This paper presents a cooperative relay strategy with a game-theoretic perspective. In multi-hop networks, each node needs to send traffic via relay nodes, which behave independently while staying aware of energy constraints. To encourage a relay to forward the packets, the proposed scheme formulates a Stackelberg game where two nodes sequentially bid their willingness weights to cooperate for their own benefits. Accordingly, all the nodes are encouraged to be cooperative only if a sender is cooperative and alternatively to be non-cooperative only if a sender is non-cooperative. This selective strategy changes the reputations of nodes depending on the amount of their bidding at each game and motivates them to maintain a good reputation so that all their respective packets can be treated well by other relays. This paper analyzes a Nash equilibrium from the proposed scheme and validates a sequential-move game by Stackelberg competition as opposed to a simultaneous-move game by Cournot competition. Simulation results demonstrate that the proposed scheme turns non-cooperative nodes into cooperative nodes and increases the cooperative relaying stimulus all over the nodes. Thus, every node forwards other packets with higher probability, thereby achieving a higher overall payoff.