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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)

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  • $\begingroup$ I (for one) have never heard the phrase "confused with" used in any sort of technical sense. "fuzzy" on the other hand is used technically...usually in the context of fuzzy sets, membership in which is not a binary affair but rather is measured by a function taking values between $0$ and $1$ (classical sets only take $0$ and $1$, the former indicating non-membership and the latter indicating membership). Do you have a particular reference in mind? $\endgroup$ – lulu Nov 3 '18 at 14:08
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    $\begingroup$ @lulu Fortunately or unfortunately, "confused with" is standard terminology in Combinatorial Game Theory, and "fuzzy games" do not really have much of a relationship to "fuzzy sets" or similar. $\endgroup$ – Mark S. Nov 5 '18 at 1:11
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    $\begingroup$ @GEdgar Fuzzy games are indeed defined on p. 73 of the second edition of On Numbers and Games. The author is John Conway. $\endgroup$ – Fabio Somenzi Nov 5 '18 at 1:26
  • $\begingroup$ @MarkS. Never knew that. Thanks! $\endgroup$ – lulu Nov 5 '18 at 10:38
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    $\begingroup$ Also, it is useful to say combinatorial game theory, at least the first time, since game theory by itself more commonly means the Von Neumann—Morgenstern theory of games. $\endgroup$ – GEdgar Nov 5 '18 at 11:47
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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 $*<1$).


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.

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