Does a logical mining pool assign harder and harder difficulty to the same miner? Although every mining pool can do things differently, I wanted to get a handle on how a logical pool would behave.  If a miner joins the pool and solves a hash at a low difficult, submits the result back to the pool, which then accepts it, would the next logical thing for the pool to be to assign that miner a slightly harder bit of work to do?  In other words, would the "set_difficulty" level be higher the next time around (assuming the difficulty solved by the miner is less than the network difficulty)?

If the pool does not assign harder work and lets the miner continue to solve problems at the same difficulty (which is less than the network difficulty), how does the overall pool benefit?
[Edit: my previous answer was incorrect in it's explanation, I've updated it since]


  would the next logical thing for the pool to be to assign that miner a slightly harder bit of work to do?


No. In a simple sense, consider this: if the difficulty is increased every time a miner submits their share, then it will increase endlessly over time, which is obviously undesirable. 


  If the pool does not assign harder work and lets the miner continue to solve problems at the same difficulty (which is less than the network difficulty), how does the overall pool benefit?


Miners in the pool will split the rewards for valid blocks amongst themselves, according to the share of hashpower each miner is contributing. The pool difficulty is a way to track each miner's share, while also dividing up the work load amongst pool participants. 

Each miner has a certain amount of hashpower available, and will set their difficulty to an appropriate level so that they can submit their work and claim their share of the rewards. Setting the difficulty too low is not a way for a miner to claim an unfair share of the rewards, it just means they'll likely find a valid hash sooner, but that valid hash will be worth a smaller share, which balances things out.