Those （who think that a man betrays his character nowhere more clearly than when he is playing a game ）might on this draw subtle inferences.

这句话的主干是，Those might on this draw subtle inferences 也就是说，Those might draw subtle inferences on this.  (那些人对此作出微妙的推论）

（who think that a man betrays his character nowhere more clearly than when he is playing a game） 这个成分是修饰Those的定语从句。  Who代替先行词those. 这句话可以理解为，those think that a man betrays his character nowhere more clearly than he is playing a game.  这个修饰those的定语从句，里面又是一个宾语从句。 

重点就是这个宾语从句的理解，我们还原为：a man betrays his character more clearly nowhere than (the man betrays his character clearly) when he is playing a game.  这里面比较的是nowhere 和when he is playing a game两个状语。 意思就是，在比赛中他表现出如此和他性格不同的一面，这是在其他任何地方他所不曾表现出的。