Speaking on the House Floor: Gaining Time and Parliamentary Phraseology, Updated December 10, 2018, pp 2-3.
Controlling Time as a Manager
Those designated to control the time often begin discussing the measure by yielding to themselves a set number of minutes or, more often, by stating:
I yield myself such time as I may consume.
The manager is then recognized and holds the floor until all of his or her available time expires or until the manager concludes by saying:
I reserve the balance of my time.
Each floor manager usually, but not necessarily, yields to Members on his or her side of the aisle. Managers do not refer to other Members by name and instead designate them by state. For example, the manager might say:
I yield two minutes to the gentleman from California.
Time is kept by the clerks sitting at the House dais, and managers often ask how much time remains available. In response, the presiding officer will announce how much time the majority and minority floor managers have left. It is not uncommon for the managers to discuss with each other how the remaining time will be distributed. For example, one manager might ask the other how many more Members on his side are waiting to speak.
Debate ends when all time has expired or all time has been yielded back. If the managers determine through discussion that no more Members wish to speak on either side, then they might, in turn, yield back their remaining time by stating:
I yield back the balance of my time.
And more ...
Is there some strategy here?
The strategy is to speak as effectively as can be done; to get the most votes on motions, amendments, bills, etc. for "their" side.
If both managers were to "reserve" then what happens then?
That is not permitted. When one side reserves, the other side speaks. If both side are done, they yield back their balance.
Does one side "win" if they end up with more time?