^^^ Okay, debate the order of my top 3...Carter might say Weber is better, but the bottom line is that Carter won more (4 All-Stars, 5 World Invitationals, 18 top five finishes in one year, etc.). Putting Weber at a close #2 is easy, though. He competed in, what, 3, 4, 5 decades at a high level? That's unbelievable.
I would stand by Anthony at #3, behind both Carter and Weber. You've got to remember that title wise, it's not like Carter was out on Tour for very long. He was from the decade or two just before Anthony.
That gets it down to the last 2 spots, and I don't think Mark Roth (as phenomenal as he was, as revolutionary as he was) stands a chance.
Pick one of the few old-timers for spot #4 -- those were the Carters and Anthonys of a bygone era. Jimmy Smith was the undisputed champ for more than a decade, and even after that, he was still the biggest name somebody could beat. Count Gengler never really got involved in the match game tournaments, but he toured with Smith, and could beat anybody, especially for the first several years after he arrived from Luxembourg. He suffered an injury, though, in 1917 or 1918, and, though he continued to bowl at a high level, I think this affected him (however, there is no mention of this in the history books; Gengler was like a ghost). It was well known that he palmed the ball when he first arrived and bowled; but in later matches, he used a two-finger ball, in a sense confirming my suspicion that his injury (an elbow related problem) may have caused him to modify his game slightly. Had he not been injured, and not been a mysterious Bavarian hustler, Gengler would be up for discussion -- the man was once seen hitting a nickel put on the five-pin spot over and over again. Marino (who pretty much took over for a few years after Smith and Jimmy Blouin) and Varipapa (winner of back-to-back All-Stars at ages 55 and 56 respectively), too, were great bowlers.
Last, Walter has to be included, no matter what (including the resin thing). It might prevent him from being considered the best ever (in my book), but he still is one of the best -- no one can deny that he has triumphed over the likes of Norm, Pete, Bohn, and others 40+ times.