Earlier quoted context omitted.
The external reward in "useful" PoW is given to all miners, good and bad. So everyone is equally incentivized. In fact it should actually increase the participation in the network, e.g. those who don't care about cryptocurrency but are interested in the "useful" problem being solved. The external reward by itself provides no competitive advantage. It reduces everyone's costs by the same amount. Rewriting history stil…
Rewriting history needs to be expensive, not just difficult. If you decrease the cost of writing history, you decrease the penalty of participating in a minority chain.
Both are extending their respective chains as fast as they can. A bad-agent, who wishes to double-spend has to rewrite history, and therefore has to start a few step behind the good agent. To succeed, the bad agent has to overtake the good agent.
Even if both have the same speed (i.e. 50% computational power each), the good agent will sill be a few steps ahead of the bad agent, and the system is secure. If the bad agent is faster(51%), then eventually it will overtake the good agent breaking security.
This is true even if the cost of computation is zero (e.g. the Govt. pays for all your computation cost). As long as the bad agent doesn't have 51% or more of the computational power, the system is secure.