A knockout tournament begins with $2^n$ players and has $n$ rounds. There are no play-offs for the positions $1,2, \cdots, 2^n-1$. What's the set of all possible outcomes? My answer: $\Omega = \left\{1,2^2,\cdots, 2^n\right\}.$
However, my book says this is only if we're interested in the ultimate winner only. Why is this the case?