Here's such an analogy designed to "prove" that the infrared radiation ("back radiation") from the atmosphere to the Earth's surface either doesn't exist or cannot have the claimed effect. Not surprisingly, it's from one of the many authors of "Slaying the Sky Dragon" who thinks he's illustrated "perpetual feedback":
Say you have a blackbody plate (think of an electric heater) radiating 1000 W/m² toward another plate which, because of distance, absorbs half of that intensity, i.e., 500 W/m². At equilibrium, the receiving plate thus radiates 250 W/m² toward the 1000 W/m² plate. Question: Does the 1000 W/m² plate thereby rise to 1250 W/m²? If so, then, by raising the radiator's temperature without adding more energy, you've disproved the first law of thermodynamics. Effectively, you've made the radiator heat itself. Moreover, now at 1250 W/m², the radiator will heat the other plate still more, absorb another dose of back-radiated energy, and will reach 1562 W/m². And so on, ad infinitum.
This seems to prove a feedback causing a runaway heating in the system. However, the author (Alan Siddons) is confusing an infinite geometric series (with a finite sum) to a never-ending sequence of feedback. Let's look closer at the diagram. Something is missing (par for the course in these analogies). In this case it's the radiation lost from the system, and examining that will directly give us the equilibrium conditions.
At equilibrium, the two-plate system is receiving the equivalent of 1000 W/m² via the heated plate, so the system must lose that energy to the surroundings. If the heated plate is now radiating x W/m², then x/2 W/m² is being lost to the surroundings. The receiver is absorbing x/2 W/m², so must be radiating the same amount, with half of that radiation, x/4 W/m² lost to the surroundings. We have a simple expression to solve:
x/2 + x/4 = 1000 so 3x/4=1000 and x = 1,333.3
The heater radiates 1,333.3/2 = 666.7 (actualy 666.66 recurring) to the receiver, which radiates half, or 333.3 back:
Balance is restored and no runaway heating. But where does the "extra" radiation come from, which as claimed above "violates the First Law"? It comes from the heat energy stored in the system during the equilibrium process, and the amount depends on the specific heat of heater and receiver. There is no violation of the First Law, and no violation of the Second Law either. Net flow between the two plates is from hotter to cooler, with both plates heating up until overall equilibrium is restored. Al the heat energy in the system came from the heated plate, which has some of that returned to it from the receiver. The returned heat reduces the net heat lost from the heater, so because the input remains the same, it heats up until the heat lost from the system equals the heat input.
Does the cooler receiver "heat" the hotter heated plate? No, it simply replaces some of the energy radiated (and therefore lost) from the heater, so it's the continual 1000 W/m² energy input which does the heating. Mr. Siddons has himself said "a cooler body cannot heat a hotter body, it just slows the rate of cooing". This is a perfect example - initially the heater radiates 500 W/m² to the receiver, reducing to a net 666.7 - 333.3 or 333.3 W/m² at equilibrium, and the result is a hotter radiator, radiating a total of 1333.3 W/m².
The misconception arises from only considering the instantaneous input to the system and ignoring the heat stored during the equilibrium process, a common thread in such discussions. If you intend waving a big stick, make sure you've got hold of the right end of it first.