and losing 3 IPC, respectively. (That is, the amount of IPC damage the Germans inflict, minus the amount of IPC damage in units that the Germans take, plus the IPCs in the bank that may be gained from captured territory).
Using the same formula, that?s ((2/16) * 0) + ((4/16) * 5) + ((10/16) * -3), or -10/16. That is, the Germans should expect to gain -10/16 IPC worth of units if they send one German infantry against one Russian infantry to gain a 2 IPC territory. That is, the Germans should expect to LOSE unit value. So, once again, unless there are mitigating factors, the Germans should not undertake the attack.
Now that we know that the battle of German fighter against Russian infantry is unfavorable, and that the battle of German infantry against Russian infantry is unfavorable, we return to the earlier given figures of what happens when a German fighter and a German infantry attack a Russian infantry in a 2 IPC territory.
Remember, in a battle of a German fighter and a German infantry against a Russian infantry, the probability that both sides will inflict a hit is 42/216. The probability that only the Germans will inflict a hit is 84/216. The probability that only the Russians will get a hit is 30/216. The probability that both will miss is 60/216.
Now, we know that if both Russians and Germans inflict casualties, the combat is over, the only decision left is for the German player can choose to lose the 10 IPC fighter, and gain a 2 IPC territory (net -8 IPC), or lose the 3 IPC infantry (net -3 IPC). If the Germans inflict a hit and the Russians do not, the combat is over. If the Germans do not inflict a hit, and the Russians inflict a hit, previously, we only knew that a new situation arose, we did not know the outcome. We now know that, ignoring mitigating factors, the German player should retreat, as pressing the attack does not result in IPC gain.
Since we now know the outcome anticipated action for each of the combat results, we can eliminate the probability of no casualties occurring. What are the new probabilities? They are: 42/156 chance of both sides inflicting a casualty, 84/156 chance of only the Germans hitting, and 30/156 of only the Russians hitting.
If both sides inflict a casualty, the German player must decide to either lose the German fighter (losing the 10 IPC in fighter value, but gaining 2 IPC in the bank from gained territory for net loss of 8 IPC), or lose 3 IPC from the infantry. Barring mitigating factors, the German player should lose the infantry. In this case, taking the amount of IPC damage the Germans inflict, minus the amount of IPC damage in units that the Germans take, plus the IPCs in the bank that may be gained from captured territory, the Germans can expect to gain 0 IPC. If only the Germans inflict a casualty, the Russians lose 3 IPC worth of infantry and the Germans gain 2 IPC worth of territory, for a net gain of 5 IPC. If only the Russians inflict a casualty, the Germans can either lose a 10 IPC fighter and retreat, or a 3 IPC infantry and retreat; in the absence of mitigating factors, this means the Germans should retreat with an effective loss of 3 IPC.
Given this, then, what is the expected change in German utility from the attack of German fighter and German infantry against Russian infantry?
The immediate result is:
((42/156) * 0) + ((84/156) * 5) + ((30/156) * -3), or 330/156 IPC. That is, the expected gain in unit value from the attack is a little under 2 IPC.
However, remember that IF the Germans capture the territory, the Germans will then have an infantry in that territory. If the Allies don?t recapture that territory, the Germans will retain that territory and the IPC income from that territory. If the Allies do recapture that territory, though, the German infantry will have some chance of inflicting a casualty on the enemy attackers, though (barring battleship support shots, which are a possibility). So, really, if the Germans DO capture the territory, there is every chance that the Germans will have at least a 2/6 chance (that is, the value of a German defender) of destroying a unit worth at least 3 IPC (that is, an enemy infantry). That adds an effective anticipated IPC gain of 1. (That is, the possibility of inflicting a casualty multiplied by the value of the casualty inflicted). In practice, the anticipated gain is really a bit MORE than 1, because there is a chance that all the opponent?s attacks will miss on the first round but the German infantry will hit, leaving the German infantry free to inflict additional casualties, and there is also a chance that the opponent will not have any infantry free to attack the German infantry with, so the Allied player may have to risk a more valuable unit, such as a tank, in case an attack is decided upon. Of course, though, if the Germans do have an infantry in that territory at that point, the Germans will lose an infantry, so will lose a 3 IPC unit.
So, the expected previous result is modified. Even though we know that the anticipated IPC gain of the German defender is worth a bit more than 1 (barring battleship support shots), we will only add 1 to the anticipated IPC gain if the Germans capture the territory with no defenders. Adding -3 to that result, for the anticipated loss of the German infantry gives us, in the 84 out of 156 cases when the Germans capture the Russian territory with no German casualties, an additional -2 IPC expected out of the German defender (that is, it is anticipated the Germans will lose a 3 IPC infantry, but have about a 1/3 chance of destroying an enemy 3 IPC infantry in turn), making the former figure of 330/156 into 330/156 + ((-168/156) * 1), that is, 162/156, or a still respectable 1.0384615384615384615384615384615 IPC gain.
Yet, the picture is still not complete. Even though the Germans stand to gain from the attack, there is a question of what the Germans stand to lose if the attack fails; this question goes beyond the immediate IPC gain or loss already described. Also, we have not yet determined whether or not some other German attack may be even more cost effective. Finally, we have not yet determined what benefit the opponent may receive from the proposed move, or what cost the opponent may yet have to pay for the proposed move.
V. Other Utility Gain Calculations
If the calculated utility of a move is to be considered completely calculated, we must figure in the fact that in the example above, the Russians can attack the Germans on the Russian turn, with the exact same expected utility gain as listed above ? but for the Russians. In turn, the Russian counter will be subject to whatever German counter the Germans can muster ? and, of course, the Russians will in turn be able to counter that German counter, etcetera ad infinitum.
