When the client figure is not defined
The client. The guy who hires and pays. Or the guy you respond to inside your company. Is he always defined? Not always. Imagine a start-up where a bunch of friends have an idea. Somehow they did some marketing, identifying a need from the users. Nevertheless, when it comes to practice, they want to use agile. Who is the Product Owner? There are some options here. Ask yourself, is there anyone in your team especially capable to understand the users? Basically, a marketing guy profile, who studies the needs of the users and try to determine the best functionality for them. Done!
Is that so? Is that possible? Sometimes we don’t count with such profile. Then why you just don’t pick a user you know? Your father, your friend, your girlfriend, etc.
Still not possible?, because of time, because lack of vision, or other reasons? Then you may want to put the source of the idea. The visionary.
Another option may be to avoid the PO figure. Not really recommended, because the no-direction problem could happen among others. Basically if you remove it, you are just spreading the role on everyone in the team. If everyone has the same vision it could be ok, but, when that doesn’t happen you need to bring the Panzers. Select your leader PO1 and a sub-leader PO2. PO1 has the last word in the importance, but everything is decided by everyone, except when there are conflicts. The PO2 is just a helper for PO1. PO2 is also elected.
When you have one guy deciding the importance, you go into a straight line. Where that line brings you is a mystery! But when you have many people, you usually don’t know where you go. Everything will depend on how good is your team.