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Morten’s guide to the States of Siege series – part 11: Math attacks

Morten Monrad Pedersen
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The previous post set the foundation for the mathematical “ANAlysis” of the States of Siege games. In this post we’ll see how we can build on that foundation to launch math attacks to get a leg up against the games.

While this post contains a lot of math, you can ignore that it if you want to and still be able to understand the conclusions.

Preliminary table of contents for the series

1) The boring introduction
2) Event deck structures
Interlude 1: Sad news about VPG
3) Designer control, storytelling, and pivotal events
4) Tension vs. variation
5) Dice, event resolution, and combat
Interlude 2: It's alive!
6) Track systems
7) Representation of player and enemy units
8) Sidetracks
Interlude 3: These posts are in the public domain
9) Embedded minigames
10) The Currency of the States of Siege
11) Math attacks
12) Defense against math attack.
13) Are the States of Siege games luck fests?
14) The three production processes
15) Sort of the end
Appendix: BGG rankings and publication order.
+ potentially a new series of posts with my ranking and mini-reviews of the games.

Revisiting ANA

Since it’s at the heart of the States of Siege math, I’ll repeat the “1d6 ANA table” from the previous post for your convenience . It shows the ANA (average number of actions) for getting 1 success at all the possible probabilities of success for the rolls of one 6-sided die (1d6).

Probability ANA
1/6 6.0
2/6 3.0
3/6 2.0
4/6 1.5
5/6 1.2

DRMs and ANAlysis

In the previous post I by mistake left out an important point: The value of applying a DRM varies depending on the base probability of success. E.g. if you attack an army with a strength of 2 a +1 DRM decreases the ANA for pushing back the army from 1.5 to 1.2 an improvement of 0.3. For an army with a strength of 4, the decrease has an ANA change from 3 to 2, i.e. an improvement of 1.

So, prioritizing attacks when you have positive DRMs can cause a significant decrease in the ANA you need to win the game and attack high-strength armies should be prioritized over low-strength one in relation to DRMs. If you know the deck well enough, you will sometimes be able to predict that a good DRM will come soon and refrain from attacking the army in question until the DRM shows up.

DRMs can become available at random times, so pushing an enemy army back to the last space of its track where it can’t be pushed further can be a bad idea, because if it’s there you can’t use a positive DRM if you get one. The same goes for games that have events, which forces an army to regress because such a regression is in most cases wasted for armies that are on the last space.

This advice can be invalid in some games such as Israeli Independence where some event cards can knock armies out of the game if they’re on the last space of their track, when the card is drawn. This will neutralize roughly one fifth of the remaining enemy advances in one go, which reduces the ANA you need to win enough that it can turn a likely defeat into a victory and if you knock out more than one army, then victory is all but guaranteed.

Use fortifications and army advance counting

Some of the games have fixed fortifications printed on specific spaces. These cause a die roll to be made if an enemy army attempts to enter the space and if the roll succeeds the army doesn’t advance.

If the fortification has a 2/6 chance of holding back the army and there’s a strength-3 army (which has an ANA of 2 to push back) in front of it, the fortification gives you an ANA of 2/6*2 = 2/3 for free every time it’s activated

This leads to two guidelines (not hard and fast rules):

1) If the army is further away from your base than the fortification it, then you shouldn’t attack it.
2) If the army is on the space of the fortification or closer to your base, then it’s advantageous to push it back.

Following these guidelines allows you to get the free ANA of the fortification multiple times. Note that you’re not wasting actions pushing it back because you’ll need to push it back no matter what to keep it out of your base in the long run.

That last statement is not quite true because at one point the army won’t have enough advances left to reach your base. You can use this to save actions:

1) Before starting the game, count how many advances, each enemy army can get in one playthrough. In many games this can be established with certainty by looking through the event cards.
2) While playing, keep track of how many advances each army has used.
3) Subtract the number of actions spent by an army from the number of advances it can get.
4) If that number is lower than the number of spaces it’s away from your base, then you can ignore it for the rest of the game and thus you don’t need to spend actions pushing it back.

If an army starts on space 5 and advances 15 times, then there are 4 advances that you can ignore. For a strength-3 army that’s a potential ANA reduction of 4*2 = 8.

