Why are combinatorial games called combinatorial?

Question

From Aaron Siegel's Combinatorial Game Theory:

Combinatorial games are two-player games with no hidden information and no chance elements.

Why are those games (and the whole field) called "combinatorial"?

EDIT: What is combinatorial in those games? Is it the number of positions? The number of possible actions?

Answer 1

Main answer

What is combinatorial in those games? Is it the number of positions? The number of possible actions?

The game position/state is, or can be, combinatorial in that it's a combination of other positions/states.

When studying combinatorial games, we typically study games that are built as a combination of other game components. For example, towards the end of a game of Go, different isolated regions of the board are often independent (or nearly so), and so it can be considered as a combination of smaller games. In general, "Two-player games with no hidden information and no chance elements" can be easily combined to form a larger game with operations like the (disjunctive) sum.

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History

I have not been able to track down the precise origin of the phrase "combinatorial game", but I do have some context that may help.

In 1966, Smith wrote a foundational paper in the first issue of the Journal of Combinatorial Theory called "Graphs and Composite Games" extending the Sprague-Grundy theory to games played on a finite digraph (which may have infinite lines of play). In it, he writes:

A compound or composite game may be defined as follows. We imagine that Abe and Barbara play simultaneously a number of impartial component games, $\mathscr{G}^1,\mathscr{G}^2,\ldots,\mathscr{G}^m$. Each player in turn (legally) makes a move in some or all of the components...

I think some of the earliest papers to use the exact phrase "combinatorial game" might be ones by A. S. Fraenkel in the mid 70s. For example, "Constructions in combinatorial games with cycles" by Fraenkel and Perl from Volume 2 of "Infinite and Finite Sets - To Paul Erdös on His 60th Birthday" and "Combinatorial games with an annihilation rule" by Fraenkel in "The Influence of Computing on Mathematical Research and Education", Volume 20 of "Proceedings of Symposia in Applied Mathematics". In the latter, he refers to the "disjunctive compound" (as did Smith) and the "contrajunctive compound".

Edit: Aaron Siegel has kindly pointed out that On a combinatorial game by Erdős and Selfridge in 1970 predates Fraenkel.

It seems that while the different types of compounds were discussed at least as far back as 1953 in Milnor's "Sums of Positional Games", perhaps the name for the games themselves evolved from "compound game" and "composite game" to "combinatorial game" (by Erdős?).

Answer 2

Each position in a combinatorial game is one of the possible states of the 'board', given the 'pieces' and their 'rules', i.e. a combination. The game usually involves moving from one state to another via a 'move', with a defined state of 'win'.