Unfortunately, stability is not always possible in this scenario. This means there can exist sets of TV shows and ratings where no stable pair of schedules exists for the two networks. To demonstrate this, let's consider a counterexample with N$=3$ shows per network:

Network A Shows:

Show $1$: Rating $3$

Show $2$: Rating $2$

Show $3$: Rating $1$

Network B Shows:

Show A: Rating $2$

Show B: Rating $3$

Show C: Rating $1$

Scenario $1$: Network A Chooses First:

Network A schedules Show $3$ in Slot $1($guaranteed win$).$

Network A schedules Show $1$ in Slot $2($ties with Show A from B$).$

Network A schedules Show $2$ in Slot $3($loses to Show B from B$).$

Outcome: Network A wins $1$ slot, Network B wins $2$ slots. Neither network has a schedule to unilaterally improve their score.

Scenario $2$: Network B Chooses First:

Network B schedules Show B in Slot $1($guaranteed win$).$

Network B schedules Show A in Slot $2($ties with Show $1$ from A$).$

Network B schedules Show C