Assign every new run to the person who created it, unless they pick a different assignee.
By default a new run is unassigned until someone picks it up. When this setting is on, a run is assigned to whoever created it unless an assignee is chosen at creation.
While the setting is on, the run creation form pre-fills Assigned to with the current user. The field stays optional, so you can pick someone else, and clearing it still lands the run on the creator.