Widespread adoption of autonomous vehicles will not become a reality until solutions are developed that enable these intelligent agents to co-exist with humans. This includes safely and efficiently interacting with human-driven vehicles, especially in both conflictive and competitive scenarios. We build up on the prior work on socially-aware navigation and borrow the concept of social value orientation from psychology —that formalizes how much importance a person allocates to the welfare of others— in order to induce altruistic behavior in autonomous driving. In contrast with existing works that explicitly model the behavior of human drivers and rely on their expected response to create opportunities for cooperation, our Sympathetic Cooperative Driving (SymCoDrive) paradigm trains altruistic agents that realize safe and smooth traffic flow in competitive driving scenarios only from experiential learning and without any explicit coordination. We demonstrate a significant improvement in both safety and traffic-level metrics as a result of this altruistic behavior and importantly conclude that the level of altruism in agents requires proper tuning as agents that are too altruistic also lead to sub-optimal traffic flow. The code and supplementary material are available at: https://symcodrive.toghi.net/