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.
Great book, would recommend. Not surr how often it has been useful for me still it is a very pleasant read. The main idea is to not give in so easily and to find mutual wins, e.g. he gives the example that he could help the other party get an article in a magazine - barely related to the deal, cost nothing for him, but was worth a lot to the other party.
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.