In the event a tournament does not go the specified number of completed holes (usually 72), and the tournament is shortened by PGA officials, the leader at the end of number of holes determined to be official by PGA officials shall be the winner:

Example 1
Tournament is scheduled for 72 holes, but only 54 holes can be played due to weather concerns. The leader at the end of 54 holes would be declared the winner of the tournament.
In the event that two (or more) golfers are tied for the lead at the end of regulation play, sudden death playoff occurs to determine the winner of the tournament. In some cases. PGA officials may also have the individuals play a specified number of holes to determine the tournament winner.

Example 2
Mike and John are tied at the end of regulation play, and sudden death occurs. Mike scores a 4 on the first sudden hole, and John scores a 5. Mike is the winner of the tournament, and John is awarded a 2nd place finish.
In the event PGA officials order a specified number of holes to be played to determine a winner, the golfer with the lowest score after the specified number of holes have been completed shall be declared the winner, and 2nd, 3rd, etc would be determined by the scores of the remaining golfers.
In head to head matchups, both golfers MUST tee off in order to have action. The most completed holes and the lowest score for either golfer is the winner of the matchup.

An example is listed below:
John -120.
Mike +100.
If John and Mike both tee off for the tournament in question, the above matchup is "in action." If both golfers fail to make the cut to advance to the final rounds of play, the golfer with the lowest score wins.

Example 3
John completes 36 holes and has a score of +8.
Mike completes 36 holes and has a score of +9.
In this case, John wins the matchup since he has a lower score than Mike. If John and Mike complete regulation play, and each have a score of -3, the matchup would be a tie, and monies refunded, EXCEPT in the case where John and Mike were tied for 1st place at the end of regulation play (then sudden death would be used to determine the winner).

Additional information on matchups
If a wager is taken on a matchup for the overall tournament, both golfers MUST tee off for action. In the event that one player or the other is disqualified or withdraws, the other golfer will be the winner of the matchup.
In the event a matchup is taken on a round by round basis (1st Round, 2nd Round, 3rd Round, etc) or (1st and 2nd rounds, or 3rd and 4th rounds), both golfers MUST tee off for action. In the event a wager is taken on the 1st and 2nd (or the 3rd and 4th rounds) for the same two players, and the player tees off in the 3rd round, and then withdraws, the player remaining would be the winner of that matchup in the 3rd round, and there would be NO action on the 4th round wager (since one of the two players withdrew).