I'm a little confused about how to correctly interchange factors in tensor products on graded vector spaces.
In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded vector space and $\bigotimes^k V$ be the graded k-fold tensor product of $V$. Then one basic convention according to the interchange of factors in the tensor product is
$v_1 \otimes v_2 = (-1)^{|v_1||v_2|}v_2 \otimes v_1$.
But this is ambiguous, isn't it?
For example take $v_1 \otimes v_2 \otimes v_3 \in \bigotimes^3 V$. Then on the one side we could first interchange $v_1$ and $v_2$ to get
$v_1 \otimes v_2 \otimes v_3 = (-1)^{|v_1||v_2|}v_2 \otimes v_1 \otimes v_3$
and then interchange $v_3$ and $v_1$ to get
$v_1 \otimes v_2 \otimes v_3 = (-1)^{|v_1||v_2|+|v_3||v_1|}v_2 \otimes v_3 \otimes v_1$
But on the other side we could first interchange $v_2$ and $v_3$ to get
$v_1 \otimes v_2 \otimes v_3 = (-1)^{|v_2||v_3|}v_1 \otimes v_3 \otimes v_2$
and then interchange $v_1$ and $v_2$ to get
$v_1 \otimes v_2 \otimes v_3 = (-1)^{|v_1||v_2|+|v_2||v_3|}v_2 \otimes v_3 \otimes v_1$
From this we get
$ (-1)^{|v_1||v_2|+|v_3||v_1|}v_2 \otimes v_3 \otimes v_1 = (-1)^{|v_1||v_2|+|v_2||v_3|}v_2 \otimes v_3 \otimes v_1$
and consequently
$(-1)^{|v_3||v_1|} = (-1)^{|v_2||v_3|}$ which is obviously not true in general.
What am I doing wrong here?