Using ANAlysis to detect red herrings

A weak point of some of the States of Siege games is that they have options that are mainly red herrings (at least if my math is correct) and you can use ANAlysis to detect them and avoid wasting actions.

I’ve mentioned the side battles in We Must Tell the Emperor briefly in a previous post but as a system that should basically be ignored. Now is the time to back up my claim. I actually did that in the comment section of a that post, so if you’ve read that you can skip until the next section.

Board Game: We Must Tell the Emperor

Each entry on this list is an important battle in the war, which isn’t handled by the main game and instead plays out as a somewhat separate microgame. Image credit Dampenon Fabien.

Let's take a look at the math for the second of those, Battle of Coral Sea, as an example: In the battle you pay 1 action, roll a die, and look up the result on the list.

* There's a 33% chance that you gain 1 prestige (sidetrack) and retreat Nimitz (enemy army) once.
* There's a 33% chance that nothing happens.
* There's a 33% chance that you lose 1 army-navy (sidetrack).

The 33% chance that you gain 1 prestige and the 33% chance that you lose 1 army-navy cancel each other out mathematically, since you have the same chance of succeeding if you spend actions to try to improve either of them.

This leaves you with a benefit of a 33% chance to retreat Nimitz once. Nimitz has a strength of 4, which according to the ANA table means that on average it takes 3 action to make it retreat once.

So if you do the battle, you pay 1 action to have a 33% chance of saving 3 actions. Since 33% of 3 actions is 1 action the math works out to this being a wash but since you can sometimes get positive die roll modifiers it will be cheaper to just take the actions then instead of doing the battle and so they’re basically red herrings that should be ignored.

That said, if you prefer prestige to army-navy, then that can change the equation a bit and if you're desperate a slightly unfavorable gamble can be worth it.

Sidetrack bonuses and free defensive actions[b]

As mentioned in a previous post Malta Besieged has 3 resource sidetracks:

* Military: Gives 1 bonus “raid” action per turn if you’re at the maximum.
* Supply: No bonus.
* Morale: Gives 1 bonus reroll for “raid” or “resource” actions per turn if you’re at the maximum.

Board Game: Malta Besieged: 1940-1942

The sidetracks of Malta Besieged on the mid-left. Image credit Ron A.

The supply track can be used to gain 1 extra action by decreasing the track once. It’s inefficient to pay actions to increase it, since on average it takes 2 actions to increase it 1 space, which works out to on average spending 2 actions to gain 1 later. Luckily the supply is increased via a convoy minigame system, so you can basically ignore it apart from using it to get an extra action when you really need it.

Getting to the maximum of the morale track, which isn’t hard because you start at space 3 out of 5, you can start rerolling if you fail an attempt to succeed in increasing the military track, which should be your next goal. Combined with a lot of actions saved by a procedure called “Operation Herkules”, which we’ll get to later, you can typically keep both these tracks at the max.

Once you also get military to the maximum you gain a free raid action each turn, which is used to increase a DRM against one of the enemy armies, Rommel’s Afrika Korps. This DRM can get up to +2 and with the free raid action potentially plus a reroll from morale, keeping it there is doable.

The Afrika Korps has a strength of 4 and so the ANA table tells us that the +2 DRM reduces the ANA of pushing back the army once from 3 to 1.5. This means that you’ll save on average 1.5 actions per time you need to push it back, which adds up to many free actions during the course of the game.

It’s of course not free to keep the military and morale tracks up and this eats significantly into the actions saved against the Afrika Korps, but those actions are not the only benefit of the resource tracks.

Malta Besieged has a free defensive system, the above-mentioned “Operation Herkules” the procedure for which can be seen in the left half of the photo above.

Each turn where one or more enemy armies end up in your base you resolve this procedure to and if you succeed the army/armies are pushed back. The sum of the resource track positions is used as a modifier for a roll of 2 six-sided dice (2d6) that has a few other modifiers. If you keep military and morale high as described above, you’ll have a modifier of roughly 15. If there’s a strength-3 army that is reduced to 15-3 = 12 but there’s also a +2 modifier because of a defensive fort on Malta, which takes us to 12+2 = 14.

