( c ) base case trivial. For induction, suppose it holds for N players and Alice is The One and an arrow means "win"

A player Bob enters. Break down all the cases.

Case 1, $Alice \rightarrow Bob$. Then Alice is still The One Case 2, $Bob \rightarrow Alice$ and $\exists y, y \rightarrow Bob$, then Alice is still The One Case 3, $Bob \rightarrow Alice$ and $\forall y, Bob \rightarrow y$. Because $\forall x \exists y, y \rightarrow x$, it must be Bob wins some y who wins x. Then Bob is The One

(b) Not sure what is being asked (just show an example or describe all the cases). A only one The One set up is like this. For N players are $Alice_1, Alice_2, \cdots , Alice_N$ and $\forall i \forall j, i< j$  set up $Alice_i \rightarrow Alice_j$ .Then $Alice_1$ is the only The One. A new N+1 player Bob enters. If the new arrows introduced by Bob are $Alice_1 \rightarrow Bob$ and $Alice_2 \rightarrow Bob$ and others arrows related to Bob are arbitrary, then still only one The One which is $Alice_1$