The 51% problem is deep. Any discussion of a solution to it should
begin with a link to an article that shows a profound discovery has
been made. Selfish mining prevention and pollution should be on
bitcoin-discussion, but it appears that list is not active.

The problem with Andrew's idea below is that it is a positive feedback
loop that amplifies oscillations. If h goes up or down due to price
changes or random solvetime variation, then the net reward goes in the
same direction, which motivates miners to cause h to go even further
in the same direction, which is a positive feedback loop until some
limit is reached. To make matters worse, miner profit motivation in
choosing which coin to mine is a non-linear function: a 30% drop in
difficulty (or 30% increase in this reward function) in an alt coin
can cause a 300% increase in hashrate.

Average of 144 past blocks to determine h are needed so that it does
not vary too much.  A selfish mine of 72 blocks would result in only a
12.5% loss compared to not using this pro-oscillation function. I've
tried similar reward functions in trying to reduce on-off mining.

There may also be a problem of issuing too many or too few coins,
depending on how fast h rises in the long term.

An alternative is to increase difficulty with this or a similar
function instead of reward. From a miner's perspective, there is not a
difference (they are only interested in the (price+fees)/difficulty
ratio. This would have the same problems.

The problem has been solved to the best of our ability by the Nakamoto
consensus. The math is straightforward, so you can't get around it's
failings unless it's a profound solution or we shift trust to some
place else. Currently we have to choose and trust a small group of
coins (or 1) to be the best choice(s), and to trust that the reward
plus fees we pay for mining (compared to coin value) is enough to
prevent a 51% attack.

> Say p is the peak hashrate for 365 periods (1 year)
> consisting of 144 blocks, h is the hashrate of the last 144 block (1
> day) period, and r is the base subsidy (12.5 BTC currently). You can
> then make the max block reward 0.5 r (1 + h/p). So if hashrate is at
> peak you get the full reward. Otherwise you get less, down to a min of
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