So in the example of the 100 share pool if you have 99 pool members and one hopper everyone does 1/100th the work and recieves 1/100th the pay I am not sure I follow how the pool is being robbed. Everye did .5BTC work and got paid .5BTC. I guess I see Paid for last N shares as being a potential for robbing people. As an example. I start a pool with 100 shares per block and mine 99 of them a pool hopper mines 1. Proportionally I get 99/100ths of the block or 49.5 BTC the hopper gets 1/100 or .05 BTC 100% of the time. Pay for last 5 shares I make 4/5ths of the take or 40BTC and the hopper that gets the last share gets 1/5 or 10BTC This means that the hopper would have a 5% chance of taking 20% of the block. Proportionaly I get paid more as I get more shares PPL5S I get paid less providing I do not take all 5 shares and only at 5 do I do better. It seems more likely to me that PPLNS rewards people disproportionately for their contributions by making faster hardware likely the only way to get one of the last shares. Yes any share can solve but I do see how mining on 100 would help your chances of being paid. Again I am sure I am missing something as even in proportional every share (work) gets and equal reward (BTC). So even by grabbing the last share the hopper would only get paid for one portion. so 50*.1*1.00 =.5BTC proportional 50*.2*.05 =.5BTC meaning that statistically hopping pays as well on either scheme. Like I said I must be missing something and these are exceedingly simple examples but I am still not seeing it.
It is far too complex and mathematically intensive topic to cover completely in a newbie thread but the reason good miners are "robbed" is because the pool hoppers exploit variance factor.
IF every block was 100 shares then yes there would be no issue w/ pool hopping.
However the simplest way to explain it is each share added to a pool becomes increasingly less valuable.
If a pool has 1000 shares and you add one share you have the potential for 1/1000th of a block.
If a pool has 10 million shares and you add one share AT BEST you have the potential to earn 1/10,000,000th of a block.
So in a long block miners get paid little. In a short block miners get paid a lot. Now if you choose your pools (and thus next block) randomly this random variance evens out.
However a pool hopper isn't choosing the block randomly. He is always looking (via scanning software) for the pool which has the smallest block. Thus on average his block size is smaller than normal and his reward is larger than normal.
I likely won't convince you in a newbie thread in 2 or 3 posts. This topic has been covered by thousands of post across dozens of threads and involving complex mathematical proofs, statistical analysis, and computer simulation. It would be like asking to prove the nuclear fission in possible in 2 form posts.
A practical example:
9 months ago when most pools were stupidly prop based and easily exploited I pool hopped. I pool hopped for roughly 3 months and submitted a total of ~12 million shares (thats 54,000 terrahashes). My revenue over that time period was 28% higher than solo mining. Remember mining is a zero sum game. There is no way to increase the size of the pie, only steal a larger share. Now you may say "mining has variance" and that is true BUT 28% variance above the mean over 54,000 terrahashes is outside realistic range for luck/variance. It is 4 standard deviations from the norm. (less than 1 in 2000 chance of happening randomly). There are many other hoppers with similar stats.
I don't hop anymore because my main reason for hopping was pools continued to use the fatally flawed proportional method even after evidence of hopping. When pool hopping first came out many pool operators denied pool hopping as flawed math, gambler fallacy, or people seeing what they wanted in variance. Today far fewer pools use prop any more. Over the last year tens of thousands of bitcoins in wealth were transfered from "fair miners" to "pool hoppers" before the community learned the lesson.
If you want more detailed info, simulations, and examples then check out the mining software subforum. Search for a thread called "learn to hop" it is a series of articles. There is also a document called "comparison of pool mining reward systems which is very detailed and includes a section on the "pool hopping exploit". If after that you still don't believe then join a proportional pool.