1. A lot of familiar icebreakers are non-competitive games. This makes sense. It is not a brilliant idea to commence battle with people you've just met and try to make them grovel. Think about your store of introductory, name-learning games, and adapt or extend them. An important plus about these games is that they are often short and fast-moving.
2. Plan games as you would any other session. Work out a running order that makes sense, giving variety of pace while avoiding jolting contrasts. Try to be explicit with yourself and colleagues about why you are doing things. Are you encouraging trust, getting exercise, building skills, or challenging expectations?
3. One of the best antidotes to competition is co-operation. The Scout Association has a useful downloadable factsheet on Co-operative Games - see www.scoutbase.org.uk. It suggests, for example, co-operative tennis - where you work together to see how long you can keep the rally going. Admittedly it won't help build anyone's tactical skills for the real game. But as the Scouts point out, it is good for mixed-ability pairings. The better player gains experience at placing shots for the less able, who then gains confidence in returning shots.
4. You can still play traditional games where players are "out", as in musical chairs. Just find ways to adapt them. You could form two separate but linked groups of players. Then when the music stops and someone is "out", they move to the other group. This keeps everyone involved as long as you play it, with none of the boredom and disruption from the losers.
5. Think about safety and play all the usual attention to risk assessing the room and the equipment. Just because something is non-competitive does not mean you can't get hurt. Some of the games can be just as fast and furious. And it would be a lot more humiliating to injure yourself playing the choo-choo train game than through scoring the all-important equaliser.
6. Avoid making speeches about the philosophy of non-competitive games. In other bits of your youth work you may want to provoke conversations about working co-operatively. But games are there to be played. Keep rule explanation to a minimum.
A good source of games of all kinds is The New Youth Games Book, by Alan Dearling and Howard Armstrong. See also www.woodcraft.org.uk/resources.