Basically Paul is saying, if you want to compete with one another, let the competition be to "outdo" each other with honor. He later says, "Do what is honorable in everyone's eyes" and live at peace with each other. Conflict is a terrible reality of life and we should seek to minimize the relational conflict around us. Yet, conflict is inevitable and cannot be avoided.
The question is: Who is responsible for the conflict in your life? Have you dealt with your sin and sought forgiveness from God and others? Have you been faithful to confess failure to others and purposed to heal the brokenness? Are you willing to have difficult conversations or do you avoid conflict and make the brokenness worse?
Peace does not come with avoidance. It comes with right choices. Sometimes the choice is to take responsibility for your own actions. Other times it is to help others take responsibility for their actions. When Paul encourages us to live at peace with everyone, he is calling for action in every broken relationship in our lives.
What a challenge!