In the match play set of rules, Four Ball is a form of competitive golf play in which 2 teams, with 2 players each, compete against each other. Many professional and club level golf tournaments around the world employ the four ball method of play in their events. The unique format calls for changes in the strategic approach compared to more traditional style formats and stroke play. In four ball, two partners on a team play their own balls on a hole and the golfer with the fewest strokes posts his score on the official scorecard. In this situation it is not against the rules for one player to sit out during a hole as long as one of them has a score to post. Basically, the better ball from both teams is the one that counts.
In Four Ball, each side has the option to play their balls in whichever order they feel is best. If one of the golfers from a side hits the wrong ball during a match, he is disqualified from that hole. His team member, however, can continue playing with no penalty and post a score for the team. The wrong ball that was hit must be replaced by the owner of the ball at the spot where it was first struck. Any penalty on a player that is related to local course rules or USGA rules 4 and 6-4 automatically penalizes the team as well.
In Four Ball, a side can be disqualified under several different conditions and scenarios: If any team member is penalized for rules regarding ball, handicap, caddie, slow play, teeing, unusual equipment, clubs, agreement to waive rules, or committee decision, the team will be disqualified. A team may also be disqualified if both team members violate rules regarding the time of starting groups, and the discontinuance of play. Any other rule violations not mentioned will only result in the disqualification for one hole only. If a team member breaches one of these other rules and it helps his partner or hurts the other team, both team members will receive a penalty. If the penalty does not affect anyone else, then only that player is penalized.
In a Foursome, there are four golfers split into two teams. Unlike Four Ball, there is only one ball played by each team instead of two. On each team the players take turns teeing off and taking shots throughout the duration of a round. Even if there is a penalty stroke, the play resumes as normal and the golfers continue to alternate shots. If the golfers play out of turn at all during a hole, they automatically lose that hole. Foursomes are commonly played in professional golf events such as the Ryder Cup and the Presidents Cup. There is also a variation to Foursomes called Greensomes in which both team members tee off. Then they choose which ball to use to finish off the hole.
If you want to learn more about the rules of golf then you should read the official USGA 2011 rulebook