It is in the interests of every individual doctor to accept the subsidy, since the extra income from the subsidy significantly exceeds the revenue they would get from having you on their books. The result is that it becomes uneconomic to treat you, since this would require a whole new doctor to be trained, and the subsidy is only available to people who will not treat you.
In this case, you would not necessarily end up going completely without treatment - there would probably be some level of medical training which it would be economic for someone to obtain this and then treat you. (That is, providing there is no occupational licensing). Still, it is obvious that this rich person's actions have made you worse off, and probably quite significantly. Yet from a natural rights perspective, it is hard to see how you could have any kind of a claim against them.
Aside from being a potential weak point of natural-rights based ethics, this has potential real-world significance. Peter Steinberger, amongst others, has argued that actively accepting benefits from the state implies consenting to it, and provides numerous examples of activities he sees as fulfilling this condition. In many cases this seems intuitive, but there is an obvious counterargument for the philosophical anarchist: that the state has actively prevented citizens from obtaining these benefits except through the state, and therefore when it provides them with there benefits it is merely compensating them, rather than actually making an implicit offer of contract. This is actually a surprisingly wide-ranging objection. The state will not allow me to go about enforcing vigilante justice, and will not allow anyone else to do it on my behalf, so when I call the police to bring to justice a man who has stolen from me, I am merely calling upon the state to do what it is obliged to do - I am in no way consenting to anything. The state does not ban private healthcare, but it regulates it to such an extent that it can hardly be seen to be respecting my rights or those of my prospective doctor, and so when I go to the NHS for medical advice I am merely exacting recompense rather than seeking benefits. Even if I claim unemployment benefits, it is unclear that I accept any duties since state measures like the minimum wage, national insurance, and income tax all violate my natural rights and make it harder for me to obtain work.
But what if the state did not restrict me from obtaining these services other than through itself? I can't think of any indisputable examples of this offhand, but a strong example is education, where there are essentially three options - state schooling, private education (which is similar to the situation described in the opening paragraphs of this post) and homeschooling. Does the fact that, by providing free schools and so making it uneconomic to run affordable private schools, the state obstructs my obtaining of private education, make using state education for one's children invalid as an expression of tacit consent? My suspicion is that it doesn't: in the classic example of invalid tacit consent ("I propose that we move next week's meeting to Tuesday. Anyone who objects to this, chop off your arm. Oh good, everyone agrees!") the objection is not that chopping off one's arm is costly or difficult (that said, how many businessmen do carry around knives ready to chop off their arms at a moment's notice?) but that one has a right to keep one's arms. If the statement had been "Any employee who objects, raise your arm. Also, if you do then you must move into a different, smaller office," then objecting would have been costly but would not have entailed unjust loss, and so the tacit consent would have been valid. So sending one's children to a state school (obviously, since they were a minor at the time, the question of whether an individual himself/herself went to a state school is irrelevant) could reasonably be described as consent to the state. Except for two problems which are a problem for basically any theory of tacit consent to the state.
The first could be overcome if there were greater awareness of political philosophy among the general public, but is currently an obstacle to, I believe, every existing state: for consent to be valid, at least one of these two conditions must be met:
- There is intent to be bound to that consent.
- All consenting parties may reasonably be expected to realise that their action entails consent.
Suppose you own a historic mansion, and are in the habit of giving guided tours around it. My joining such a tour does not of itself imply that I agree to pay you for it; however, if you have a notice by the door indicating that there will be a charge, then I may reasonably be expected to pay even if I would rather not.
If someone intends to be bound to obeying the state, then sending their kids to a state school is probably unnecessary to achieve this. Hence, it is the second condition which is more likely to be useful for demonstrating that people consent to the state. But I think it is fair to say that the vast majority of people do not realise that sending kids to school can entail consent, and given that it is not clearly stated anywhere that "sending your child to a school owned by Her Majesty's Government implies that you agree to obey the laws set down by Her Majesty's Government" this is rather a problem for the validity of tacit consent.
That problem is tough but not impossible to overcome. The real problem with tacit consent is very similar to the problem with benefit theory: the idea that the state can obtain consent by providing benefits presupposes that the state had a right to provide those benefits. This in turn presupposes that the state had a right to the resources with which it provided those benefits, which presupposes that the taxes with which it gained the resources had been consented to by the people of the nation. Thus there is an infinite regress unless you have a situation in which either the state legitimately held assets without acquiring them from an outside source, or the taxed population consented in advance of receiving benefits. But no-one seriously believes the state began as anything other than a local warlord, which rules out the first option, and no-one seriously believes in an explicit contract with explicit consent, either present or historical, as the second option requires. Therefore tacit consent cannot provide a basis for political obligation in any existing state.