Adversarial processes seem to be at the core of the Western idea of "fairness", which encourages their use in potentially violent contexts as a basic means of conflict reduction. They are especially common in politics, where they're encouraged by the current use of democratic voting systems. Legal courts often employ adversarial processes to evaluate a defendant's alleged wrong-doing and decide appropriate restitution. Legal ethical codes and standards of evidence are needed in order to ensure that the overall process approximates truth-seeking.
The operation of a prediction market may also be seen as a kind of adversarial process, in which bid and ask prices are constantly compared.
Some view Wikipedia as an adversarial process where alternating edits to an article seek to balance what is seen by each author as an extreme point of view. Wikipedia's policy of a "neutral point of view" implicitly assumes that such adversarial edits will eventually converge to an equilibrium.