Abstract.
Game Theory Analysis of The Settlers of Catan: The Optimal Solver Through Theoretical Model
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Jefray Ding, Mark Ma, Thienkingkeaw Porpun
May 20, 2024
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Game Theory has seen increasing applications in economics, social and behavioral sciences, and, more recently, artificial intelligence. To explore how AI combined with Game Theory can model and enhance the strategies of colonists during the Partition of Africa, which was driven by new imperialism, we ran simulations using the framework of the board game ”The Settlers of Catan.” This non-deterministic, multi-player game involves numerous branching options that satisfy trading, ports, settlements, and roads. To determine the optimal course of action, we utilized the MiniMax algorithm as the primary decision-making structure, supplemented by Alpha-Beta Pruning to analyze multiple moves ahead (search depth) and identify actions leading to the most advantageous board state (closest to victory). We assigned assumed values for various board conditions to construct payouts for each board layout (value functions) and adjusted them based on win rates during benchmarks. We developed a Catan game in Java featuring two types of play-testing players: the Complete Random player and the Weighted Random player. Based on our solver, our results show that the player achieved a 84% win rate in 2000 games against a weighted random player using a normal distribution. Expanding the scope of data-driven value functions in different contexts, particularly with multiple agents instead of just two, will further enhance our understanding of colonial economics and expansion strategies.
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Data Visualization
In Catan there are generally two possible ways to win : Road Rush (wood/brick early) vs City/Army (wheat/ore mid-late). Wheat resource tends to be king once cities/dev cards come online. This is a weighted data snippet taken from the model with different strategy profiles on 4 different map seeds. Pips means the opening access of that resource production weight from dice probabilities on the starting adjacencies. TurnsTo10VP means the average player turns it took to reach 10 Victory Points (1v1v1v1 winning condition). For each winning game, we tag which axis (LongestRoad_%, LargestArmy_%, BaseVP_%) was decisive at the moment of victory (which is than attributed and normalized to 100.)
Chart 1: How Each Strategy Wins (100% of Decisive Axis)
Road Rush wins are dominated by Longest Road (~60%), while City/Army leans heavily on Largest Army (~60%). Port Play and Ore Monopoly show the highest Base VP shares (>50%) reflecting slower, build heavy paths. This may sound intuitive, however, it points an insight on that players should wisely choose the in game strategy based on its starting bases. For example, if you are a player that has a playing style leaning toward road building than you should start heavily on brick and trees. If the map doesn't has a good brick/tree starting pip than you should probably choose a different strategy rather than insisting on your personal playing style.
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Chart 2: Scatter (OpeningPipSum vs TurnsTo10VP, colored by strategy)
Higher OpeningPipSum generally reduces TurnsTo10VP, showing the expected more production, faster victory pattern. At comparable pip totals, City/Army markers tend to sit lower (faster finishes) than Road Rush, consistent with cities/devs converting pips into VP more efficiently. The spread by strategy suggests playstyle moderates, but doesn’t erase, the pip-sum effect.
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Chart 3: Clustered Columns (Cities_T12 & Devs_T12 by WheatQuartile)
City and dev activity climbs with wheat access. Average Cities_T12 rises from ≈0.75 in the low wheat quartile to ≈2.4 at the top quartile. Devs_T12 also peaks highest in Q4 (~2.6). This supports the mid game mantra: wheat unlocks cities and dev cards, accelerating the City/Army path. This actually make sense if we look at Catan's production card, Wheat can be used in all three scenarios of building a village, upgrading to a town, and buying a dev card. That is also why in our mini-max model wheat has the highest utility rating compared to all other resources. This finding is also consistent with the monopoly strategy has everyone needs wheat. ​
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