The existence of hoppers changes the % of time the pool spends in young rounds - hoppers make the pool grow old faster. This does decrease the total benefit that can be achieved. However, if a hopper does find a pool with a young round, this does not change the amplification factor for shares he submits. The payout per share will be B/N, where N is the total number of shares which will be found in this round, which is distributed geometrically conditioned on I shares having already been found. Time plays no role with this part.
In the limit where everyone hops, all proportional pools will be stuck at 43.5%, and thus no more hopping can be done.
The payout in absolute terms does not decay. The rate of return decays. In my opinion, time should always be a component when evaluating human behavior in an economic context.
It doesn't work this way. If you condition on the total number of shares in the round of course you'll find there's no benefit. You need to condition on the number of shares already found.
It was just a counter-example to discount a claim that a function was always continuous and positive. It was not meant to disprove the existence of rewards for pool hopping. They do exist, but I believe they have been quantified in a way that is misleading. Vladimir brings up an excellent point with mtred where it's hashrate changed from 100Gh/s to 400Gh/s to 100Gh/s. There are only a few pools available and the longer that mtred spends in the unfavorable share of a block, the longer it cannot be hopped into. This is a corner case where the number of pools is small and the length of rounds is potentially very large because of increased difficulty and dramatically lower hash rate.
If you say they hop at 50% of the total shares in the round (and it doesn't matter if it's 50% or some other number, if they know it or not, or whatever), you're leaving open the possibility that they will mine until 1.5D shares were found in a round whose total is 3D, which is something they will not do, since they leave at 0.435D.
Yes. But it was supposed to be a single counter-example in the form of "what if this occurred once when the stars are aligned, the result will be zero". It was not meant as a template for behavior in mining.
If everyone hops, the result is known - all proportional pool will end as no one will mine past 43.5%. The situation I analyze is the situation in practice, that hoppers are a minority. The hopper's rewards don't decay over time as the pool's reward system has no consideration of time. The amplification factor as a function of round age doesn't change. What does change is how young the rounds are expected to be. If the hopper does find a pool with a young round he should choose it over fallback mining (in all cases it is assumed that the hopper has a solo / fair pool option available for which he mines if there are no milkable proportional pools).
Hoppers are a minority in the overall population, at least I think they are (/checks under bed and in closet). However, in a small pool they can easily form a temporary majority. The reason why I bring up this "corner case" is that I believe we are actually living it right now.
As for solutions, I think smoothing the rewards over a varying number of blocks found would make it impossible to know the dimensions and boundaries of a "round". That would make it very hard to hop effectively unless the hoppers knew the algorithm
and inputs to the smoothing function. The issue is that the reward for the block is arbitrary with respect to the work completed.