We introduce a dynamic game of outbidding where two groups use violence to compete in a tug-of-war fashion for evolving public support. We fit the model to the canonical outbidding rivalry between Hamas and Fatah using newly collected data on Palestinian public support for these groups. Competition has heterogeneous effects, and we demonstrate that intergroup competition can discourage violence. Competition from Hamas leads Fatah to use more terrorism than it would in a world where Hamas abstains from terrorism, but competition from Fatah can lead Hamas to attack less than it otherwise would. Likewise, making Hamas more capable or interested in competing increases overall violence, but making Fatah more capable or interested discourages violence on both sides. These discouragement effects of competition on violence emerge through an asymmetric contest, in which we find that Fatah uses terrorism more effectively to boost its support, although Hamas has lower attack costs. Expanding on these results, we demonstrate that outbidding theory is consistent with a positive, negative, or null relationship between measures of violence and incentives to compete.