The Welfare Theorems

Person $n^*$ is a dictator over any $A,C$ pair not involving $B$. To see this, choose, say, $A$ from the pair, and construct $\succeq'''$ from $\succ''$ by letting $n^*$ move $A$ above $B$, so that $A\succ'''_{n^*}B\succ'''_{n^*}C$, and letting all other voters arbitrarily rearranging their rankings, only leaving $B$ in its extreme position. Independence implies $A\succ B$, because all $A,B$ rankings are as they were in $\succ'$, where $A\succ B$. Independence also implies $B\succ C$, because all $B,C$ rankings are as they were in $\succ''$. Transitivity thus implies $A\succ C$. By independence, society must rank $A\succ C$ whenever $A\succ_{n^*}C$.

Person $n^*$ is also a dictator over all $A,B$ pairs. To see this, choose a third alternative $C$, and repeat the preceding arguments to conclude that there is an $n(C)$ who is a dictator over all $A,B$ rankings. But $n^*$ can affect the $A,B$ rankings in some profiles, in particular $\succ'$ and $\succ''$, so $n(C)=n^*$.∎