Distinction between "fuzzy" and "confused with."

Question

In the terminology of game theory, "fuzzy" and "confused with" signify different things. How are their associated concepts alike and distinct?

EDIT: My initial encounter with the terms was here: https://en.wikipedia.org/wiki/Star_(game_theory)

Answer 1

More recent textbooks such as Lessons in Play: An Introduction to Combinatorial Game Theory and Combinatorial Game Theory have consistent and distinct definitions for the two terms.

If $G$ and $H$ are (partizan combinatorial) games, then we say $G$ is confused with $H$ when neither $G\le H$ nor $H\le G$ hold. In other mathematical contexts such as order theory, one might say "$G$ and $H$ are incomparable".

In Lessons in Play, this is written with notation like $G\shortparallel H$ or maybe $G\parallel H$. In Combinatorial Game Theory, this is written $G\not\gtrless H$.

If $0$ is the game with no moves (or the equality class thereof), then we say that a game $G$ is fuzzy if $G$ is confused with $0$. For example, the game $*$ mentioned in the original question is "fuzzy" because it is "confused with" $0$, even though it is not confused with $1$ (we have $*<1$).

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There are also a few sources that use "fuzzy with" to mean "confused with" (e.g. Will Johnson's thesis and the Wikipedia page for "Fuzzy game"). I personally would recommend against that usage.