We have to adopt a wider perspective, and always find common things between the people of north, east, south, and west. Conflict comes from the basis of differences.
-- Tenzin Gyatso, the 14th Dalai Lama
Because conflict is often rooted in our differences, finding common ground is often the way to resolution. If common ground is too hard to find, a trusted third-party can serve the same purpose as a common point of contact, a mediator.
Research into mediation has shown that it is more successful when it comes after a test of strength between those in conflict. Sporting events are relatively harmless tests of strength which can serve several related purposes, including building shared experience and stress relief. The competition itself can play the role of a common struggle for those in conflict, providing opportunities for direct competitors to work together -- like two opposing players who fall to the ground, then help each other up. A player may have opportunities to build trust with the competition by admitting to having broken a rule, or by offering equipment, a drink of water, or other help.
The shared struggle for athletic excellence is a powerful common ground. Sports have had historic success in mediating conflict even between nations who had no formal relationship. "Ping Pong Diplomacy" between the United States and the People's Republic of China in the 1970s led directly to more serious diplomacy.
Even without an active conflict, finding common ground is a compassionate action which benefits everyone involved. The Olympic Games and soccer's World Cup may be the two most popular events on Earth. They and millions of other regularly scheduled sporting events keep friendly relations going.