Splitting Secret Key
The mentioned approach of splitting the keys in pair as $(k1,k2)$ for first person, $(k1,k3)$ for second and $(k2,k3)$ for third person for the given situation is correct! And the PRF can be computed as follows:
First person computes: $F1((k1,k2),x)=F(k1,x)⊕F(k2,x)$
Send this F1 to the second person and compute F1⊕F2 where F2 is $F2(k3,x)=F(k3,x)$
$F1⊕F2 = F(k1,x)⊕F(k2,x)⊕F(k3,x) = F'((k1,k2,k3),x)$