For a boolean function f:{0,1}n⟶{0,1}, Havg(f) is a function from N⟶N, termed as the average case hardness, if ∀ circuit Cn of size Havg(f)(n), Prx∈Un[Cn(x)=f(x)]<1/2+ϵ, ϵ>0.
Similarly, for a boolean function f:{0,1}n⟶{0,1}, Hwrs(f) is a function from N⟶N, termed as the worst case hardness, if ∀ circuit Cn of size Hwrs(f)(n), Prx∈Un[Cn(x)=f(x)]<1.
This I know from the definition of average and worst case hardness. My question is what is the motivation behind these definitions. Can anyone help?