The verifier would normally have a financial stake in the correctness of the computation. For example, 2 parties could use this model to play a game on-chain for some amount of bitcoin. Then for every move, the party making the move commits to some move and new game state (acting as Prover) while the other party checks that the new state is correct (acting as Verifier). So the roles of Prover and Verifier are swapped at every move.