On Amae Koromo, the stats aggregator for Mahjong Soul, it lists the average starting shanten in Jade room as being around 3.482. People thought this sounded low, so let's see the averages in Houou.
Let's start by simply looking at the average shanten of all the starting hands in Houou Hanchan games. Tenhou's wall code is public and we know it's fair.
This is a high sample size, with 1.5 million games. The numbers won't deviate much. For example, here's the same chart for East-only games, which has a sample size of half a million.
It's only different by roughly 0.0003. These numbers are a full 0.1 shanten higher than Amae Koromo's numbers.
However, this was calculated with thirteen tile hands. Perhaps they're including the dealer's first draw into the equation. When I last looked at Mahjong Soul replays, they included it. Though I don't think that's the case any longer, let's see how the data changes when we include the dealer's first draw.
This is closer, but it's still off by 0.01. Another possibility is that it's calculating the shanten after the first discard. This sounds very strange, but perhaps they wanted all hands to be thirteen tiles, but didn't want to just remove a random tile from the dealer's hands. That would slightly increase the shanten, as people will sometimes go for strange hands or fold from turn one.
There are other explanations for the numbers being different than Mahjong Soul being rigged in some way. For example, the algorithm Amae Koromo uses could be incorrect. Shanten calculation is fairly complicated, especially if you want it to be performant. The algorithm I use has been verified to be correct for every hand.
Amae Koromo also lists the average starting shanten for three-player games. It's listed as 3.278. In Houou, this number is 3.394, or 3.249 while including the dealer's first draw. The difference is even bigger than the four player numbers.
Without knowing how Amae Koromo calculates the shanten it's hard to put blame anywhere for the differences.
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