It's a very common situation at work - if two team members have serious difference of opinion, the manager, instead of looking into the matter or taking sides, asks both to duel it out and come out with an agreed solution. Manager doesn't care about who is right and not willing to take the risk of judging. Easier for manager to is just let them fight it out.
In reality, the manager already has a perception for both and knows the outcome. The team is aware of that perception, and would not dare to correct it or give a surprise because that is not going to benefit them. Bosses don't like their hard-earned perceptions being corrected. No surprises please. The weaker would realize this perceptional power and bow out even if their view point has merit. Manager would thank both for sorting it out.
You can't engage in gladiator fights against bad and powerful people. You are more likely to hurt yourself than influencing the decisions. The audience may clap and encourage the fight, but they are not completely neutral.
Each person gains their power by perception (from their boss or co-workers). They can use this power to shut you off. So, from an individual perspective, you need to choose what's best for you based on how powerful you are. Simply getting into politics without such assessment is foolish.
The whole point of collaboration is to build the perceptional power. Also, being in higher position keeps you in loop for all communication giving you more access to the on-goings and opportunity to get visibility. This increases the perceptional power.
People hoard information. It's a treasure and sometimes their whole career depends on keeping it as secret. Also people hoard work items. Just having a long queue of work also gives power. Atleast you have a lot to talk about and possibly grow your team.