In this paper we analyze a simple adaptive model of competition called the Minority Game, which is used in analyzing competitive phenomena such as the operation of the market economy. The Minority Game is played by many simple autonomous agents, which develop collective self-organization as a result of simple behavioral rules. Many algorithms that produce the desired behavior in the game have been proposed. In all work to date, however, the focus has been on the macroscopic behavior of the agents as a whole. We focus on the behavior of individual agents, paying particular attention to the original form of the Minority Game. We suggest that the core elements responsible for the development of self-organization are (i) rules that place a good constraint on the behaviors of individual agents and (ii) the existence of rules that lead to effective indirect coordination. We also show that when efficient organization is formed, a power-law can be seen among behavior of individual agents.