Accelerated pairings are a way to try to cut the number of potential perfect scores in half. This is generally used in tournaments with a wide range of ratings, large number of players compared to the number of rounds, and most often due to the existence of indivisible first place prizes (like trophies). The easiest variation to understand is where the top half of the field (by rating) is paired as if the players had an extra point in the first two rounds. Round three and later pairings are normal.
As an example, take a three-round 16 player tournament with ratings distributed every 50 points from 1000-1750. If the ratings are not accelerated then you have the following pairings (assuming no upsets and ignoring color conflicts). The final result is two perfect 3-0 scores.
Round 1 (all 0-0)
1750 beats 1350
1700 beats 1300
1650 beats 1250
1600 beats 1200
1550 beats 1150
1500 beats 1100
1450 beats 1050
1400 beats 1000
Round 2 (first four games with 1-0 players, second 4 with 0-1)
1750 beats 1550
1700 beats 1500
1650 beats 1450
1600 beats 1400
1350 beats 1150
1300 beats 1100
1250 beats 1050
1200 beats 1000
Round 3 (2 games with 2-0 players, 4 games with 1-1, 2 games with 0-2)
1750 beats 1650 (winner is one of 2 perfect scores)
1700 beats 1600 (winner is one of 2 perfect scores)
1550 beats 1350
1500 beats 1300
1450 beats 1250
1400 beats 1200
1150 beats 1050
1100 beats 1000
With acceleration, if there are no upsets then there is only one perfect 3-0 score at the end.
Round 1 (the top half - for this tournament the 1400+ players - are paired as if they had an extra point in the first two rounds)
1750 beats 1550
1700 beats 1500
1650 beats 1450
1600 beats 1400
1350 beats 1150
1300 beats 1100
1250 beats 1050
1200 beats 1000
Round 2 (1400+ players are paired as if they had an extra point, 2 games between 1-0 players, four games between a higher-rated 0-1 and a lower rated 1-0, 2 games between 0-1 players)
1750 beats 1650
1700 beats 1600
1550 beats 1350
1500 beats 1300
1450 beats 1250
1400 beats 1200
1150 beats 1050
1100 beats 1000
Round 3 (no more extra points used for pairings, 1 game between 2-0 players, 6 games between 1-1 players, 1 game between 0-2 players)
1750 beats 1700 (winner is the only perfect score)
1650 beats 1350
1600 beats 1300
1550 beats 1250
1500 beats 1200
1450 beats 1150
1400 beats 1100
1050 beats 1000
Acceleration has been used in the open sections of National Scholastics for some years.
Questionable tournaments to accelerate are:
Those where the number of perfect scores can be limited to one without acceleration;
Those where the ratings are all fairly close (such as a class tournament) since there is a greater risk of having a number of round 2 “upsets” with lower-rated 1-0 players beating higher-rated 0-1 players (this may give you up to 50% more 2-0 scores than if you did not accelerate)
Those where the ratings are uncertain (such as many unrateds, or many provisional ratings)
Tournaments where you might expect a significant percentage of draws between the top-rated players, as that would also reduce the number of perfect scores