How wide is a Settlers turn?

The engine's legal-action list is the raw width: every legal build site, every unit bank exchange, every playable card, and every robber hex paired with each victim. The search does not branch over all of it. Its candidate generator keeps builds, purchases, card plays, roads that shorten the path to a reachable site or extend a contested longest road, and exchanges that reduce a build deficit, then narrows later steps of the turn to the best few by static value.

What a planned turn costs

The measurement analysed every third decision with expectimax-v2 at depths 1 to 4 under a fixed 400,000-transition budget and no clock. Medians rise from 2,110 transitions and 15 ms at depth 1 to 180,066 transitions and 1.65 s at depth 4; a quarter of the depth-4 analyses hit the budget and are excluded from the medians. The ratio between consecutive depths is about six, then four, then three and a half: the opponents' round costs a fixed number of scenarios, and the transposition memo catches more as trees grow.

Where the work goes

In a depth-4 analysis the current turn is a small tree, a median of 179 own-turn nodes ending in 19 positions. Each later level starts from the positions the previous level followed through the opponents' round, so the counts multiply: a median of 709 nodes at level 2, 3,984 at level 3, and 17,251 at level 4. The share of own-turn positions already seen falls from 21% at depth 1 to 5% at depth 4, because dice scenarios spread later levels over many distinct boards.

In the browser

The same search compiled to WebAssembly runs 1.26 to 1.38 times slower than the native release build on 40 sampled main-phase positions (medians by depth in the chart), so a one-second budget in a browser worker usually completes depth 3 and always completes depth 2.

One decision, term by term

At turn 59 of an engine-arena game on seed 5, the corrected search at depth 3 chose to buy a development card with two wool in hand rather than extend a road. The bars show each candidate's value as the stack of its additive leaf terms; the point term dominates because a fifth of the deck is a victory card and the game is close.