Another angle.
In general solid state amp try to hold the voltage constant as the load changes.
This formula wheel shows derivations of Ohm's Law and Watt's Law.

click for larger image.
So if a solid state amp holds the Voltage constant, the power will double if the impedance is cut in half.
For example assume the voltage is 2V and figure the power at 4 ohms and 8 ohms using P = (V^2)/Z
(2^2) = 4
At 4 ohms we get P = 4/4 = 1W
At 8 ohms we get P = 4/8 = 1/2W
This shows the power at 4 ohms is double the power at 8 ohms.
The limiting factor with solid state is the minimum impedance the amp is rated for.
So if our amp is rated for 2 ohms and can hold the voltage constant at 2V we get.
P=4/2 = 2W
If the amp is not rated for 2 ohms it will either go into protect mode, or it may incur damage.
Tube amps don't work this way. The transformer uses a ratio to match the high output impedance of the tubes to the low impedance of the speakers. A consequence of this is the voltage and current are varied so the power remains constant.
Let's say your amp is rated for 100W at 4 and 8 ohms. We will figure the Voltage and Current (I) for each impedance. To do this we need to solve for I or V first. You can do it either way.
I will solve for V first using V = sq rt(PZ)
For 8 ohm V = sq rt(100x8) = ~28.28V
For 4 ohms V = sq rt(100x4) = 20V
Now we can use I = P/V to solve for the current.
For 8 ohms I = 100/~28.28 =~3.54A
For 4 ohms I = 100/20 = 5A
Now inserting the values back into P = VI
At 8 ohms 100W = ~28.28V x ~3.54A
At 4 ohms 100W = 20V x 5A
In this example you get 100W at either 4 or 8 ohms. At 8ohms the voltage is higher and the current is lower.
In the event of an impedance mismatch, not all of the power is transferred from the tubes to the speakers. So available power is decreased.