Working paper · Catan simulation project
Opening placement, the middle-game race, and the endgame, measured across 100,000 simulated games and a search bot
Rule-exact bots played 100,000 four-player games of Catan. Each game was tracked from the first settlement to the winning point. The opening decides more than anything that follows. The player with the best starting spots wins 33% of games; the player with the worst wins 15%. A starting pip of ore or wheat is worth about 1.5 pips of wood, and covering all five resources adds about 5 points of win probability on top of the pips. The snake draft gives every seat about the same total pips, but not the same kind. The first pick takes the ore; the last seat gets two picks in a row and the most variety. In the middle game the start becomes a production lead, and by round 12 the player in front wins almost half the time. In the endgame the first player to 8 points wins 63% of games, usually about two rounds later. Cities, a hidden victory-point card, and one of the two awards make up most of the gap. A search bot (MCTS) that beats the best tuned bot draws most of its edge from a handful of decisions per game: where to build roads, when to play a knight, and which trades to refuse.
What wins a game of Catan, and when is it decided?
Catan mixes luck with judgment. The dice decide what each hex produces. The players decide where to build, what to trade, and when to spend. The real question is how much each choice matters, and at which point in the game. Concretely:
To answer these, a complete rules engine ran with bots that negotiate trades. Every placement, roll, trade, build, and steal was logged, then related to winning. Effects are reported in points of win probability. In a four-player game an average player wins 25%, so "+5 points" means 30%.
The board. Each hex produces one of five resources: wood (forest), brick (hills), sheep (pasture), wheat (fields), and ore (mountains). There are four hexes each of wood, sheep, and wheat, three each of brick and ore, and one desert. Every producing hex carries a number from 2 to 12.
Dice and pips. Each turn starts with a roll of two dice. Every hex showing the rolled number pays one card to each settlement on its corners, and two to each city. Some numbers come up more often than others. The number token shows this with dots, called pips: the number of ways two dice can make that total.
Table 1. Pips per number token
What things cost. Every build has a fixed recipe:
Bonus points. The longest continuous road of at least 5 segments earns Longest Road (2 points). The most knights played, at least 3, earns Largest Army (2 points).
The opening draft. Before play starts, each player places two settlements, each with a road, in a snake order: 1, 2, 3, 4, then 4, 3, 2, 1. Seat 1 picks first and last; seat 4 picks twice in a row in the middle. The second settlement also pays one card of each hex it touches, so players begin with a small hand.
Trading. On their turn a player can trade cards with others on any terms both accept. They can also trade with the bank at 4 cards for 1, or at 3:1 or 2:1 if they have a settlement on a port.
The robber. When a 7 is rolled, anyone holding more than 7 cards discards half. The roller then moves the robber onto a hex, which stops it producing, and steals a card from a player there. Knights move the robber too.
How we measured. The games are played by bots built on an exact implementation of the base rules. Six bot styles (balanced, ore-and-wheat cities, expansion, development cards, port trading, and hoarding), each with randomly varied weights, give 100,000 games of varied but competent play. For the search bot in Section 5, a Monte Carlo tree search player is also pitted against the strongest tuned rule-based bot over 600 paired games.
The two opening settlements fix about half a player's production for the whole game, and they decide more than any later choice. Rank the four players in each game by the total pips of their two starting spots (Figure 1). The best start wins 33.4% of games and the worst 15.4%, more than double. The gap between them averages five pips.
Figure 1. Win rate by the quality of the start, ranked within each game
Measured another way, each extra starting pip adds about 3 points of win probability. A player opening with 25 or more pips wins 31–47% of games depending on the mix, against 6–10% for 15 or fewer.
Pips are not all equal. A logistic regression of winning on starting pips by resource, holding everything else fixed, gives the value of one more pip of each (Figure 2):
One ore pip is worth about one and a half wood pips. The reason sits in the recipes. Cities and development cards, which produce most of a winner's points, need ore and wheat. The board has only three ore hexes, and almost all ore gets built into something (93% of it, against 43% of wood). The player market prices ore and wheat highest for the same reason, at about 1.15 average cards each against 0.85 for wood. Ore is the resource you cannot easily buy, so it pays to own it.
Figure 2. What one more starting pip of each resource is worth
Figure 3 shows the same pattern in the raw data. Within every band of total pips, the share of the start that is ore and wheat lifts the win rate. At 19–21 pips, a start with under 20% ore and wheat wins 18%, and one with over half wins 31%. At 22–24 pips the range is 24% to 40%.
