This series of guides exists because as far as I know there has been no systematic approach to Axis and Allies as a discipline since Don's essays. Much remains to be addressed.

There's more to cover than just what I've put in these guides, and others are better qualified to write better guides. My writings are limited by my understanding. But perhaps you will find these guides of use.

Some references in this guide are incorrectly phrased for convenience or custom. For example, a binomial distribution is a discrete probability distribution but I reference its shape as a "curve". An attack action with intent to retreat is called a "strafe" here and by some in the community, though that's not the modern usage of the word.

Thanks to Bernard (who taught me analysis always misses something and there's always someone better), Hobbes (whose analytic approach I try to emulate), Minh (who introduced me to the game), Craig (who taught me threat multiplication), Ralf, Steven, and too many others to list here.

Articles and forum posts I read for previous versions of the board game helped me develop my thoughts on this version. Those articles and posts can be found at

http://donsessays.freeservers.com/axisand.htm

https://www.axisandallies.org/

Finally, thanks to the reader. Players that work to improve their game make the community what it is.

Understanding theaters and board states = knowing where you are and where you want to be. Understanding how to transition = knowing how to *get* from where you are to where you want to be.

Player-defined contested areas of specific significance(s). Conditions in one theater affect operations in other theaters, theaters may overlap.

E.g. a player defines Norway-Finland-Karelia as a theater. UK and US land units transported to Norway and Finland can't break through Karelia's German stack reinforced with Japanese fighters. Germany defends Karelia's industrial complex, but pushing Finland then Norway would leave big openings for Russia. Neither side retreats or advances, the area is contested. The more Germany uses to defend Karelia, the less it has to push Ukraine. Axis fighters defending Karelia threaten UK/US fleets in the Northeast Atlantic theater.

Under different conditions Norway might be a theater of itself and held by Germany to park air units threatening Eastern Canada's sea zones. Or Norway might not be part of any theater if the area is not contested.

Player-defined understanding of a real or imagined board position. Very different positions may be considered the same board state.

E.g. a player defines a good Germany fourth turn (G4) board state as forces on Karelia and Ukraine that can together take West Russia G5. In one game Germany might have a huge stack on Karelia and near nothing at Ukraine, in another a mid-sized stack on Karelia and a mid-sized stack on Ukraine. The positions are different, but the player considers them equal as both boards will see a G5 capture of West Russia.

Error is essential to strategy. If players knew exactly what to do, there would be no plan, no art of war, only a list of instructions.

In Axis and Allies, player strategy consists of attempting to successfully transition from one board state to another. The entire process is subject to error from beginning to end.

Even players with decades of experience make errors, made worse by reinforcement from years of play. E.g. veteran infantry chain players that haven't learned principles of threat multiplication.

Specific, Measurable, Achievable, Relevant, Time-bound.

Change, and how it happens *exactly*. Think SMART (see above).

E.g. Japan can attack India with 8 units this turn, but Allies defend with 11 units. By Japan's next turn, Japan can attack with 16 units but Allies can defend with 16 units.

Doesn't look good, but suppose Germany's pressing hard on West Russia and suppose Allies can't add 5 units to India *and* get a good defense at West Russia. The Allies have to choose, and they drop India. Why?

If Allies defend India, the expected stack shifts are UK stack at India, Japan stack to Burma, Germany's stack to West Russia, UK stack to Persia, Japan stack captures India, Germany stack captures Caucasus and reinforces India with fighters. Then the UK stack is cut off and the Axis end up destroying a load of UK units for very little loss of Axis units.

If the Allies abandon India *immediately*, the UK stack can escape into Russia without being cut off.

The tactics concepts section gets into transitioning mechanisms, and I'll get into applications in the third part of this guide series. For now remember - transitioning is about the *process* of getting to a winning game state.

Error is inherent to strategy, and transitioning is complicated. Others think about the game differently to you, take different risks, and make different mistakes.

Watch opponent plays to get an idea of their experience, risk preferences, and playstyle. How you test your opponent, and how your opponent responds, determine game outcomes.

If your opponent seems to be responding well to your moves, consider changing playstyle. You may gain an advantage if your opponent doesn't respond correctly.

Axis and Allies is not only about optimization of probability distributions. Future opponent action is an unknown variable.

Applying practical mathematics reduces errors and increases ability to exploit opponent errors.

