I haven't read the book, but, in general, it's not a good move to split the difference from the start, because there is usually at least one more round of back-and-forth. So once you've split the difference, you're already going to end up closer to the other person's initial bid.
e.g. "I want 10,000" -> "I'll offer 5000" -> "Let's spilt the difference and say 7500" -> "How about 6000?" -> ... now you're already between 6000 and 7500.