Cobordism-valued intersection theory on $\overline{\mathcal{M}}_{0,n}$

We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued psi-class intersections on $\overline{\mathcal{M}}_{0,n}$, and in particular the cobordism classes $[\overline{\mathcal{M}}_{0,n}]$, and for their images in $K$-theory. Explicit formulas are given up to $n = 8$.