Split-proof probabilistic scheduling

If shortest jobs are served first, splitting a long job into smaller jobs reported under different aliases, will reduce the actual wait until completion. If longest jobs are served first, the dual maneuver of merging several jobs under a single reported identity is profitable. Both manipulations can be avoided if the scheduling order is random, and users care only about the expected wait until completion of their job. In the natural class of separable scheduling rules, Merge-proofness holds if individual delay is monotonic in own job size. Split-proofness is more demanding. The Proportional rule stands out among rules immune to splitting and merging. It draws the job served last with probabilities proportional to sizes, then repeats among the remaining jobs. It is the only split-proof scheduling rule constructed in this recursive way, that minimizes the worst expected delay of individual jobs; or such that an agent with a longer job incurs a longer delay.