🏝️ Strategies in Catan
Strategies in board games with elements of chance
*Catan* is a classic board game in which players build settlements and cities by collecting resources from hexes marked with specific numbers. Each hex bears a number from 2 to 12 (excluding 7); when a roll of two dice results in that number, any player with a settlement adjacent to that hex receives the corresponding resource. The probability distribution of the sums follows the triangular pattern discussed in the section on dice. Consequently, the numbers 6 and 8 appear more frequently (each with a probability of \( \frac{5}{36} \)) than 5 and 9 (\( \frac{4}{36} \)), and even more so than 4 and 10 (\( \frac{3}{36} \)), 3 and 11 (\( \frac{2}{36} \)) or 2 and 12 (\( \frac{1}{36} \)).
Optimal placement of settlements. At the start of the game, each player places two settlements. The mathematical strategy is simple: players should choose locations where the sum of the probabilities of resource production is maximized. For example, if they place a settlement at the intersection of three hexes numbered 6, 8 and 5, the average number of resources per roll will be \( \frac{5}{36} + \frac{5}{36} + \frac{4}{36} = \frac{14}{36} \approx 0.389 \) resources. If, on the other hand, they choose hexes numbered 2, 12 and 3, the result is \( \frac{1+1+2}{36} = \frac{4}{36} \approx 0.111 \) — nearly 3.5 times less.
However, the number 7 does not yield any resources. Instead, rolling a 7 triggers the "robber", who steals resources. For this reason, it sometimes makes sense to avoid having too many settlements concentrated on popular numbers, so as not to attract the robber’s attention. Furthermore, players should also consider the relative scarcity of different resources (for example, stone or ore might be more in demand than wood).
Therefore, it is recommended to always place your settlements on combinations with the highest probability at the start of the game. Experienced players use probability tables to plan their expansion, for example, by building a road toward a hex marked with the number 6, even if it is far away.
Another aspect to consider is players' development strategy in relation to how the game unfolds. If they notice that their numbers are coming up less frequently than they should theoretically (meaning they are on a losing streak), they should not panic. While the law of large numbers will eventually even out the frequencies, the game might end before that happens. It is more rational to adapt: build ports to trade resources or try to secure new settlements associated with different numbers.
Mathematics does not guarantee a win in any single game, but over the course of many games, a player who uses probabilistic calculations will win more often than anyone else.
Expected value of resources in Catan
A key value for decision-making is the expected amount of resources a settlement yields per die roll. If a settlement is located on three hexes numbered \(a, b, c\), then the expected yield is: \[ \mathbb{E}[\text{resources}] = P(a) + P(b) + P(c), \] where \(P(n)\) is the probability of rolling a sum of \(n\) with two dice. A city doubles the yield from the hexes it is adjacent to. Therefore, when a settlement is upgraded to a city, the expected increase in resources from that hex is equal to \(P(n)\).
For example, suppose a player has a settlement on a hex numbered 6 (\(P=5/36\)). Upgrading it to a city will yield, on average, an additional \(5/36 \approx 0.139\) resources per roll. Compare this to building a new settlement on hexes 8 (\(5/36\)) and 5 (\(4/36\)). The combined expected yield from a new settlement across two hexes (while a settlement typically touches three hexes, if one of the numbers is already covered, the gain comes from the two new ones) might be higher. This type of analysis helps decide how to spend the resources: whether to build a city or to expand.
There are 5 types of resources in the game: wood, brick, wool, grain and ore. They do not have equal value: ore and grain are important for cities and development cards, while brick and wood are used for roads and settlements. Therefore, the expected value (EV) must be weighted to account for scarcity. We can introduce weights \(w_i\) for each resource; the expected "utility" of a settlement then becomes: \[ \mathbb{E}[U] = \sum_{\text{hexes}} w_{\text{type}} \cdot P(\text{number}). \] Experienced players use this formula implicitly, prioritizing ore and grain in the late game.
