  We formulate a theory of agent-based models in which agents compete to be in
a winning group. The agents may be part of a network or not, and the winning
group may be a minority group or not. The novel feature of the present
formalism is its focus on the dynamical pattern of strategy rankings, and its
careful treatment of the strategy ties which arise during the system's temporal
evolution. We apply it to the Minority Game (MG) with connected populations.
Expressions for the mean success rate among the agents and for the mean success
rate for agents with $k$ neighbors are derived. We also use the theory to
estimate the value of connectivity $p$ above which the Binary-Agent-Resource
system with high resource level goes into the high-connectivity state.
