Before revision:  We give an algorithm that requires on average 4n \log  2 n+O(n) bits to determine the order type of P, and show that any algorithm requires at least 4n \log  2 n -- O(n \log\log n) bits.
After revision:  We give an algorithm that requires on average 4n \log 2 n+O(n) bits to determine the order type of P, and show that any algorithm requires at least 4n \log 2 n - O(n \log\log n) bits.