Combinatorial Game Theory
who wins, and by how much
Field

Sums and comparison

Real games break into independent parts. Adding them up is the whole method, and comparing them is how it is checked.
2+-1+1/4+∗={5/4 | 5/4}outcome Leach sprig is a separate game; a move is a move in one of themthe total was computed by adding the games, not the labels

The sum is the object

Real positions come apart into independent regions, and a move happens in exactly one of them. That operation — the disjunctive sum — is what the whole theory is built to survive, and it is the reason values exist at all.

 ↑ − 0= ↑outcome L↑ > 0∗ − 0= ∗outcome N∗ ‖ 0⇑ − ↑= ↑outcome L⇑ > ↑1/2 − 1/4= 1/4outcome L1/2 > 1/4↑∗ − ∗= ↑outcome L↑∗ > ∗the differencethe verdict‖ means confused: neither greater, nor smaller, nor equal — and no amount of care removes it

Comparing positions

One position is worth at least another when the second player wins their difference. That is the only definition there is, it is a computation rather than a judgement, and it produces an order in which some pairs are simply not comparable.

 ∗ + ∗N + N0outcome P∗ + ∗2N + N∗3outcome N↑∗ + ↑∗N + N⇑outcome Leach part is a first-player winthe sumsame outcome classes going in, different outcomes coming outso a position has to be given a value, not merely a winner

Outcomes do not add

Knowing who wins each part of a position tells almost nothing about who wins the whole. Two first-player wins can sum to a second-player win, or to another first-player win, and no rule distinguishes the cases from the outcomes alone.

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