Optimal taxing in case of negative externalities
I would have thought the social planner might want the value of $x_i$ to be such that $f'(x_i)=k(n-1)$ at which point the marginal gain to each individual offsets the marginal total cost to the others 

while individual driver $i$ might want the value of $x_i$ to be such that $f'(x_i)=t$ at which point the marginal gain offsets the marginal tax

To me that suggests the tax rate should be $t=k(n-1)$