First, an important point. Polo isn't Chess. Chess is black and white, there is no score. Podium uses the fact that Polo has much more information built into the result than just W/L. When a game ends 5-0, that says more about the skill difference between those two teams than if the game ends 4-3. Systems using Buchholz flatten each result into 1 or 0, and ignore the meaningful data inherent in the game itself. I think this is a major flaw
Podium's ranking criteria goes like this: [points, differential, total_goals] (points: W3/L0/T1)). This is easy to understand, and it is packed with information. Anyone can look at the results table and see why their team is listed where it is, and what they have to do to move up. With Buchholz, you never know what is going to change after the round. Some team you played first round might win, giving you more tie-break points, or some team your opponent played in some round might win and you get bumped down and have no idea why. There is no easy way to figure out why your tie-break score is what it is.
Another advantage is that Podium can stage games before a round is over - something that Buchholz simply can't do. That means you wind up with courts empty for long periods of time at the end of each round - waiting for the last game that was delayed for whatever reason, wasting precious daylight. Podium only needs the results from a few games to make a proper match, because the tie-breaks aren't dependent on knowing the results of every other team in the session. As long as teams have played the same number of games, they can be paired.
To sum it up: sure, Buchholz sounds smart, but it 1) flattens results to W/L. 2) is not human readable. and 3) wastes court time every round.
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Podium also has a more sophisticated staging algorithm than just straight swiss (1v2, 3v4, etc). The current system was developed over the course of a year at a handful of tourneys, with valuable input from the organizers and especially from kevin walsh and jonny hunter.
-The first round is what I call a "suicide" round. For example, in a session with 10 teams: 1v10, 2v9, 3v8, 4v7, 5v6. With a decent pre-rank, that means we get the most meaningful information. The top few teams are gonna beat almost anyone anyway, and the bottom are going to loose most games. I pair top v bottom to get those teams started in the right trajectory. The middle teams will have meaningful matchups, and that helps start sorting out the middle.
-The second round I call a "group suicide" round. I pair top v bottom, but inside groups of matching records. For example, say 4 teams are 1-0... inside that, the top team plays the bottom team, the two middle teams play... Same goal: create the most meaningful matchups in the middle, thats where you don't have as much information yet.
Also, this method prevents the top teams from playing too early. You want to hold that off for later in the rounds, when the result of the game will give you more refined information about skill difference.
Starting in Round 3 and on, pairings are done in ranking order. The only reason a potential matchup is skipped is if teams have played already.
Which brings up a final point. By round 5 in an appropriately sized session, a team will have played most of the similarly ranked teams, and the teams around them will have played each other as well. This becomes a more significant pairing criteria than the rankings themselves. Finding a set of pairings that avoids any duplicates means you'll have matchups between teams that aren't right next to each other in the rankings list every single time.
This basically means that the staging algorithm is only so important. There are many other factors that are of equal or greater concern: ease of use, understandablility, flexibility, etc. Podium provides a simple info-packed leader board display. Podium lets you enter scores from the court on a phone. Podium manages multiple courts for you. Podium lets you manually stage and re-arrange games. Podium handles the DE bracket, and seeds it with a single click. I could go on, but thats not the goal of the thread.
OK, that was fun:)
Challonge apparently uses Median-Buchholz.
By the way, i don't think that the swiss rounds algorithm is the only thing that makes a tourney management tool the right choice, it's just one factor.