You then roll the 2d6 and if that’s lower than or equal to the number calculated above, the army is pushed back. Since 2d6 cannot roll more than 14 you’ll be home safe unless two armies get in at the same time. On rare occasions it happened to me that there was “only” a probability of 35/36 of the army being pushed back.

This means that you can completely ignore one enemy army and just let it bounce off Malta at no cost to you. If it’s a strength-3 army that has 15 advances during the game and starts at the 5th space of a track it has 11 advances that need to be countered and since Operation Herkules takes care of all of them that’s 11 advances you don’t need to counter by spending actions. According to the ANA table that saves you an ANA of 11*2 = 22. Often you can even get away with using the system to hold two armies back saving you even further actions stand right outside your base without much risk.

As you might imagine such a high number of free actions is a game changer and I won my handful of plays without breaking a sweat.

As always with this kind of thing it might just be me playing a rule wrong but after checking on the forum and asking in a blog post, I didn’t find any errors. That’s of course not a guarantee and I didn’t do enough plays to rule out this being a statistical fluke but one other player who’s good at this kind of thing also reported an extremely high win rate once the game was grokked.

The clear lesson for States of Siege designers is that mechanisms that allow the player to gain free actions repeatedly should be avoided unless it comes with a high risk or high cost for the player to keep the game in a state where the free actions are generated.

[b]Keeping “pet” armies alive keep tougher enemies away

In Dawn of the Zeds (2nd edition) gamebreaker Wes Ernie has coined the term “pet zeds” (zombies): Weak zombie units that are kept alive in front of a powerful defensive position to prevent more powerful zeds from reaching the frontlines because the number of zombie units on each space is limited to 2. Thus, you can increase your ANA and decrease risk by fighting an easy enemy instead of a hard one. The issue was fixed with Wes’ help in the 3rd edition and IIRC also in the Director’s cut expansion for the 2nd edition.

I don’t remember Wes’ explanation of how to pull it off, but I found a similar exploit in Empires in America (2nd edition).

In that game both you and the enemy have leaders of different power levels that can be used to fight battles. If at the beginning of your turn you have 2+ leaders more than your enemy, then you have to discard leaders until you only have a surplus of only one.

That’s of course not a good thing but you reduce this negative effect with the following strategy: In a situation where the enemy has a weak leader in play and you have two more powerful ones.

If you wiped out that enemy leader you would have 2 more armies than the enemy and you’d be forced to discard a good leader. If you instead keep the enemy leader alive in an army in front of a fortification, you can keep up a significant leader power advantage while letting the weak leader bang his head against the fortification.

Board Game: Empires in America: The French and Indian War, 1754-1763 (Second Edition)

A “pet army leader” candidate in Empires in America. Card shown from two different production processes. Image credit Tracy Baker.


The mechanisms described in this post works to different extents in the various games. In most they’re not a problem just part of the strategic deliberations, while in the most extreme example is a game that’s basically broken. That game is Levée en Masse.

I’m sorry to say I no longer remember the details because I haven’t played it in 10+ years but since I played the game in the official app, I’m pretty sure that the issue isn’t a rule error on my part and I got it confirmed by another player I trust in this regard that he had found the same breaks as me.

Leveé en Masse features a system like Operation Herkules that can be abused and a helpful sidetrack-based DRM that is very powerful. I have a vague recollection about a fortification system that also worked well but I’m not sure about that.

Once I had grokked the challenge of the game was basically gone. I'm not quite sure whether I won every time or whether I lost once or twice but I do know that several times I won by the largest margin possible in the game.

I played enough times and won by large enough margins that the odds of it being a statistical fluke are astronomical.

Board Game: Levée en Masse: The Wars of the French Revolution

An almost perfect win in Levée en Masse. Image credit Kin Hassar.

While my criticism of Malta Besieged and Levée en Masse might seem harsh, I actually enjoyed them. How much the issues will affect you depends on how you approach the games.

Next up: Defense against the dark arts math attacks

At its core the SoS engine is very susceptible to this kind of analysis because of the simplicity of the track system, the action point allowance system, and the dependency on rolling single d6s. All the games add mechanisms to the core engine, though, that counter the math attacks.

In the next post we’ll look at some of the mechanisms the designers have used to defend against math attacks.
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