Settlers of Catan Strategies Part 1

In college, my friends and I would get into board gaming spells that would result in us playing a single game exclusively. I used to hang out at Games of Westwood after school hours and could almost guarantee getting a board game in nearly every other night.  One of my favorite board games is called Settlers of Catan. Since Settlers is a quick game (2-3 hours) and the gaming store had an irregular cycle of players, I got the opportunity for many games against repeat and one off players. I must have played it at least 40 or 50 times and developed a small reputation for being a very strong opponent. Before I get into some of my strategies, let’s talk a little bit about the game.

The basic game consists of 19 hexagonal tiles that represent the island of Catan. Each tile, except for the sole desert tile, produces a single type of resource- grain, sheep, clay, ore, and wood. These resources are used to build settlements, roads, cities, and development cards. Each settlement and city is worth points, with bonus points for the longest road, largest army, and certain development cards. The first player to 10 points wins. What makes the game interesting, from a strategic point, are two elements. First the game has a turn-wise random but game-wise static probability distribution to distribute new resources within the game, and secondly the players have the ability to trade resources forming a functional market with specific spot prices.

Each player places their game tokens at the intersection of three hexagonal board pieces. Each board piece has a number that when rolled produces that resource for anyone on the intersection. The possible numbers for each board section range from 2 to 12 (with no 7); this coincides with the sum of two six sided dice. There are four board sections for grain, sheep, and wood and three board sections for clay and ore.  The 18 numbered board sections have two numbers from 3 to 11 and only a single 2 or 12. The basic game has a set number for each board section, but most players play with a random drop of tiles and then follow the assigned drop of numbers. This makes sure that you can never have a 6 by and 8 (the two best numbers), and that the 2 and 12 are diametrically opposite of each other. Surrounding the land is the Sea of Catan that contains trade ports that allow players to exchange certain resources for others in more favorable ratios.

What is very interesting about this setup is that you have certain amount of probability that can be determined at the start of each game. The sum of two 6 sided dice has 36 distinct combinations with 7 being the most possible combo (1 in 6 or 6 in 36) and 2 and 12 the least possible at (1 in 36). Since it’s very easy to think in parts of 36, I like to think of each board section in their respective probability. A board section with a 6 has a 5 (in 36) probability of being rolled each turn and a 2 has a 1 (in 36) chance of being rolled. The creators made sure to give this value to everyone as each number is listed with 1 to 5 circles depending upon the probability of being rolled per turn. Most players are savvy enough to realize this and determine their city placement by trying to get sections with the 6 and 8. You’ll notice in the section (since placement proceeds in a circle), each of the four players has a single city on a 6 or an 8 intersection. White probably was the fourth placement and got to place both cities at the same time. This is probably why it has two 6 placements.

While this is an effective strategy, what most players miss is that the clay and ore board sections are rarer than the others. While they may have lower values, their ‘trade value’ is actually much higher than other resources. I like to multiply their value by a 1/3 (+33%) to represent their true trade value. If you look at the blue player’s placement, his two choices have a combined value of 10 and 9. This is lower than other players’ cities on the 6 and 8 board sections which have a combined value of around 11. What isn’t usually calculated is that the blue player has the clay and ore sections and you should internally multiply their value by 1/3. If we do this, the left placement comes out to a trade-weighted value of 11 and the right placement has a trade weighted value of 10.5. This raises his placement value significantly. If you calculate the un-modified values, the red player has a placement value of 22 and all of the others are at 19. Red definitely has the best placement, but no other player has a disadvantage. If you recalculate the trade-weighted value of the placements, Red has placement value of 23.6, Blue is second at 21.5, Orange is at 21.3, and White is far behind at 19.3. Even though white was able to capture two 6’s (the best placement), his trade-weighted placement puts him far behind the others. He probably has the worst chance of winning of all of the players. I would also subjectively raise the value of blue’s ore positions, since the third ore spot is a twelve which no one has. If you want you could instead think of the ore values with a 50% bonus. In this case, red and blue are the clear outliers and placed strongly with placement values of 24 and 22.5 respectively.

… to be continued in Settler’s of Catan Strategies Part 2.

Lizardmen Leadership Defined

In Warhammer Fantasy Battle the lizardmen army has a special rule known as ‘cold blooded’ which allows them to roll three dice instead of two for leadership, drop the highest and only sum up the other two. While this gives results similar to the basic two dice leadership roll (between 2 to 12), I’ve always known intuitively that the results are a lot better. Today I’m going to quantify the actual difference and compare them to a different high leadership army (the dwarves).

Firstly, I built a simple spreadsheet in Excel to represent the results of the rule. I know, I know, right about now you’re wondering why I didn’t write a program to solve the dice roll for me. Well to tell you the truth, I’m not that great of programmer. Actually, I’m a horrible programmer. I got through my dissertation on Labview and writing my monochromator automation program with goto statements…. yes the program was in Basic. But it worked, and I got the spectroscopy data I needed so it’s all win-win in my eyes.

With a simple Excel spreadsheet emulating the three dice roll drop the highest leadership rule, I decided to do some basic statistics to determine what the ‘cold blooded’ rule does for the army versus a standard leadership roll of 2d6. The average leadership roll is 5.54 (vs. 7) with a standard deviation of 2.22 (vs. 2.45). So you gain a base of 1.46 points in leadership and have a tighter spread meaning that what you’re troops end up rolling is closer to the average they should be rolling. All in all a plus for leadership and a deviation that definitely boosts them into a high leadership army (even though the numbers in the book don’t look to be that high). But that’s not all that happens. The distribution is left skewed also, meaning it’s really hard to roll a 10, 11, or 12 with the army and rolling a 2, 3, or 4 is more probable than it the numbers suggest. The two graphs below show what I mean. The solid fill is the relative chance of rolling a particular result, while the line is the cumulative chance of rolling the stated leadership value or less.


As you can plainly see there is a significant probability of rolling low numbers. For leadership, which happens to be the most important roll you make in a battle (other than cannon sniping shots), this actually skews the odds towards the lizardmen side. I’m also including the standard leadership distribution below so you can visually compare the two.


For a standard army testing on 7, you’ve got a 58% chance of passing. The lizardman army has a 52% chance of passing on a 5 (nearly 2 points lower). As you get a higher leadership the number becomes even more skewed. A lizardman leadership of 9 (which the slann has) has a pass rate of 94%, the equivalent standard leadership number is 11 (at 97%)! The same applies for saurus oldbloods and scar-vet characters, vanilla saurus and temple guard infantry, and saurus cavalry. Their leadership of 8 (90% pass rate) is equivalent to a 10 (92% pass rate) meaning that lone units can act as if they have leadership equivalent to other armies generals!! What this means for lizardmen troops, if your leadership is 5 or below you can safely say you add 1 to 1.5 points to it. As soon as you’re leadership is 6 and above, it’s as if your leadership score just went up by two points. That’s a fairly powerful leadership boost hidden within the cold-blooded rule.




FYI: Dwarf units and heroes are 9 with the fighting lord at a 10. A lizardman army made up of saurus units has a higher leadership than dwarfs, has a similar cost, strength and toughness. They do have a bit lower armor save, but a much higher movement speed. All in all, I’m thinking the ‘cold blooded’ rule isn’t priced correctly, if you consider a dwarf well priced (which they aren’t).