Figure 3. Win rate by starting pips and the share of those pips that are ore or wheat
The next rule is to cover the resources you lack. Players whose two settlements touch all five resources win 32.6% of games, those touching four win 25.4%, and those touching three win 18.9% (Table 2). Some of this is pips, since spots touching three different hexes have more of them. But the regression holds pips fixed and still credits each extra resource with about +5.3 points (Figure 4).
Missing a resource means buying it, usually at 4:1 from the bank or at a premium from other players. One combination matters most. A start that can make both a road (wood + brick) and a city (ore + wheat) wins 30.2%. Ore and wheat without the wood–brick pair wins 27.8%. Wood and brick without ore and wheat wins 18.9%. Neither pair wins 13.3%. About a quarter of players (24%) open with no ore at all. They win 18.0%, against 27.2% for those with some.
Table 2. Win rate by variety and by resource pairs in the start
New players are often told to grab the 6s and 8s. In this data the red numbers are worth exactly their pips and no more. Players with zero, one, or two 6/8 hexes win 24.6%, 25.0%, and 25.0%, even though those with two have two more pips on average. With pips held fixed, each 6/8 hex is worth about −3.3 points.
Why? Concentration and the robber. Pips on one big number pay out in bursts, which pushes hands over 7 and invites discards. Those hexes are also the ones the robber targets first. A 5 + 9 + 10 spot produces as much as a 6 + 8 + 2 spot, and more steadily.
Ports are worth a lot only when they match production. A 2:1 port in the start adds about +5.2 points with production held fixed, and a 3:1 port adds +1.5. In the raw data, though, port openers win less often, because port spots on the coast usually have few pips. A port is a good second settlement when it trades the resource your first settlement produces in bulk.
Figure 4. Other features of the start, with pips held fixed
The snake order means the spots get worse as picks go by, but not evenly (Figure 5). The first pick of the game takes a spot worth 11.2 pips on average, and the eighth pick 8.5. The best spot still available falls from 12.3 to 9.7.
Bots take the single highest-pip spot only about 44% of the time. They usually give up a pip or so for the right resources or for variety, which Section 3.3 shows is worth it.
Figure 5. Pips of each setup pick, in snake order
The snake order evens out the totals. Seat 1 gets the best first spot and the worst second one; seat 4 gets the reverse. All four seats end setup with 19.5–19.7 pips (Table 3). What differs is the kind of pips each seat gets (Figure 6).
Figure 6. Resource mix of each pick, in snake order
Table 3. The opening by seat
So the best plan depends on the seat.
The seat effect on winning is small. Seat 1 wins 25.9% and seat 4 24.0%, and the regression puts seat 4 at −1.3 points against seat 1 with the start held fixed. Part of seat 1's edge is simply moving first in the main game.
The second settlement matters nearly as much as the first. Players whose second spot has 6 pips or fewer win 14.6% of games, and those with 10–12 win 30.4% (Figure 7). For seat 1 the second pick is the last of the draft, which is why its first pick has to be strong.
Figure 7. Win rate by the pips of the second setup settlement
The opening sets the production a player will have, and with it the plan they can follow. Two experiments test whether that plan should follow the board.
Choosing a plan before placing. In the strategy lab, each bot chose a style before setup. The balanced placement (pips, ore and wheat, variety) won the most in every seat, from 36.6% in seat 1 to 34.0% in seat 4 (Table 4). Committing early to a narrow plan, such as port trading or pure expansion, cost 14–17 points in every seat.
Changing plan after placing. In the replan lab, every bot placed the balanced way and then switched style for the rest of the game. Playing in the port-trader style after a balanced placement was the only switch that helped: +1.3 points (±0.8). It was also the best follow-up for weak, average, and strong starts alike. Every other switch cost 4.6–8.4 points.
Place for balance first. Afterwards, use the bank and ports to turn the surplus your spots produce into what they lack, rather than chasing one resource.
Table 4. Placement style by seat (strategy lab) and follow-up style after a balanced placement (replan lab)
Early leads are fragile. The player with the most points at round 3 wins only 37.8% of games. By round 12 the leader wins 47.7%, and by round 18 54.6% (Figure 8). Last place goes the other way: 21.6% at round 3, close to an even chance, and 10.2% by round 18.