I reference binomial distributions as bell-shaped "curves", but actually they're discrete probability distributions. Please bear with it. Each unit in a homogeneous group hits or misses; those hits or misses taken together generate a binomial distribution.

Mixed forces contain separate homogeneous groups, each generating a binomial distribution that combine to create a distribution that resembles a bell shape.

An attacker generates a single bell-shaped curve, a defender generates a single bell-shaped curve. Many think combat results over time are also a single bell-shaped curve, but actually the typical result is a two-peaked curve with peaks at very different locations.

Mixed forces combine curves that add to one another. But combat results combine curves that work against one another. With combat results, for a single peak to result, every attacker hit would have to be exactly balanced by an appropriate number of defender hits, but since defender hits are distributed over a curve rather than a single point that almost never happens. The combined effect of mismatches over repeated rounds pushes results into two distinct peaks.

Don't expect results, especially in initially close battles, to converge on a single result over time. A player might think a calculated 60% win battle "safe", reasoning even if they'll lose surely it'll only be by a small margin, but *actually* they could lose by quite a bit.

A player may initiate an attack with an expectation of a particular range of results, but each round of rolls will likely deviate from the prediction. Calculations should be done anew each round.

Pre-emptive shots generate a separate discrete probability distribution on which each point corresponds to an entire curve corresponding to the opposing side's composition after pre-emptive shot casualties are removed. Each of those single-peaked curves is multiplied by the matching point on the discrete probability distribution, then the entirety run against the curve of the opposing side and evaluated over multiple rounds of combat.

Pre-emptive shots are powerful and volatile.

Players using automated aids or simple manual calculations must remember simple predictive models do not correctly account for the multi-peak reality, or discrete probability functions not actually being curves.

Predict outcomes without automated aids by adding attack/defense values then dividing each by 6 to estimate hits for each side.

Remember also unit count and "skew". A side with a higher unit count does better than a simple attack/defense power calculation predicts. A side with units with different attack/defense values (skew) does better over time compared to a side with units with mostly the same attack/defense values.

E.g. 2 infantry attacking 1 infantry, the 2 infantry are slightly favored to win. 10 infantry 10 tanks attack 20 infantry, the mixed force is much more likely to win.

Volatile, as in unpredictable explosion.

Players using automated aids to assist with calculation read results like "75% attacker wins" and think a battle is safe, but such evaluation does not account for multi-peaked models or discrete probability functions not actually being curves. Watch out especially for:

• Close battles - the peak the results push towards may not be in your favor
  
• Combats with low dice count - the fewer dice involved in a round, the less a discrete probability function resembles a smooth curve
  
• Combats with pre-emptive shots
  
• Combats with skew

Tools I currently use

http://calc.axisandallies.org/
(based on older ruleset, doesn't allow casualty assignation of antiaircraft artillery)

https://www.aatoolkit.com/conflict
(limited, but does allow casualty assignation of antiaircraft artillery)

Neither are proper binomial distribution calculators, both run PRNGs then total results.

Adding units and/or changing quality (i.e. using an artillery instead of an infantry) can make attacks safer in the short term. But spending resources to make short term attacks safer leaves fewer resources to deal with opponent pressure later, and eventually the price is paid.

Against competent opponents every single IPC makes a difference, in a real and practical sense. Every purchase, move, and risk should be considered to the finest detail, weighing potential cost against potential gain, both short-term and long-term. Remember you have to take short-term risks to conserve resources for the long term. Figure for yourself which short-term risks have rewards worth additional investment, and which do not.

A single unit can change projected victory/loss outcomes by 16% or more even in battles of near a hundred units. Do not make the mistake of thinking micromanagement can be ignored to pursue "grand strategy"!

Tactics is about transitioning between and correctly identifying board states.

The first guide in this series emphasized controlling infantry flows. That doesn't necessarily mean *building* infantry.

E.g. Allies have no real navy in Atlantic, Germany and Japan push Russia with loads of tanks. Axis are attempting to cut off Russia's infantry flow by grabbing territory to choke off Russia's income then capture capital before sufficient infantry can be built up to resist.

What should Allies do? UK and US could build transports and escorts to ferry cost-effective infantry to Europe. But Russia will be overrun before UK / US are relevant.

Instead, the Allies may do something different. Say Russia managed to hold West Russia. UK builds fighters at London and infantry at India, lands London fighters on West Russia. US builds carriers and fighters and/or destroyers to chase German submarines out of the Atlantic.

