Theorem (Erdos 1963) If (R)(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. se(l) 0 se()If n k ⇥ 1 2k⇥nk < 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