The example given at the beginning of section III shows that if your opponent has considerable forces, an attack that is profitable for you in the short term may be very unprofitable in the long term.
However, what that example does not show is that if your opponent does not have considerable forces, an attack that is profitable for you may end up being even more profitable.
Say that instead, there were one German infantry and one German fighter in Eastern Europe, one Russian infantry in Belorussia. Also assume that Russia cannot bring any forces to Belorussia on the Russian turn. Now, the utility calculations change.
Previously, we assumed that if the German infantry captured the territory, that the Russians would counterattack. At that point, it was anticipated that the Germans would lose a 3 IPC unit, and gain about 1 IPC (for a 1/3 chance of destroying a 3 IPC Russian infantry). In this example, though, the Russians cannot counterattack. Nor can they gain back that territory on their turn.
So in the 84 out of 156 cases that we assumed a German infantry would be destroyed, we instead now know that the German infantry cannot be destroyed, so we can add back the ((84/156) * 2) IPC that we subtracted. Furthermore, the Germans will still be collecting income from that 2 IPC territory on the next German turn, as the Russians cannot recapture that territory, so we add an additional ((84/156) * 2) IPC.
Instead of 162/156 IPCs anticipated, then, the Germans in this case anticipate a gain of 498/156 IPCs. That is, 3.1923076923076923076923076923077 instead of 1.0384615384615384615384615384615 IPCs, a fairly considerable difference.
Usually, there will not be a case in which the Germans have units with which to attack, and the Russians have both insufficient defense, and no units in the surrounding attacks that can counter the German attack next turn. However, it may well be the case that the Russians will be unable to respond in a cost-effective manner to all of the German attacks. For example, if the Germans control Karelia, Belorussia, and Ukraine, each with one infantry, Russia will very likely have the infantry required to respond, but it is very possible that Russia will not have the necessary fighters, as the Russians would need three fighters, but fighters are very expensive and Russia only starts with two.
VI. Assessing the Board Situation
When you look at the board at the start of your turn, you know how many IPCs you have, where your opponent?s units are, and where your own units are. With this knowledge, you can plan your attacks, and in turn, what units and/or tech research you will purchase.
Deciding what the most cost-effective allocation of your units for your combat move phase will be is difficult. Sending an additional infantry to attack on this turn will mean a better chance of succeeding at the immediate battle, but will probably mean that you won?t have that infantry available next turn to respond to your opponent?s countermove. Sending an additional fighter to participate in a naval battle may save valuable naval units, but may also mean that you don?t have good odds in a land battle, which can potentially be more important in the long run.
As difficult as this is, though, it is even more difficult to assess what the most cost-effective allocation of your opponents? units will be, and what your opponent is therefore likely to build. It may be that even though you have a move that will greatly increase your utility, that your opponent has a countermove that will greatly decrease your utility.
The complexity is increased another degree by the fact that units can be produced. Although the units your opponent is going to build on his or her turn can?t counterattack any attack you made this turn, the units your opponent is going to build on his or her turn can counterattack any counter that you make to your opponent?s counterattack to your attack, and vice versa.
Even yet, there is more to the situation. The units that you build on your turn may not be immediately usable to attack. However, even if the units you build on your turn cannot be used to attack on your next turn, or even the turn after that or later, those units can still be moved into position so that you can attack later. This last is the reason why it is effective for Germany to produce mostly infantry on the first two turns, but produce tanks starting about three turns before serious pressure is to be exerted on Russia. Infantry that are produced early can march towards Russia, and the tanks? speed means that the German infantry and the German tanks can hit the Russian lines at the same time, creating a difficult situation for Russia to deal with.
There is still one more thing to address; the use of friendly powers? units to help defend your own. The most common examples of this are for the Allied fleet in the Atlantic, and the German attack on the Ukraine and/or the Caucasus. In both cases, the powers in question unite their forces to make themselves more difficult to attack. Specifically, when the Allies control 1 Russian sub, 2 UK transports, a UK battleship, a US destroyer, a US carrier, and 2 US transports, that is quite difficult for Germany to take down, if the Allied players make sure the German navy and airforce can?t both hit the Allied fleet. Or, if Germany puts a lot of units in the Ukraine or Caucasus, Russia can very likely make a very damaging attack, but if Japan lands some fighters in the Ukraine, any Russian attack becomes considerably more expensive.
VII. Long-Term Goals
The ability to figuring out the short-term risks, costs, and benefits of a decision is important, particularly when the decision contemplated is crucial to the course of the game. However, the short-term risks, costs, and benefits of a single decision must be viewed in light of the long-term risks, costs, and benefits, which are not easily calculable.
For example, it may not immediately be obvious that if Germany loses fighters, the Allies will have a far easier time moving infantry and other cost-effective ground units into Europe and/or Africa. However, that is the case; if Germany has few fighters to threaten Allied transports, the Allies won?t need to build as many escort ships for their transports. In turn, that will mean the Allies will have more IPCs to build transports and ground units to transport to Europe and Africa.
It may also not be immediately obvious that Germany should purchase some number of infantry on Germany?s first turn in response to a Russian infantry build. After all, if Germany produces tanks on Germany?s first turn, Germany?s position against Russia will become much better very quickly. However, if Germany produces nothing but tanks first turn, the German player won?t have the numbers of units needed to absorb Russian counterattacks, once the Germans reach the Balkans.
On the other hand, it may be that Germany should purchase nothing but tanks, in response to a Russian double fighter build and overly aggressive combat move. Russia will have more units that it can use immediately on its next turn, and the German front will be depleted, but Germany?s counterattack can be very costly to the Russians, and the German tank build may well stop the Russians from counterattacking. In the end, German numbers and speed may mean that the Germans may be able to secure more territory on the Russian front, and together with the Japanese, eventually secure Moscow.