The Axis are concentrating on destroying Russia's infantry flows instead of building their own, and UK and US are concentrating on disrupting the Axis plan to interrupt Russia's flows.

Attacking then retreating. Forces capturing a territory may be counterattacked; it may be better to retreat instead of capturing.

E.g. Russia has to decide whether to retreat from Ukraine with 1 infantry, 1 artillery, 3 tanks, and a fighter. Germany has a fighter on the territory. If Russia captures the territory, Russia destroys the German fighter (10 IPC) and gains income (2 IPC), Germany can destroy 25 IPC of units (all the Russian land that had to stay on the territory) on the counter, say at a cost of 6 IPC worth of infantry. Russia loses 25 IPC worth of units for 18. IPC is a bad measurement of utility, but you get the idea.

Putting units where an opponent could but doesn't want to defend or counter. If opponent doesn't defend or counter, pressuring units continue for gains. If opponent defends or counters, resources are drawn from another area.

E.g. a USA submarine moves towards Japan, where Japanese transports are operating, say most of Japan's navy is off India pressing towards Mediterranean to relieve pressure on Germany. If Japan doesn't deal with the USA submarine, its undefended transports are destroyed. If Japan builds anything or sends anything to deal with the submarine, that's resources not directed towards the Mediterranean.

E.g. of *not* pressure: Russia captures Poland with a couple units and Germany wants to march a load of infantry from Germany into Poland on its turn. Russia's capturing Poland doesn't create any pressure on Germany as Germany wants to send its units into Poland anyways and the Russian forces aren't stopping them.

In combat, the side with superior force does better. If you have more forces in an area than an opponent, you may "stack" those units on one territory; an opponent can't attack with advantage so may not attack at all. But what if you have less forces in an area?

If you "split" forces, your opponent may hit any or all of your splits with superior force, but in so doing they break their main stack into smaller stacks that may be vulnerable. They can shift their stack to attack a single territory, but then they shift off the location they were guarding. Or an opponent's stack may not move at all, then you'll likely gain income while they lose it from the territories you hit.

Neither stacks nor splitting should be used indiscriminately. Splitting only works if implemented properly as part of a pressure tactic. Poor splitting just leaves a player open to strafes or defeat in detail.

It's often best to attack with minimum units. "Minimum" depends on board position; with identical forces available a player may attack a 2-IPC territory held by 1 opponent infantry with 2 infantry, 1 infantry 1 artillery, 1 infantry 1 fighter, 2 infantry 1 fighter, 1 artillery, 1 infantry 2 fighters, 1 infantry 2 tanks - each being "correct" for a different situation. (If the opponent could counterattack the position, using fighters would be better so land units don't need to be committed to where they may be counterattacked and destroyed. If the opponent can't counterattack, more valuable units like artillery or tanks might be used; tanks would be able to hit the target yet still be in range to threaten other targets next turn).

Sometimes a player may attack with maximum units to inflict maximum casualties, to suffer minimum casualties in turn. The mass of units may also be too large for opponents to attack safely. Large attacks may also happen as a matter of course, as in Germany sending forces to Ukraine and destroying anything that gets in the way.

Potential unit counts in an area determine what tactics are likely to be successful - whether a player should attack with minimums, maximums, try to apply pressure, etc.

Simply counting units isn't enough - potential builds and moves need to be accounted for, and quality of units makes a difference. But experienced players can consider those in their calculations, and use a quick unit count to get an idea of what moves may be effective.

For a given attack, calculate probability of each attacker and defender outcome, projected attacker costs, defender costs, net expected IPC change in units, consider IPC gain from potentially captured territory and expected change in IPC after opponent counter.

E.g. 1 infantry 1 fighter attack 1 infantry on a 2-IPC territory. 7/26 both sides hit (+3 destroy enemy infantry, -3 lose infantry), 14/26 only attackers hit (+3 destroy enemy infantry, +2 capture territory, +1 figuring on 1/3 chance of destroying 3-IPC opponent infantry when opponent counterattacks, -3 lose infantry on counterattack), 5/26 only defender hits (-3 lose infantry). (In this simple example, if both sides miss the attack just goes into the next round, so that outcome is eliminated). Net +27/26 IPC, so the attack is a good risk, ignoring other factors.

