I played with this idea quite extensively recently. Note that all my testing was done with a system based purely on probability with a target profit and stop loss, exiting end of day if no other exit was triggered. I considered each series of entries as one "trade". I maintained the same risk for each trade throughout my testing. That is to say if I set my risk at $100 and had 3 entries, my risk for each entry would be some fraction of the $100 with the sum of all 3 equaling $100. I compared all my testing to just one entry at the defined risk.
Although I tested many concepts, the main ones I investigated were as follows:
Concept 1: 2 entries of equal size (number of shares). Second entry at various percentages to the stop loss. Maintaining the initial profit target for both entries. Through some testing I found the second entry at 50% to the stop loss to be close to optimal.
Concept 2: 2 entries of equal risk. Second entry at 50% to the stop loss (so twice the size of entry 1). Maintaining the initial profit target for both entries. My testing found this to be inferior to Concept 1.
Concept 3: 3 entries of equal size. Second entry at 25% or 50% to stop loss, third entry at 50% or 75% to stop loss. My testing found this to be inferior to Concept 1.
Concept 4: 2 entries of equal size. Second entry at 50% to the stop loss. Profit target lowered to various percentages of the initial profit target if the second entry was taken. My testing found this to be inferior to Concept 1.
Concept 5: 2 entries of equal size. Second entry at 50% to the stop loss. Profit target of trade lowered at various rates as the trading day progressed (for example, maintain initial target for first 30 minutes, lower target by x amount for next hour, lower target by x amount for next hour, etc.). My testing found this to be inferior to Concept 1.
When compared to the simplest system of 1 entry, target profit, stop loss, or end of day exit, Concept 1 wasn't that much better. I did not feel the difference was statistically significant. So my main conclusion from this exercise was that for intraday trading with a fixed risk per trade (set of entries), there is no real benefit to do multiple entries, and certainly no benefit to doing more than 2 entries. Furthermore, the trade management becomes exponentially more complicated with each additional entry (with purely automated systems). From extensive testing I basically found that adding more complexity in simply reduced the expectancy...so took me about a month and a half of extensive testing to learn to K.I.S.S.
Over a longer timeframe, I do believe there is some benefit to multiple entries. A while back (8 months or so) I was playing with multiple entry systems over a couple weeks timeframe and multiple entries did seem to be beneficial moreso than a single entry system.
Anyways...those are my conclusions for the systems I gravitate towards...certainly won't be true in all cases I'm sure.