Amplifiers have output impedances themselves. Picture it as a virtual resistor on the speaker wires, through which the power flows.
if this resistor wasn't there, and your amp had beefy enough powersupply inside, then what
@staticV3 has explained is correct, well theoretically.
But if the resistor (output impedance) is high in value, then power that flows through it gets shared between the speaker and this resistor. The closer the speaker impedance gets to the output impedance, the more this
sharing of power becomes noticeable.
Another related issue is
current limitation of the powersupply inside the amp. when it runs out of juice, then the amp can not output anything beyond.
To answer your specific question, it depends!
If the amp in question, has very low impedance, then yes it would double to 100W, if not then it would be 80W. You see the original 160W at four ohms, may be due to PSU's limitation.
For an amp to be able to almost double its output power at 4 ohms, it needs a good PSU
and very little output impedance, which costs money and expertise.
Also, consider the above golden reply!