Theorem (Erdos 1963) If ()(1-2-k)"<1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Event As:no player in \S beat all players in S. Pr V As <1 se(g)」 s∈()If ￾n k ⇥ ￾1 ￾ 2￾k⇥n￾k < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Event AS : no player in V \S beat all players in S. Pr < 1 ￾ ⇧ ⇤ ⌥ S￾( [n] k ) AS ⇥ ⌃ ⌅ Pr[T is k-paradoxical] = 1 ￾ Pr ￾ ⇧ ⇤ ⌥ S￾( [n] k ) AS ⇥ ⌃ ⌅ > 0