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Combinatorial game theory studies positions in alternating-move games with fully specified legal moves, usually perfect information and no chance. A position can be defined recursively by the positions available to each player. Under the normal-play convention, a player with no legal move loses.

An impartial game gives both players the same moves; Nim is the standard example. A partizan game allows different moves for Left and Right. The disjunctive sum lets a player move in exactly one component on each turn, making it possible to study how game positions combine algebraically. Infinite play or misère play requires additional conventions; normal-play results do not automatically transfer.

In a finite impartial game with the normal-play convention and no draws, a losing position is one from which the player whose turn it is cannot force a win against best play. It is also called a P-position, since the previous player wins.

A terminal position is losing. Recursively, a position is winning if it has at least one move to a losing position, and losing if every legal move goes to a winning position. Induction on the finite remaining game tree proves this classification. For a single Nim heap, only the empty heap is losing. Infinite plays and misère play need separate conventions and can change the base case.