The RPI is in many ways a relic of a different time, a missing link between hunches and KenPom. There is a fracture between what the RPI is capable of doing and what people try to use it for, and that often leads to frustration with and/or open denigration of the RPI as a tool. Nothing you read below is going to change your mind about the RPI, but hopefully it will give you a little better understanding of how the tool came to be.
To understand the RPI, you have to understand the era it came out of. The brain child of a staff led by a man called Jim Van Valkenburg, the formula came to be for the 1980-81 season. It was a combination of the three factors we're familiar with today - winning percentage, opponents' winning percentage, and opponents' opponents' winning percentage - but it also including a team's road winning percentage in the calculation. As the NCAA tournament was expanding and allowing for more at-large bids, the selection committee needed a tool for comparing teams whose schedules often hardly overlapped. Now they had one.
The RPI kind of serves as a gateway drug to analytic stats, the way OPS functions in baseball. It's simple, it's easy to understand, and it can be calculated by any doofus with a smart phone and some time on his hands. Do the calculations, don't forget to apply greater weight to the road What's not to love?
Plenty, actually. Where most people go awry with the RPI is trying to use it to tell how good a team is. There is probably some coincidental correlation between a team's RPI and how good they are, but that is not the purpose of the tool. It doesn't really stand up favorably to KenPom or the Massey ratings or the BPI in terms of judging how likely a team is to win its next game. Those tools all incorporate margin of victory and various other factors in an attempt to not just record the result of the game but to also try to integrate that information into the picture of how good the teams involved were.
Where the RPI does shine is as a record of how good a team has been at winning basketball games. Just for a recent example that I know off the top of my head, Seton Hall's one-point win at St. John's hurt the Pirates' KenPom ranking. Pomeroy's numbers saw a team that should have run the Johnnies off the court instead struggle to escape with a victory and dinged it accordingly. The RPI saw a team whose best player shot 1-12 from the floor and turned the ball over 4 times find a way to win on the road and gave it credit for that. Actually, it doesn't even go that deep; the RPI just saw a win.
If we concede that the point of playing the game is to win - and I think Herm Edwards will back me up on this one - then teams should be rewarded for finding ways to do so. The NCAA tournament at its heart is a challenge to win six times without losing; how you get it done is immaterial. Entry to the tournament should be based at least in part on how well teams have done at winning up until that point. The RPI is far from perfect and isn't even really that good at a lot of things, but it has one thing in common with the greatest event in sports: the final score is the only thing that matters.