Positive expected net IPC does not mean an attack is safe or good. In the above example, there *is* a chance attackers miss and the defender hits, resulting in loss.

A player shouldn't always try to maximize net IPC. On different board positions you might want to pick worse net IPC, losing a bomber instead of a fighter, or losing a fighter instead of a tank.

It's common for players to attach too much or too little significance to net IPC calculations. Players ignore board position to maximize short-term net IPC, not considering mid-to-long-term board states. Or players don't consider projected cost-to-benefit after an opponent's immediate counter.

Use net IPC calculations responsibly, remembering IPC value often does not reflect utility.

In combat, the side with superior force does better. It's often best to send everything that can reach to a single major battle. More units means more opponent casualties meaning fewer casualties of your own. The combat is more likely to succeed and more attackers likely to survive, this usually more than offsets any gains that may have been made from splitting forces to attack elsewhere. There are exceptions.

Under certain conditions an attacker may retreat all land or sea units to one adjacent territory or sea zone an attacking land or sea unit moved from. This can "teleport" units.

E.g. A stack of UK land on Finland and a UK land unit on West Russia attack German-controlled Karelia. After a round of combat, if Karelia was not captured, all UK land units in the combat may retreat to Finland or West Russia.

Mass infantry produced each turn move by the shortest route to a key area. This leaves a "chain" of infantry across the board.

E.g. Germany's Baltic States-Karelia-West Russia / Poland/Ukraine/West Russia chains, US's East Canada-Finland/Karelia chain, Japan's Yunnan-Szechwan-Kazakh chain.

A unit at one location threatening multiple locations.

E.g. German tanks at Baltic States protect that territory from invasion and threaten Karelia, Archangel, Finland, West Russia, Ukraine, and France - all likely spots for Allied pressure. German fighters on Northwest Europe threaten all key Allied landing zones in the Atlantic and points in Europe as well, the Allies must defend multiple places from a single threat.

In the first example, a single stack of tanks poses the same threat as least four stacks of infantyr/artillery. In the second example, a single stack of fighters poses the same threat as multiple infantry/artillery stacks and navies.

German mass air tactic from Axis and Allies Europe edition. Combines mobility of air, force concentration, and threat multiplication.

Building, moving, and coordinating to push an area effectively at a planned time. E.g. two rounds of German infantry buys followed by German tank buy can push Ukraine on the fourth round; this would be vulnerable to Russian counterattack except Japanese fighters reinforce.

Using different powers in combination to achieve what cannot be done individually.

Powers may defend together but attacks allow only one power. The power on a side controlling the "main offensive stack", may deliberately avoid engagement while other powers on the same side attack territories controlled by the other side's main stack controller. The opposing power then must choose to "bleed out" reinforcements that could otherwise go to its stack, or continue building its stack leaving territories unclaimed so having less income.

Players quickly learn to react to luck e.g. if an attack goes poorly, a player may retreat and move up reinforcements. Even that requires some proactive thought (enough reinforcements must be available to deter the counter). But good practice may require much more thought.

E.g. with LHTR setup, Germany's first turn may have 2 submarines, 2 fighters, and cruiser attacking battleship, destroyer, and submarine. Germany could send a third submarine to that battle, or against East Canada's UK destroyer / transport.

• 3 submarines to battleship, East Canada destroyer and air can attack naval survivors so Germany probably has no navy starting Germany's second round. This allocation reduces probability of bad outcomes for the first battle, but requires lucky dice to avoid losing everything on the counter.
  
• 2 submarines to battleship, East Canada destroyer destroyed. If Germany is lucky and sustains only one casualty, the cruiser may be assigned; the Allies cannot destroy Germany's submarine survivors.
  
• 2 submarines to battleship, East Canada destroyer not destroyed. If Germany sustains only one casualty, a submarine may be assigned; the cruiser has a better chance of inflicting a casualty on UK's counter.
  
• 2 submarines to battleship, East Canada destroyer status unknown, Germany sustains 1 casualty. Germany doesn't know what to do.

Being prepared to take advantage of good luck or offset bad luck is a deliberate proactive process. The Germany player could send all three submarines to the battleship and hope to get lucky. Or the Germany player could split off a submarine, do the battleship battle first, then hope to be lucky at the East Canada destroyer battle and hope they picked the right choice. Or the Germany player could split a submarine, do the East Canada destroyer battle first, then know what to do in case they get lucky dice at the battleship battle.