Choice, chance, and looking ahead

Every Drop7 turn is two things glued together. First a choice: you pick one of seven columns. Then chance: the game deals the next disc (any of 1–7, equally likely) and, if your move opened a gray disc, shows you a hidden number you could not have known. A strategy that asks only "which column scores the most right now?" ignores half the game.

The examples build the idea of looking ahead one piece at a time on a real position, using numbers computed by the repository's rules engine. The only "evaluation" used here is points scored, so you can check every figure by hand.

A real position

Column 6 looks wonderful: the 1 lands on the stack, clears, the 4 underneath finds itself in a column of exactly four and clears too, and the 2 falls beside the 3 and clears as a second wave. The move earns 53 points. A player who only looks at the present takes it immediately. We will call that player greedy.

The disc you do not know yet

Now suppose you are a little more patient. You know the current disc is a 1. You do not know what comes after. But you know the odds exactly: seven possibilities, each one in seven. So for each column you could ask: after I play here, what is the best I can do with each possible next disc, and what is that worth on average?

That question is a chance node, and the average is its expected value. Here it is for the greedy move:

And here it is for a quieter move, dropping the 1 in column 1 where it simply clears on its own for 7 points and leaves the chain structure standing:

That is the whole idea of expectimax in one comparison. The greedy move banks more now; the patient move is worth more on average, because it keeps the future open. Neither number required a clever trick. They came from enumerating what chance can do.

Four ways to handle the same uncertainty

Averaging is not the only option. You could assume the best disc will come (optimistic), or the worst (pessimistic), or ignore the future altogether (greedy). On this position, these four attitudes pick three different columns:

Across the research ledger, the repository's status summary lists "fair chance handling matters" as its first durable conclusion: optimistic, worst-case, and tiny reused reveal samples ranked moves incorrectly in measured games. The optimist chases jackpots that usually do not arrive; the pessimist refuses to build anything; averaging is the attitude that survives contact with a random disc stream.

Looking further ahead: depth

Everything above looks one move ahead and then stops. Nothing prevents you from continuing: after each possible next disc, consider each column again, then each disc after that, and so on. How many moves ahead you go is the search's depth. A depth-4 search, written "D4", plays out four of its own choices interleaved with four rounds of chance before it evaluates what it sees.

The price is steep:

Because the tree explodes, two tricks are universal:

• Evaluate, don't finish. At the bottom of the tree the game is not over, so the search needs an opinion about how good each leaf board is. That opinion is the leaf evaluator: a hand-written scoring function in the reference policy, a learned model in many experiments. The concept page on evaluating a board is about how hard that opinion is to get right.
• Sample the chance. When enumerating every disc and every hidden reveal is too expensive, the search looks at a fixed handful of representative outcomes per chance node. The repository calls these samples strata, usually five or seven. Fewer strata is faster and noisier.

A live tree you can search yourself

Everything above was drawn from one recorded position. The explorer below builds the same kind of tree live, in your browser, for any seeded game: the position is the MAX node, each legal column is a choice whose number is the exact average over every chance outcome of that drop, and the outcomes under the expanded column are real boards the engine produced, each with its probability and the leaf's opinion of it. Click a column to expand it, click an outcome to watch the engine play that transition and carry on from the board it leaves behind. Set "below each outcome" to one or two plies to see the values move as the search looks further.

Walk the tree yourself

Pick a column, then a next disc, then see every reply ranked. This is exactly the tree a depth-2 search walks, and you can verify any of the numbers above by following the same path.

What "fair D4" means

The repository's reference policy, fair D4, is a depth-4 search that takes the best column at every choice node, averages over chance at every chance node (no optimism, no pessimism), samples five or seven strata per chance node, and scores leaves with a hand-tuned evaluator. "Fair" also means it uses only what a real player can see. It never reads hidden gray values or the random seed.

Fair D4 is a careful, patient, slightly short-sighted player. On its broad reference cohort it averaged about 308,000 points per game over 64 games, compared with a target of one million. It is the yardstick every other idea in this repository is measured against.

Why deeper is not automatically better

The natural next question (so go to depth 5, 6, 7?) has been asked and tested. The status summary records that selective depth 5, full depth 5, and cycle-boundary variants "often spent much more work, sampled chance outcomes too noisily, or overrode good D4 actions on unstable estimates." Deeper trees need more chance samples to stay fair, and a deep search with noisy chance can talk itself out of a good move. Depth interacts with chance sampling, with the quality of the leaf evaluator, and with the work budget; it is a dial, not a ladder. The concept page on whether more computation is the answer goes through the evidence.