I would like to know if run in this context means organize (like organize an event).
"I watched four volleyball games running simultaneously" Thanks
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To run an event, meaning to organize it and oversee it while it is underway, is a transitive verb that takes a direct object:
Are you running v.t. the presentation d.o. tomorrow?
The volleyball games themselves are not organizing anything. Someone is running them.
The volleyball games are being run simultaneously (by someone, e.g. by the school athletic department, or by the league).
In your example, the verb is intransitive, and it does not have the meaning of the transitive verb. Intransitive run means "to be underway, to be in progress, to be operating (properly)".