Here’s a self-enforcing protocol for determining property tax: the homeowner decides the value of the property and calculates the resultant tax, and the government can either accept the tax or buy the home for that price. Sounds unrealistic, but the Greek government implemented exactly that system for the taxation of antiquities. It was the easiest way to motivate people to accurately report the value of antiquities.This makes a lot of sense to me for things like Greek antiquities, where the owner of the object has a lot of private and expensive knowledge about the true market value of the good. I'm not sure that it makes sense for housing. Right now, city councils hire assessors to make their tax assessments; it's unclear that distributing the burden of assessment more efficiently extracts information about market prices. I'd expect that the administrative cost minimizing solution is the one where the council hires assessors rather than having every household privately either hire assessors or otherwise come up with a figure.
I wonder if anyone's ever done formal modeling of this arrangement. In contrast to the case with antiquities, the valuer/owner experiences massive transactions costs if the government unexpectedly decides to exercise its option to buy. Consequently, folks would have an incentive to bid willingness to accept rather than expected market price. And that means we'd then be taxing folks' sentimental attachment to pieces of property. However, if market value (presumably the government's strike price for the call option) is below willingness to accept, folks will shade their valuations down to market value. If there's uncertainty about what the government thinks actual market value is, risk-neutral folks will shade their bids such that the probability-weighted sums of utility across keeping the house with a lower tax bill and losing the house are maximized; risk-averse folks might overvalue their house for fear that the government will exercise the call option, though they shouldn't go above willingness to accept.
Another complication is whether the government would really ever be able to exercise its call option. Could they really evict the little old lady who came up with her valuation number using the recommended rule-of-thumb but whose property experienced idiosyncratic appreciation? If not, the system falls apart.
Leaving aside those problems, I can see two reasons for wanting to move to such a system, but I'm not sure that they're sufficient bases for such a move.
First, if the government wanted to base property taxes on the owner's willingness to accept rather than on market value, then this system would be preferable. In a world where official assessments are updated infrequently but are always updated in the case of a market sale, tax assessments can induce inefficiently low turnover in housing: if your property has appreciated since the last assessment, you'll be more reluctant to move than if your property were taxed based on a current assessment. Moving to regular owner-assessment could solve this problem, but so too could more frequent official assessments.
Second, if the system did induce truthful revelation of willingness to accept, with a call option for government based on that price, eminent domain would be a heck of a lot simpler. But again, I'm not sure that these benefits would beat the costs. You could argue that it's an erosion of property rights in giving the government a call option on all our property; it's unclear to me that this system has that much effect at the margin. Kelo v. City of New London is the current precedent in the US; New Zealand also seems plenty able to expropriate property owners for public works as well. At least this system would force payment closer to actual willingness to accept.
Finally, if valuations were in the public domain and folks were posting their real willingness to accept, the real estate market would be a heck of a lot more interesting. If you always fancied that house down the road, you'd know what you'd need to pay for it.
A more worrying downside would be that it would allow an unscrupulous government to force its opponents to pay higher property taxes: if your property tax is based on how far from average valuation your house is, and the government uses a rule of thumb that it exercises its call option if reported valuation is say 5% below market value, you might rightly fear that the rule of thumb in your case would be actual market value, or some higher number. You then have to report a valuation closer to your true willingness to accept than do others; in equilibrium, you pay slightly higher property taxes.
Working out the formal modeling and likely effects of such a rule would be an interesting project for someone, if it hasn't already been done. I have a hard time seeing it being a worthwhile move in property taxation though.