Consider the following two player game based on the Euclidean algorithm: Positions are given by $(a,b)$ in $\mathbb N^2\setminus\{(0,0)\}$ (where $\mathbb N=\{0,1,2,\ldots\}$) defining a greatest common divisor in $\mathbb N$. Moves are given by
$(a,b)\longrightarrow (\min(a,b),\max(a,b)-k\min(a,b))$ for
$k$ in $\{1,\ldots,\lfloor \max(a,b)/\min(a,b)\rfloor\}$ where we assume $\min(a,b)>0$.
Two players move in turn until arriving at $(d,0)$ or $(0,d)$.
At this point, the player who can no longer move announces the
greatest common divisor $d$ of $a$ and $b$ defining the initial position $(a,b)$ and wins.
Winning positions are easy to describe: $(a,b)$ is
winning if and only if $\min(a,b)/\max(a,b)$
is smaller than the inverse $(-1+\sqrt{5})/2$ of the golden number $(1+\sqrt{5})/2$.
Has this game be described somewhere?
The variation where the last player with a move (i.e. playing a position $(a,b)$ with $\min(a,b)$ a divisor of $\max(a,b)$) wins
is also easy to describe: A position $(a,b)$ is winning if either $(a=b)$ or if $ab>0$ and $\max(a,b)/\min(a,b)>(1+\sqrt{5})/2$.
Best Answer
Following up on Peter Taylor's answer, here's a fairly complete annotated bibliography.
What you describe as the variant came first:
A. J. Cole and A. J. T. Davie, A game based on the Euclidean algorithm and a winning strategy for it, Math. Gaz. 53 (1969) 354-357. https://doi.org/10.2307/3612461
The analysis of your initial game was posed and answered in Mathematics Magazine:
J. W. Grossman, A Nim-type game, Problem #1537, Math. Mag. 70 (1997) 382. https://doi.org/10.1080/0025570X.1997.11996580
P. D. Straffin, Solution to Problem #1537, Math. Mag. 71 (1998) 394-395. https://doi.org/10.1080/0025570X.1998.11996686
Using continued fractions in the analysis:
T. Lengyel, A Nim-type game and continued fractions, Fibonacci Quart. 41 (2003) 310-320. https://www.fq.math.ca/Scanned/41-4/lengyel.pdf
Refining the N-positions of your initial game*:
G. Nivasch, The Sprague-Grundy function of the game Euclid, Discrete Math. 306 (2006) 2798-2800. https://doi.org/10.1016/j.disc.2006.04.020
Using the structure of the Calkin-Wilf tree in the analysis:
S. Hofmann, G. Shuster, J. Steuding, Euclid, Calkin & Wilf--playing with rationals, Elem. Math. 63 (2008) 109-117. https://doi.org/10.4171/EM/95
Refining the N-positions of your variant game*:
G. Cairns, N. B. Bao, T. Lengyel, The Sprague-Grundy function of the real game Euclid, Discrete Math. 311 (2011) 457-462. https://doi.org/10.1016/j.disc.2010.12.011
*The N- and P-position analysis of the two versions of the game is the same, but refining the N-positions by their Sprague-Grundy values differs.