Production predicts the winner sooner than points do. The player with the highest production at round 3 wins 36.8%, and by round 12 that reaches 48%. The lowest producer falls to 9%.
Figure 8. Chance of winning by standing at each round
The lead still changes hands. Only 20% of games keep the same sole leader once one emerges. In 27% the lead changes once, in 29% twice, and in 24% three times or more.
Distance to the leader at round 10 is a good guide to a player's chances (Figure 9). Level or ahead, they win 38%. One point behind, 26%. Two behind, 19%. Four or more behind, 6%.
Figure 9. Win rate by VP behind the leader at round 10
Take a snapshot of every player at round 10 and ask which features predict winning, holding the others fixed (Figure 10). The strongest are:
Starting pips add nothing once current production is known (−0.3, not significant). That is the link between the opening and the middle game. The start matters because it seeds the production engine. By round 10, what counts is how far that engine has grown.
Figure 10. Round-10 position and the chance of winning
A city doubles a spot's output, and it is the single most telling milestone. Players who build their first city by round 4 win 33.5%. Those who build it in rounds 5–13 win 29–31%, and from round 14 on 28.9%. Players who never build one win 4.9% (Figure 11). Timing matters less than doing it at all. More than half of players (55%) have no city at round 10, and they win 19.9%.
Figure 11. Win rate by the round of the first city
Falling behind is rarely one bad roll. It is usually one of four slow leaks.
The game also pushes back against the leader. When a player steals with the robber, they pick the points leader 77% of the time, against 50% at random. Many players also refuse to trade with someone close to winning. Both slow the leader down, which is part of why early leads are fragile.
Player-to-player trading peaks around round 10, at about 0.9 trades per round. After that, trading with the bank takes over. Trading is how players fix a one-sided start, and it helps everyone. Without player trading, games take 14% longer and players discard 37% more. But on any single trade, the player who asks usually pays a small premium for getting what they need now.
The first player to show 8 public points wins 62.8% of games, typically about 2.1 rounds later. The last two points rarely come from a new settlement, because most spots are taken by then. They come from a city, an award, or a hidden victory-point card. Three in four winners (74%) finish holding at least one victory-point card that the others could not see.
The rest of the table pushes back on a leader. From 7 points on, other players' trade refusals start to matter. A proposer at 7 VP gets 18% of offers through, at 8 VP 12%, and at 9 VP 3% (Figure 12). The leader turns to the bank instead, which costs time but rarely stops them.
Figure 12. Share of trade offers accepted, by the proposer's points
Winners average 10.1 points and the others 6.5 (Figure 13). The difference of 3.5 points comes from:
Settlements are about the same for both groups (2.0 against 2.2), because winners upgrade theirs into cities. That is the opening's ore-and-wheat preference, paying out at the end.
Figure 13. Sources of victory points at game end
The two awards are close to decisive (Table 5). A player holding Longest Road alone wins 45.5% and Largest Army alone 40.2%. The rare player holding both wins 91%. Without either, a player wins only 8.3%, and only 18% of winners manage without one.
Table 5. The two awards at game end
The Monopoly card is an endgame weapon. Hands are biggest late in the game, so a Monopoly takes more then: about 6 cards in rounds 1–5 against 7.4 in rounds 21–30. Holding the card until it will take 7 or more is worth about +3 points against playing it early for a small haul.
The strongest player here is a Monte Carlo tree search (MCTS) bot. At each decision it tries the legal moves, plays each forward many times with simulated opponents, and scores the outcome with a learned value model. Its main rival is the best rule-based bot, tuned by evolution, which already beats the balanced style by about 9.6 points.
Over 600 paired games, MCTS beats the tuned bot by +7.2 points (95% CI 2.5–11.7). A decision-by-decision attribution, which credits each point where the two bots would choose differently, puts the edge at +9.1 points (7.6–10.6).
MCTS disagrees with the tuned bot only about 3.2 times in a game of roughly 126 decisions. In a typical game, two of those decisions account for 80% of the swing. Strong play here is mostly about recognizing the few moments that matter.
Figure 14. Where the search bot gains, by the move it chose instead
Its priorities, in order of the value they add:
Figure 15. The search bot's gains by game stage
Table 6. The search bot's most valuable disagreements
Put together, the search bot plays the same game this paper describes. It values the road to the next good spot, uses the robber against the leader, and trades only when the trade helps it more than the partner. It leaves the opening and city-building to the rules that already work.