The victim chooses who he or she wants on the team - the two worst bullies and four respected classmates.
The school counsellor convenes the team and reads out a statement by the victim on what they have suffered and how it has affected them, without naming the bullies.
"When I read out the stories, sometimes kids will cry," Mr Williams says. "This has a powerful impact because for the first time someone is listening to [the victim]."
He then invites the team to be part of a secret operation to support the victim. If they agree, he asks them to develop a five-point support plan, allocating tasks to team members.
"They come up with simple things," he says. "Sit next to her - who could do that? Say hi. Hang out with her. Ask her if she's okay. Stick up for her. Stop mocking her."
Surprisingly, even the worst bullies jump at the chance to change.
"They don't like it, they want to change," Mr Williams says. "Kids that bully - other kids steer away from them. So the undercover team not only gives the kid who is bullied a chance to change their reputation, but gives everyone a chance to change their self-concept."
Auckland University psychologist Dr Ian Lambie is evaluating the system, but Mr Williams is confident.
"It always works," he says. "I haven't had one single team where it hasn't worked."
Its success illustrates a general conclusion: that children who bully have felt "put down" by adults in their lives, so schools can help by making all children feel valued and good about themselves, at least while they are at school.
That is partly a matter of putting more emphasis on praise than on punishment, and using tests to lift all students over the line rather than labelling some as failures.
But anti-bullying programmes can also help. The police programme Kia Kaha, the Peace Foundation's Cool Schools and Roots of Empathy, the Virtues Project, Character Education, and mountaineer Graeme Dingle's programmes Kiwi Can, Stars and Project K all aim to build up students' self-esteem and respect for others.
Thirty-pupil Waiharara School, north of Kaitaia, surveys its children every term about bullying, helps the bullies to apologise to their victims and modify their behaviour, and keeps them in a "bully zone" in the playground until they stop bullying.
"We are a 'Virtues' school so the virtues of friendship or forgiveness may be studied for the next fortnight and sent home with the school newsletter," says principal Kathy Cotching.
The surveys show that the number of children bullying others has dropped over the past nine years from two-thirds to less than 5 per cent.
Some critics such as Steve Taylor, an independent counsellor in Auckland, cite American studies showing that no anti-bullying programmes actually have any statistically significant effects on the rate of bullying.
He advocates building the victims' abilities to stand up to bullies emotionally and verbally and to defend themselves physically. He says bullies should face "meaningful consequences".
"There is a school of thought that we mustn't stand down children who are bullying because we might affect their education. Well I would call that the law of natural consequences."
But a comprehensive review by the Children's Commissioner in 2009 found that "whole-of-school" programmes can reduce bullying by changing a school's culture to "discourage bullying and encourage students to care about each other".
Safe and Peaceful Schools: Addressing Conflict and Eliminating Violence, by John Winslade and Michael Williams, Corwin, 2012, US$36.95 ($46.50).
