Summary: We introduce the concept of \(varepsilon\)-consistent equilibrium where each player plays a \(varepsilon\)-best response after every history reached with positive probability. In particular, an \(varepsilon\)-consistent equilibrium induces an \(varepsilon\)-equilibrium in any subgame reached along the play path. The existence of \(varepsilon\)-consistent equilibrium is examined in various repeated games. The main result is the existence in stochastic games with absorbing states.