This paper formulates a new control problem of achieving the global objective of a discrete event system with networked control agents that pursue only their own objectives and compete with each other. In this framework, each agent makes a decision to meet its own private objective with a given mask during state observation of the system and such a decision can be changed by the influence of other agents connected through a directed network. Agents with an initial decision of enabling or disabling of an event do not change their decision, but those agents reserving their decision in an undetermined way can change their decision to either enabling or disabling depending on the influence from other agents. We derive a condition for convergence of the decision of all agents. Based on the individual decision of each agent, the final decision for enabling or disabling of an event is made by majority rule. We present the network-controllability of a global objective with respect to a given mask set as a main condition with which the networked agents can meet the global objective while they pursue only their own private objectives.