Must sign expansion definition of surreal addition use induction on the natural sum? In Gonshor surreals and their arithmatic are defined using sign expansions. Addition of surreals $a,b$ is defined inductively by
$a+b = (a_L+b,a+b_L)\mid (a_R+b,a+b_R) $
Where the induction is on the ordinal $\alpha $ in  $dom (a)\oplus dom (b)<\alpha $ and $\oplus $ is the natural/Hessenberg sum.
My question is why can't we use normal ordinal addition in the induction. Gonshor seems to imply that using the natural sum "permits" the definition to work?
 A: $\DeclareMathOperator{\Noo}{\mathbf{No}}$For $a' \in a_L$, $l(a') + l(b)$ might not be strictly smaller than $l(a) + l(b)$ (for instance if $l(a) = 2$ and $l(b) = \omega$), whereas $l(a') \oplus l(b) < l(a) \oplus l(b)$: this is why Hessenberg sum is preferred.

The choice of considering Hessenberg sums is still somewhat arbitrary: the smallest well-founded order on $\Noo^2$ that works for most of those proofs is $(a,b) \prec_0 (c,d)$ iff $(a \triangleleft c$ and $b \trianglelefteq d$) or $(a \trianglelefteq c$ and $b \triangleleft d$) where $\triangleleft$ denotes the simplicity relation.
If one wants to deal only with length, then the natural order would be $(a,b) \prec_1 (c,d)$ iff $(l(a) < l(c)$ and $l(b) \leq l(d)$) or $(l(a) \leq l(c)$ and $l(b) < l(d)$).
Both of these are contained in $(a,b) \prec_2 (c,d)$ iff $l(a) \oplus l(b) < l(c) \oplus l(d)$.
So in a way, the last one is a more powerful tool for induction: most of the time one has to deal with elements $(a',b), (a,b')$ and so on, those are $\prec_0$ strictly smaller than $(a,b)$, and thus $\prec_1$ and $\prec_2$ strictly smaller than $(a,b)$, which allows induction on $(\Noo^2,\prec_2)$. 
As to why $\prec_2$ is broader: for example one could prove that $a\star b:= (a_{LL}\star(b+1),(a+1)\star b_{LL}) \ | \ (a_{RR} \star(b+1),(a+1)\star b_{RR})$ is commutative (here for instance $x_{LL}$ denotes the set of surreals $z \in y_L$ for some $y \in x_L$) with $\prec_2$ when neither $\prec_0$ nor $\prec_1$ would apply.
So my guess is Gonshor (and Conway) chose $\prec_2$ for its versatility, probably for the sake of concision.
