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The problem with public goods and free riding, as you describe, is that the expression of demand is conditional. If I can get what I want for free (because others have already paid enough), I won’t pay for it. This is, strictly speaking, a distortion of the system you propose: the aggregate importance is greater than it seems.
I think this is why a market as proposed needs modification before it can work for public goods.
You should also not just look at allocation of financial resources, but at regulation and legislation. This is not about allocating public funds, but about enacting restrictions (e.g. on carbon emissions). Policy interventions like this do not compete for resources in the same way that space exploration and cancer research compete with public investment to mitigate climate change.
If such measures are not taken it is not because financial resources are going elsewhere (higher priorities), but for example
- because parliaments are not devoting enough time to it (if other issues get a higher priority) — this could in principle also be the subject of a market, in which voters bid for the amount of time parliament must devote to certain topics, but it is distinct from the allocation of public funds
- because political parties are not including them in their manifestos, which means they do not even compete for legislative time.