A tunable expectimax that beats the builders on paired boards

What changed from the reference

The frozen reference searches a beam of single actions and evaluates a fixed additive leaf; it loses to both builders and does not improve with budget. The new search plans whole turns instead:

1. It enumerates the distinct end-of-turn positions a turn can reach through builds, card plays, purchases, useful roads, and bank exchanges, treating positions reached by different action orders as one.
2. Purchases, thefts, and the player's own roll branch exactly over their outcomes. Discards enumerate every legal multiset and keep the four that best preserve the current build target.
3. Between planned turns, every other seat plays a fixed builder on its sampled hand under common random dice scenarios, so every candidate is judged on the same dice. Deeper turns expand only the best few first actions.
4. Iterative deepening adds one planned turn at a time under a wall-clock budget and a depth cap; the deepest completed depth is played.

The leaf prices points, cards in hand, and expected production in one currency: a card is worth a fixed number of points, and production is worth its expected cards over the rolls that remain. A plan term values the best build target by its gain divided by one plus the turns needed to afford it at current production, which is the same acquisition-time idea the ETA builder uses. Every weight is a setting, and five inclination sliders (aggression, risk, development, expansion, trade) scale weight groups so a browser opponent can have a recognisable style without changing the search.

The leaf was the problem, and how it was found

The first leaf treated a hidden victory card as a full point and production as nearly worthless. In unregistered 8-seed exploration the search bought 5.7 development cards per game and built 0.4 cities; ETA built 2.4 cities. Pricing production over the remaining rolls did not change the pattern, because a city that needs five specific cards lies beyond a two-turn horizon. Adding the acquisition-time plan term raised cities to 1.95 per game and lifted mean points to ETA's level. Those exploratory rounds guided design and carry no evidential weight; the registered cohort above is the first retained comparison.

Does more search help now?

With the reference, extra depth and samples changed nothing. With the new leaf, unregistered 16-seed sweeps (64 games each, about ±12 points of noise) ordered the configurations as expected: depth 1 with no lookahead won 3%, the depth-2 default 25%, sixteen scenarios with eight sampled worlds 36%, and depth 3 with a 1.5 s budget, sixteen scenarios and eight worlds 42% at 589 ms per decision. Depth 4 under a 3 s budget won 36% at 1.3 s per decision. The registered depth-3 cohort below tests the strongest of these.

Depth 3 on 64 boards: not distinguishable from depth 2

Run 9dc44463-a29d-4d38-af29-91483f109370 seated the same lineup with the search at depth 3, a 1.5 s budget, sixteen scenarios and eight sampled worlds. It won 82 of 256 slot-games (32.0%) against ETA's 24.2%; the paired contrast is +0.078 (95% interval −0.022 to +0.178), which includes no difference, so the registered rule is not met. Decisions averaged 629 ms and never exceeded the budget by more than a few milliseconds. The 42% seen in the 16-seed sweep was the optimistic end of its noise.

Both v2 cohorts beat ETA on points and wins by a similar margin. What the deep search does not fix is visible in the per-game means: it still buys 4.2 development cards and discards 10.5 cards per game, against ETA's 1.2 and 9.2. The next tuning question is the development-card and discard-risk weights, and the next evidence question is whether the gain survives the protocol arena. A frozen depth-3 candidate, expectimax-v2-deep in the policy registry, was registered for an 80-game protocol cohort.

Through the protocol arena: a positive direction, not a confirmation

Run e4949ebe-8132-4428-aae9-37ae1e0c3748 played the frozen candidate through the authoritative server against two ETA builders and one fast builder, twelve games at a time, with every one of the 80 games completing and no timeout moves. The candidate won 31 of 80 (38.8%); the ETA builder in the compared slot won 25 (31.2%). The registered rule asked for a 10-point gap and a one-sided exact test at p ≤ 0.05; the gap is 7.5 points and p = 0.25, so the cohort is inconclusive. The other ETA slot won 13 and the fast slot 11, a spread that shows how much seat and randomness move an 80-game unpaired cohort.