Games

This is Unit Three of Artificial Intelligence: games. The object here is the tree from Unit Two with exactly one thing added. Control of the branching alternates — at some forks one player chooses, at the rest the other one does. That single change is what makes a game tree a game tree. The machinery is fenced in three ways, and the course names the fences rather than hiding them: two players, both able to see the whole position, and one player's gain being exactly the other's loss. Each is a declared scope. Lift any of them and you are in a different field. The unit's central technique runs the tree backward — score the endings, then work upward giving each position the value its controller would choose. All the intelligence is in the direction of travel, and none of it is in knowing anything about the game. By the end you will be able to do five things. One: score a tree backward, and say where the intelligence sits. Two: say what a minimax value promises, and the two errors about it. Three: say why pricing a branch by its best leaf is wrong. Four: say why pruning is exact rather than approximate. Five: say what the champion-beating machines actually did.

Three entries of setup. The object, the two roles, and the three limits — and the entry on the limits is unusually honest, calling them declared scope rather than pretending they are properties of games in general.

The object.

AI 0141 Games as Striped Trees

A tree where control of the branching alternates between two players. The tree is Unit Two's familiar object — the alternation is the only new ingredient.

The two roles.

AI 0142 MAX and MIN

One number scores the whole outcome, with one player trying to raise it and the other trying to lower it. The names label roles rather than people — whoever moves at a fork controls it.

The three limits.

AI 0143 The Three Fences

Exactly two players, both seeing the whole position, and one player's gain being the other's loss. Each is a declared scope rather than a law about games, and lifting any one of them opens a different field.

Four entries, and the technique is fully contained in the first. Score the endings, then move upward giving every position the value of whichever child its controller would pick. Then it runs on a game small enough to finish completely.

The technique.

AI 0144 Score the Tree Backward

Score the endings, then give each position the value of the child its controller would choose. All the intelligence is in the direction of travel, and none of it in any knowledge of the game.

Run whole.

AI 0145 Pebbles, Solved Whole

Backward scoring solves a small game completely, on a card. No cleverness about the game itself appears anywhere — the rule alone does it.

A position with no escape.

AI 0146 The Poisoned Position

In a solved game, some positions lose for whoever must move — every available action hands the opponent a winning position. Lost does not mean badly played.

The capstone. A policy, not a move.

AI 0147 The Seven-Pebble Strategy

Solving from a particular starting position produces not a move but an answer prepared for every reply the opponent could make. The policy has to cover each branch — count the coverage, not the moves.

Four entries on interpreting the number. What it actually promises, an error about it, a worked example with real numbers rather than wins and losses, and a misreading that appears constantly outside games.

What the number is.

AI 0148 The Minimax Value

The score best play locks in from both sides — the most one player can guarantee and the least the other can hold them to. It assumes a capable opponent, so against a weak one it is a floor.

The error, and its reverse.

AI 0149 "Perfect Play Means Winning"

Perfect play does not mean winning — it means the best outcome achievable against best resistance, which may be a loss. And the reverse error matters more: the recommended move is the best guarantee available, not a promise.

Worked with numbers.

AI 0150 The Stall Duel, Scored

A scored duel showing the method with quantities rather than outcomes. One option dangles the largest number available and is the worst choice on the board.

And the trap.

AI 0151 The Best-Leaf Trap

Pricing a branch by its best leaf is the classic misreading — a branch is worth what alternating control will actually produce, not its most attractive ending. Every lottery advertisement is this trap in retail clothing.

Three entries on reading fewer leaves, and the third makes the claim that matters. This is not a shortcut that trades accuracy for speed — it returns the identical answer, which is a rarer property than it sounds.

The method.

AI 0152 Alpha-Beta

Track the best guarantee found so far and abandon any branch already certain to fail against it. It changes which leaves get read, and never which move gets chosen.

Watched.

AI 0153 Alpha-Beta at the Stalls

The same duel with a running best guarantee, showing a leaf go unread. It was not guessed about — it was proven unable to change the outcome.

The claim.

AI 0154 Pruning Is Not Approximation

The pruned search returns the same move and the same value as the full one. Elimination here is exact rather than approximate — ruling out what provably cannot matter is how an exact answer gets cheap.

Two entries on where the technique stops. Real games are too large to solve this way — the same counting law from Unit Two — so programs search partway and substitute a judgment for the depth they cannot reach.

The wall.

AI 0155 The Real-Board Wall

No machine solves chess or go the way small games are solved — those trees exceed every computer that will ever exist. This is Unit Two's counting law, not an engineering shortfall.

The substitute.

AI 0156 The Evaluation Function

Programs search partway down and score unfinished positions with a judging function standing in for the rest. The play is only as sound as that function at the frontier.

Three entries on the two most famous results in the field, and what separates them is where the judging function came from — one was written by people, and one was grown. The third entry says what neither machine did.

The first.

AI 0157 Deep Blue: Judgment Written

Massive search with a judging function crafted by hand, and no learning anywhere. Industrial scale did not change the kind of machinery — it was striped-tree search throughout.

The second.

AI 0158 AlphaGo: Judgment Grown

Tree search combined with judgment learned from games and then from self-play. And note the difference that creates: a written judge can be read and argued with, and a grown one cannot.

And what neither did.

AI 0159 "It Considered Every Possibility"

Neither came close to considering everything — each searched selectively and trusted a judge at the frontier. They did not out-think every line; they were more disciplined than humans about what to skip.

One entry closing the unit, and it returns to the three limits from Topic One. Each one, removed, opens onto a real field — and it names what that costs, which is every guarantee this unit established.

What each fence opens onto.

AI 0160 Lifting the Fences

Remove full visibility and you get hidden information; remove opposed interests and you get shared ones; remove the two-player limit and you get many. Every guarantee here was a purchase made inside the fences.

Five things you can now do. You can score a game tree backward — endings first, then each position taking the value its controller would choose — and say that the intelligence is the direction of travel rather than any knowledge of the game. You can say what a minimax value promises: the best result against best resistance, which may be a loss, and which is a floor against a weaker opponent. You can say why pricing a branch by its best leaf is wrong, since alternating control decides what a branch is actually worth. You can say why pruning is exact — it returns the identical move and value, by ruling out only what provably cannot matter. And you can say what the champion-beating machines did: searched selectively and trusted a judge, one written by hand and one grown from play.