We consider a non-cooperative game between N relays in Delay Tolerant Networks with one fixed source and one fixed destination. The source has no contact with the destination, so it has to rely on the relays when it has a message to send. We assume that the source has a sequence of messages and it proposes them to relays one by one with a fixed reward for the first transmission for each message. We analyse a symmetric mixed strategy for this game. A mixed strategy means a relay decides to accept relaying the k th message with probability $q_k$ when it meets the source. We establish the conditions under which $q_k = 1$; $q_k = 0$ or $q_k ∈ (0, 1)$, and prove the existence and the uniqueness of the symmetric Nash equilibrium. We also give the formula to compute this mixed strategy as well as the probability of success and the delay of a given message. When $k$ is large, we give the limiting value of the mixed strategy q and the probability of success for the messages.