With self-driven particle models, like the social force model, most of the physics of moving crowds can be modeled. However, it has not been fully unraveled why large crowds evacuating through narrow bottlenecks often act against their self-interest. They form jams in front of the bottleneck, that slow down the evacuation, and fatal pressures build up in the crowd. Here, we take a novel approach, and model the local decision-making in an evacuating crowd as a spatial game. The game is coupled to the social force model, so that different strategies alter the physical parameters. With our integrated treatment of behavioral and physical aspects, we are able to simulate when, why and how typical phenomena of an evacuation through a bottleneck occur. Most importantly, we attain non-monotonous speed and kinetic pressure patterns, in contrast to the monotonous patterns predicted by the pure social force model. This is a result of impatient agents in the back of the simulated crowd pushing and overtaking their way forward. Our findings give insight into the origin of crowd disasters, since the build-up of kinetic pressure has been related to the risk of falling and crowd turbulence.