Salutations ~
Part of last month’s goals were to make a Risk board game of the American Civil War.
The goal fell short due to the game not giving the right feel, but I sure-as-heck did the math to make the map 😁
For your reference, the game Risk has a map made of connected continents with various territories in each. If you control a continent, ie have a game piece in every territory, you get the continent bonus, which you usually spend for more game pieces.
The Data Set
It’s the continent bonus I calculated. To do so, I analyzed top-rated Risk games for the number of territories in each continent and how many connections every continent had with other continents. Here’s the list of games (pardon the formatting; yet to look into adding tables to WordPress):
- Classic
- Classic w/ a common community modification to connect the Australian continent and rebalance bonuses (ie “Connected”)
- Star Wars Clone Wars
- Starcraft
- Halo (Ring, Forge, Hammer, and Anvil maps treated separately)
- 2210
- Mass Effect
- Star Wars Original Trilogy
Online forums talking about Risk usually base the bonus on a continent’s connections (one territory in one continent connects to one territory in another continent). I feel we need to add territories to this calculation, however, as to control a larger continent requires the spending of more game pieces, thus larger continents are more expensive to get the bonus, regardless of connections (connections being a means for other players to disrupt your control of a continent).
The Equation
Because territories (required to get bonus) and connections (required to keep bonus) are so different in what they mean for a continent, I started my work with a linear equation for each continent for each game:
Nt * Ct + Nc * Cc = B
Nt = Number of territories
Ct = Territory constant for a bonus
Nc = Number of connections
Cc = Connection constant for a bonus
B = Continent bonus
We have Nt, Nc, and B for every continent. We need to solve for Ct and Cc, which we can do by combining the equations to eliminate those variables one at a time.
Note: Nc is the number of connections regardless of which territories are connected. 1 territory with 1 connection is 1 Nc; 1 territory with 4 connections is 4 Nc.
The Calculations
I assumed this would be straight forward for at least one of the Risk games. Spoiler: It was not 😑
Saving you some of the nitty-gritty calculations (you can do this yourself), let’s look at Risk Classic:
- Continent – Territories – Connections
- N. Amer. 9 3
- S. Amer. 4 2
- Europe 7 8
- Africa 6 6
- Asia 12 8
- Aust. 4 1
This leads to getting multiple values for Ct and Cc, meaning how bonuses were calculated was a seemingly arbitrary affair 🤷♂️
OK! No problem! I’ll try the same thing on the other games…
The Problem
OK. We have a problem. They also churn out obviously tiered continents (some being better than others). For instance, the Connected modification to Classic Risk, while better, leaves us with 3 distinct groups:
- Cc = 1.167 * Ct
- Cc = Ct
- Cc = 0.571 * Ct
To get around this, I tried averaging, normalizing, and a few other pen-and-paper solutions to make this work out.
Nothing worked out 🤦♂️
UNTIL I REMEMBERED:
~simplify~
The Solution
How does one simplify this sticky situation across multiple games? Some grossly off in their bonuses? (*ahem* Halo Risk 😐)
The solution is to combine territories and connections 🎉 Doing that, we get:
(Nt + Nc) * C = B
Nt = Number of territories
Nc = Number of connections
C = Constant for a bonus
B = Continent bonus
That equation allows for each game to get to C = B / (Nt + Nc), so a constant can appear. Here’s what I pulled out, also weighting each with BoardGameGeek ratings:
-
- Game – Constant – Weight
- Classic .400 5.58
- Connected .411 6.00 (Classic rounded up)
- SW CW .419 6.01
- Starcraft .389 6.37
- Halo* (Ring) .398 6.44
- Halo (Forge) .396 6.44
- Halo (Hammer) .407 6.44
- Halo (Anvil) .383 6.44
- 2210 .411 6.69
- Mass Effect .391 6.81
- SW OT .391 6.84
- * Halo needed extensive recalculation of its bonuses – they were incredibly low compared to any other Risk game. I may update BBG someday with a rules correction for improved and more consistent gameplay.
The Answer
We are left with two numbers: The weighted average (.399) and the median (.398). For simplicity’s sake, let’s call it .4 for:
(Nt + Nc) * .4 = B
Nt = Number of territories
Nc = Number of connections
B = Continent bonus
I adore when numbers come together ❤
TLDR; To get a fair continent bonus, add each territory and territory connection to another continent together, then multiply that by .4 to get the bonus for control of the continent.
The Other Observations
Looking at a fair number of Risk games, I noticed some trends between the versions. (We will skip looking at copy-paste Risk games that only do a reskinning of the theme.)
- The bonus constant 40% (.4) can be ‘flexed’ down to 33% (.33) or up to 42% (.42) without skewing the fairness of the continent. Whatever percent is used, keep in mind that higher percentages are preferred (more reward for the ‘risk’ of controlling a continent).
- 6 continents is expected on a Risk map.
- Each continent has a minimum of 2 connections and 5 territories (4 territories is doable but extreme and should remain only thematic).
- Good design means connections are greater than 25% of the territories in a continent. (Bad design examples: Australia in Classic, North Atlantic in 2210.)
- Good design means there are more territories than connections in a continent. (Bad design examples: Africa, Europe, and Asia in Classic.)
- More game pieces means better player experience and faster play (long games is a common critique of Risk).
- Capping either the number of game rounds, putting in a score tracker, limiting the number of game pieces per territory, or all of these things and more also assist the slow play problem.
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This was fun 😁 I may share later how I would “fix” each Risk game. Let me know if I should get on that sooner 😉 Cheers for now~
Jimmy Chattin 
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