They might have an incentive to compete, but I think they also have an incentive to collude. Last time I shopped around for insurance it seemed like the latter was taking place.
You're required to get homeowner's insurance if you have a mortgage, and you're required to have car insurance if you want to drive a car on public roads.
So the majority of Americans are forced to purchase at least one of these in order to live their normal lives, which makes demand inelastic.