The second civilization would just go ahead and build them anyways, since doing so maximizes their own utility function. Of course, there is an additional question of whether and how the first civilization will try to stop this from happening, since the second civ's Catastrophe Engines reduce their own utility. If the first civ ignores them, the second civ builds Catastrophe Engines the same way as before. If the first civ enforces a ban on Catastrophe Engines, then the second civ colonizes space using conventional methods. But most likely the first civ would eliminate the second civ (the "Berserker" scenario.)
The second civilization would just go ahead and build them anyways, since doing so maximizes their own utility function.
Then why isn't there an Engine on every star?
The Fermi Paradox leads us to conclude that either A) intelligent life is extremely improbable, B) intelligent life very rarely grows to a higher-level civilization, or C) that higher-level civilizations are common, but are not easy to spot. But each of these explanations are hard to believe. It is hard to believe that intelligent life is rare, given that hominids evolved intelligence so quickly. It is hard to believe that intelligence is inherently self-destructive, since as soon as an intelligent species gains the ability to colonize distant planets, it becomes increasingly unlikely that the entire species could be wiped out; meanwhile, it appears that our own species is on the verge of attaining this potential. It is hard to believe C, since natural selection favors expansionism, so if even a tiny fraction of higher-level civilizations value expansion, then that civilization becomes extremely visible to observers due to its exponential rate of expansion. Not to mention that our own system should have already been colonized by now.
Here I present a new explanation on why higher-level civilizations might be common, and yet still undetected. The key assumption is the existence of a type of Matrioshka brain which I call a "Catastrophe Engine." I cannot even speculate on the exotic physics which might give rise to such a design. However, the defining characteristics of a Catastrophe Engine are as follows:
The Catastrophe Engine is by no means a conservative explanation of the Fermi Paradox, since only the very most speculative principles of physics could possibly explain how an object of such destructive power could be constructed. Nevertheless, it is one explanation of how higher civilizations might be hard to detect as a consequence of purely economical motivations.
Supposing this is a correct explanation of the Fermi paradox, does it result in a desirable outcome for the long-term future of the human race? Perhaps not, since it necessarily implies the existence of a destructive technology that could damage a distant civilization. Any civilization lying close enough to be affected by our civilization would be incentivized to neutralize us before we gain this technology. On the other hand, if we could gain the technology before being detected, then mutually assured destruction could give us a bargaining chip, say, to be granted virtual tenancy in one of their Matrioshka Brains.