Territorial battles among ants exhibit temporal and spatial patterns that self-organize, arising spontaneously from distributed decisions by large numbers of individuals. We describe agent-based models of inter-group fights in ants and show that two behavioral mechanisms that are rarely quantified have large effects on the dynamics of intraspecific battles; specifically, the pattern of search by unengaged ants, and assessment of relative numbers. In the absence of assessment, recruitment by both colonies rises to steady averages. Alternatively, if ants tend to lay trails only when they detect that their nestmates outnumber opponents, fights can be rapidly resolved as one colony ceases recruiting. If ants tend to lay trails when their nestmates are locally outnumbered, the position of the battle may oscillate. We show that the collective ability of fighting ants to accurately compare group sizes may be high even if each ant has limited perception and memory. However, amplification of small initial numerical advantages can lead to priority effects favoring the first colony to recruit even if it is the smaller colony.