What is stopping pool managers to fake a miner? What if a pool manager introduces a miner that is not there and then uses the reward distribution?
The pool members will notice that the pool is not mining as many blocks as it ought to, given its supposed total hash power as reported by the manager, and that they are therefore not earning as much money as they should.  A certain amount of this could be attributed to "bad luck"; but over time, or if it becomes extreme, they can tell statistically that something is fishy, and then they will switch to other pools.