(Of course it is not the bare tautology 'A=A'
that is at fault, but the attempt to derive
metaphysical principles from it, in particular
a form of hard determinism which insists that
there is only ever one course of action available
to an entity at any one time).

Considering the claim:
'..there can not be a "variety of results from a given cause".
That would contradict the meaning of the word "cause".
Leonard Peikoff makes this
point on page 14 of OPAR, "A thing cannot act against it's nature, 
i.e., in contradiction to it's identity, because A is A and 
contradictions are impossible. In any given set of circumstances, 
therefore, there is only one action possible to an entity, 
the action expressive of it's identity."'
                   
                     
The statement: "A thing cannot act against it's nature/identity"
is justified by the tautology A=A,
which would seem to mean that
"A thing cannot act against itself",
or rather,
"A thing cannot act except in the way it acts",
which is just as true, and just as pointless as
saying
"A thing cannot be any colour except the colour it is". 
                     
And of course a thing can be more than one colour,
and nothing in the foregoing justifies the conclusion:
                     
"...there is only one action possible.."
                
                     
The Objectivist argument seems to be that for an entity
to behave differently under the same circumstances but
at different times is self-contradictory -- or that
it contradicts the 'nature' or 'identity' of the entity.
However, the statement 'A=A' does not establish the 
'nature' as being any different from the entity itself.
Therefore, any behaviour that an entity displays will
be the behaviour of that entity, and will be 
consistent with its' nature' because all 'A=A' tells 
us about the behaviour of an entity is that its behaviour
is its behaviour is its behaviour.
                     
It would certainly be contradictory if an entity where to
perform certain combinations of actions at the same time,
eg. if your ballon were to simultaneously inflate and deflate.
But  Dr Peikoff has not established that failing
to perform the same action under the same circumstances
but at different times is self-contradictory.
Nor has has shown that the fact that some
actions are forbidden implies that only one course
of action is permittd.
                     
For instance a thrown die is forbidden from hovering in mid-air
or landing with a '7', but it still has 6 perfectly
possible and non-contradictory actions available to it.
                    
                     
 All this talk of 'A=A' is an would appear to be an attempt
to establish physical necessity (hard determinsim)
on the basis of logical necessity (A=A), but it
just doesn't work. The argument pivots on the
'nature' of the entiry, which is established on the
 basis of A=A (meaning the nature of the entity
simply *is* the entity), but subsequently treated
as though it was some determining feature above
and beyond the entity itself.
This sleight-of-hand is then used to support the
idea the question of an entity acting 'contrary to its nature'
is not a contingent fact to be established,
empirically, but a rather a logical necessity.
                     
                   
That the Objectivist argument for determinism,
as phrased by Leonard Peikoff, is patchwork of
ambiguities and non-sequiturs.