We study the dynamics and resulting score distribution of three-agent games where after each competition a single agent wins and scores a point. A single competition is described by a triplet of numbers p, t and q denoting the probabilities that the team with the highest, middle or lowest accumulated score wins. The three-agent game can be regarded as a social model where a player can be favored or disfavored for advancement, based on his/her accumulated score. We study the full family of solutions in the regime, where the number of agents and competitions is large, which can be regarded as a hydrodynamic limit. Depending on the parameter values (p, q, t), we find six qualitatively different asymptotic score distributions and we provide a qualitative explanation of these results. We also compare our analytical results against numerical simulations of the microscopic model and find these to be in excellent agreement. It is possible to decide the outcome of a three-agent game through a mini-tournament of two-agent competitions among the participating players and it turns out that the resulting possible score distributions are a subset of those obtained for the general three-agent games. We discuss how one can add a steady and democratic decline rate to the model and present a simple geometric construction that allows one to obtain the score evolution equations for n-agent games.