The Robber and its effect on expected value
When a 7 is rolled (a probability of \(6/36 \approx 16.7\%\)), the robber is activated. Any player with more than 7 resource cards must discard half of them. This significantly reduces the actual EV for players who have accumulated a large number of cards. Formally, if you have \(k\) resources, the expected loss due to the robber is: \[ \text{Loss} = P(7) \cdot \frac{k}{2} \cdot \mathbf{1}_{k>7}. \] Therefore, a rational strategy is to spend the resources so that one doesn’t exceed the limit when a 7 is most likely to be rolled (i.e., after one's turn, but before the next player’s roll). Calculating the expected value of the resources becomes more complex once one factors in the robber, but the core principle remains: diversify the settlements to avoid an overabundance of any single resource and build ports to allow for flexible resource conversion.
Players often underestimate the fact that the robber reduces EV by approximately 2–5%, depending on their playstyle. If a player is in the lead, their opponents are more likely to place the robber on their best hexes. This is another argument for avoiding a strategy of placing all the settlements on 6s and 8s. It is better to have at least one "safe" number (such as a 3 or an 11) that is rarely blocked.
Comparing strategies: Maximum EV vs. Balanced portfolio
Consider two initial placements:
- Strategy A: a settlement on three hexes with the numbers 6, 8 and 5 (a combined probability of \(14/36 \approx 0.389\)). Resources: wood, brick, ore (a good combination).
- Strategy B: a settlement on 6, 4, 10 (probability \(5/36 + 3/36 + 3/36 = 11/36 \approx 0.306\)), with two hexes providing grain and one providing ore (great for cities).
In terms of pure EV, Strategy A wins. However, if there is a grain shortage in the game and one plans to build cities rapidly, Strategy B might prove more effective, as grain and ore are the key resources for city construction. Over 20 rolls, Strategy A yields 7.8 resources but with very little grain whereas Strategy B yields 6.12 resources, almost all of which are the ones one actually needs. In practice, one should calculate not just the raw sum, but the weighted utility.
A mathematical approach is to define a goal (for example, building a city using 4 resources: 2 ore + 2 grain). The EV of the utility for strategy B will be higher if ore and grain appear with the necessary frequency. Therefore, before the game begins, it is useful to assess which resources are most important on the given map (for instance, if there is a port for wood, then wood becomes less valuable).
Large numbers law and Long-term strategy
In a game of Catan, the dice are typically rolled about 60–80 times. Over the course of a single game, the frequency of specific numbers appearing can deviate significantly from their theoretical probabilities. However, across a tournament series consisting of many games, a player who consistently chooses positions with the highest EV will hold a statistically significant advantage. Professional Catan players employ pre-match analysis: they determine the placement of their starting settlements by calculating the expected yield for the game, while also accounting for the likelihood of the robber blocking their resources and the availability of ports.
The formula for the expected total resource over the entire game (given \(N\) rolls): \[ \mathbb{E}[\text{resourses total}] = N \times \sum_{\text{your hexes}} P(\text{number}) \times \text{factor (1 for settlement, 2 for city)}. \] With \(N=70\) and two settlements having a combined probability of 0.7 (for example, 0.389 + 0.311), one can expect \(70 \times 0.7 = 49\) resources. This makes it possible to plan, on average, how many roads, settlements and cities one will be able to build.
Therefore, it is recommended to always keep a probability table in mind. If a trade is offered, evaluate it not just based on the current needs, but also on the resources expected to be received in the coming turns. Players who have an intuitive grasp of EV win more often.
Mathematics does not replace luck, but it increases the odds.
In Catan, as in any game involving randomness, it is impossible to guarantee a win in any single match. However, by systematically applying expected value calculations, accounting for variance and adapting to the situation at hand (such as the presence of the robber or a shortage of specific resources) one can transform the game from a guessing exercise into a process of rational planning. Mathematics is the best advisor, even when the dice do not roll in one's favor.