Today’s post marks a milestone in this blog series: It’s the 10th post, if we don’t count the three interlude posts that weren’t about the games themselves and we’re well over halfway through what I think will be a 16-post series – again excluding interludes.
That aside, let’s move on to what I think is one of the most interesting and crucial topics for understanding the States of Siege games. Not only that, it’s also important for game design in general. So, I hope you’ll read along even though this post is more complicated and theoretical than any of the previous ones in the series.
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
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.
The currency of the States of Siege
One thing I’ve learned as I’ve studied game design is that at their core, all games have an internal economy and understanding that economy is a major factor in understanding a game and in playing it well. If you’re a game designer, then understanding the economies of your games can be the difference between balanced and broken.
In some games, that economy is easy to identify, while in others it can seem completely opaque, but luckily for us the States of Siege tend to be fairly simple in this regard.
So, what do I mean by games having an internal economy? I mean that games have one or more currencies and every action and resource in the game can be assigned a value in one of those currencies and you can exchange from one currency to another. By currency I don’t mean money. Well, in some cases it is money, but in others it’s food, spaceships, penguins, or actions.
In States of Siege games, the currency is actions. The value of anything you do in the game can in principle have its value calculated or estimated as a number of actions it will on average cost to achieve your.
We can call this the average number of actions required to succeed. I’ll use the acronym ANA for average number of actions.
Expected number of die rolls
In order to use actions as the currency of the States of Siege games, we need to talk about math for a moment.
Typically, what you need to know is the ANA of achieving something, that is how many actions you on average need to spend in order to succeed at doing the thing. This could be pushing back an enemy army 1 space.
For each action spent, you get 1 die roll and have a specific probability of succeeding on that roll. Let’s call that probability p.
So, we need to ask how many times do we on average have to roll a die before we succeed if the probability of success on each roll is p?
This is a case of what mathematicians call Bernoulli trials and it can be proven that the number of rolls is 1/p.
So, for everything that has a fixed success probability in the game, we have: ANA = 1/p.
Let’s illustrate the concept by an example that is common to almost all of the games: If you want to push a strength-3 enemy army 1 space back you must roll 4, 5, or 6, i.e., you have a 50% chance of success on any single attempt, said in another way, p = 0.5.
Therefore, the ANA of pushing back the army is 1/0.5 = 2 or said in another way, the cost of a 1-space enemy advance has the required ANA = 2 because that’s the number of actions taken to undo the advance.
Had the strength of the army been 2, your chance of success on any single roll would be 4/6, which means that p = 2/3 ~ 0.67 and therefore the ANA would be 1/0.67 = 1.5.
The 1d6 ANA table
The “1d6 ANA table” below shows the ANA for getting 1 success at all the possible probabilities of success for the rolls of one 6-sided die (1d6). The table assumes that the game doesn’t allow certain success or failure, i.e. there are no probabilities lower that 1/6 or higher than 5/6.
A die roll modifier (DRM) of +1 takes you from a row to the one below it, +2 takes you to from a row to the one 2 steps below, etc. and it goes in the opposite direction for negative DRMs.
So, the ANA of pushing back a strength-3 army with a +1 DRM is 1.5 or said in another way: On average it takes 1.5 actions to push back an army with a strength of 3 one space back if you have add 1 to your die rolls.[ImageID=medium]
Counters from Israeli Independence. A +1 DRM against the Jordan army gives you an ANA increase of 1 while the same DRM against the Lebanon army gives you an ANA increase of 0.3. Image credit Alan Emrich.
Putting the 1d6 ANA table to good use
Let’s take a slightly more advanced example of ANA analysis:
1) You decide to build a fortification which has a 4/6 chance of being built for each action spent. According to the 1d6 ANA table, the ANA for this is 1.5.
2) The fortification has a 0.5 probability of blocking an attempt of an enemy army to move 1 space forwards and a 0.5 probability of being destroyed instead of preventing the advance.
3) According to the 1d6 ANA table, the ANA for the army to advance and destroy the fortification is 2 or said in another way, the fortification will on average prevent the enemy from advancing 1 time before it’s destroyed.
4) If it’s a strength-3 army that’s attacking the ANA worth of the fortification is 2 because the ANA for pushing back a strength-3 army is 2.
5) Since the fortification costs 1.5 ANA to build the “profit” of building it has an ANA of 2 – 1.5 = 0.5. You can then compare this to the profit of other actions you have available to see which action is worth the most.
Of course, there are confounding factors especially in the more complex games in the series, but the actions-as-currency and “ANAlysis” approach is still extremely useful if you want to win the game, study its balance, or if you’re a designer making a SoS game.
Applying the ANA concept to the entire game
Such calculations are not only useful when applied to a single event there’s also a long-term aspect. Let’s take an example of that: A track in a game has a strength-3 army that starts on space 5. You count the number of advances that army has in the event deck and find that it will in total move 15 spaces forward during the game.
Since it starts on space 5 it can do 4 advances without making you lose. Thus, you need to push it back at least 15-4 = 11 times during the game to avoid losing because of that army. Given that it’s a strength-3 army the ANA for pushing it back once is 2.
Thus, the ANA for keeping the army out of your base is 11*2 = 22 if we ignore DRMs.
So, we can simplistically reduce success in a States of Siege game to this equation:Number of actions you get >= Sum of the ANA to keep each enemy out of your base.
Let’s call it the ANA equation.
Thinking about the game in this way gives the designer a quick way to roughly estimate of how many actions they should give the player compared to the events they want to include in the game.
Next up: Math attacks
In the next post, we’ll take a look at how ANAlysis can be used create “math attacks” on the States of Siege games.
A blog about solitaire games and how to design them. I'm your host, Morten, co-designer of solo modes for Scythe, Gaia Project, Wingspan, Glen More II, and others.
16 Jun 2022
- [+] Dice rolls