Is multiplication of games that are equivalent to numbers well-defined? It's well-known that if you take the definition of surreal multiplication and attempts to generalize it to all games, the result is not well-defined, in that it does not respect equivalence of games.
What I'm wondering is this: What if we consider games that are equivalent to numbers, such as $\{*|*\}$, which is equivalent to $0$?  Or to put it differently, what if we consider surreal multiplication, but we allow representing our numbers in non-numeric ways, such as representing a $0$ by $\{*|*\}$?
Is multiplication well-defined (equivalence-respecting) in this intermediate setting?
(I ask because I noticed that, although multiplication is well-defined for impartial games, it's not well-defined for games equivalent to impartial games.  I'm wondering if something similar happens with numbers, or whether it remains well-defined; I haven't been able to find a counterexample.)
Thank you!
 A: (the counterexample has been worked out in collaboration with Harry Altman. All errors are mine)
The counterexample provides a game $K$ such that $K=0$ but $K^2\neq 0$. 
Here $\cong$ is identity of games, $=$ is Conway equivalence 
and juxtaposition (or sometimes $\cdot$) denote Conway product,
 and $K^2$ is $K \cdot K$.
Consider  $G \cong^{def} \{-1,0|0,1\}$ and  $H \cong^{def} * \cong^{def} \{0|0\}$. 
$G$ and $H$ are Conway equivalent, and moreover $*+*=0$, hence $K\cong^{def} G + H =0$.
Let's compute 
$* \cdot * \cong *$
$GH  \cong G \cdot * \cong \{*|*\} =0 $
$K^2 \cong (G+H)(G+H) = G^2 + GH + HG + H^2 = G^2 + \{0|0\}$.
Considering the left options $-1$, $0$ 
 in the definition of the Conway product $G \cdot G$ we get that 
$ G \cdot (-1) + 0  \cdot G - 0 \cdot  (-1) = -G = G = * $
is a left  option of $G^2$; similarly
$G =*$  is a right  option of $G^2$.
There is no need to fully compute $G^2$ to see that 
$G^2 + \{0|0\}$ is a first-winner game (hence not $=0$),
since the first player can always move in  $G^2$
choosing the option equivalent to $*$. The second player is left to play with
a game equivalent to
$*+*$, so he loses.
