The tournament without a referee

1. Proof by induction on n: If X beats P_1, place it at the beginning.
2. Proof by induction in n: Inductive step: suppose a row valid for n-1 players: P_1to P_2to·sto P_ n-1.
3. Final section: Proof by induction in n: Inductive step: suppose a row valid for n-1 players: P_1to P_2to·sto P_ n-1. Then, in another case, there exists some index i such that: P_ito X and Xto P_ i+1 , and it is inserted between P_i and P_ i+1.

Back to problem
Answer: Yes, it always exists.
Proof by induction in $n$:

• Base $n=1$: trivial.
• Inductive step: assume a row valid for $n-1$ players:

$$ P_1\to P_2\to\cdots\to P_{n-1}. $$

Add a new player $X$.

1. If $X$ beats $P_1$, place him at the start.
2. If $P_{n-1}$ beats $X$, place it last.
3. Otherwise, there exists some index $i$ such that:

$$ P_i\to X\quad\text{y}\quad X\to P_{i+1}, $$

and is inserted between $P_i$ and $P_{i+1}$.
In all three cases the property “everyone wins the one on the right” is preserved.
By induction, it exists for all $n$.