I was curious if hands with higher starting shanten had higher average values, so I ran some analysis to gather the data. Also included are the win rates, riichi rates, average riichi turns, and call rates for each starting shanten.
As usual, the analysis was run on 1246147 Houou replays, covering a total of a bit under 53 million hands. After the player drew their first tile, their shanten was calculated and the events in the game collected. If they called a tile before drawing their tile, then the tile they called was counted as the tile they drew. If the round ended before they drew a tile, their hand was not counted.
Most hands are 3 or 4 shanten, with 20% being better and 10% being worse. This only considers the standard shanten, so the 7- and 8-shanten hands could be quite close to kokushi (Thirteen Orphans). Some of the -1 shanten (complete) hands might have had someone call before they drew a tile, ruining their chihou (Blessing of Earth) chance.
The average riichi turn column illustrates the difficulty in improving the hand the lower the shanten gets. From 5 to 4 is a 0.84 turn difference, from 4 to 3 is a 1 turn difference, from 3 to 2 is a 1.3 turn difference, and 2 to 1 is a 2 turn difference. Of course, they'll sometimes choose to delay their riichi for various reasons, so this isn't truly an accurate representation of speed.
Looking at the average value column, it seems my thought about higher starting shantens having more value if they won was false. The EV column is just the win% times the average value. Let's dig a bit more into the winning values to see the difference between riichi, dama, and open hands.
Interestingly, when it's like this, the values do increase along with the starting shanten. The percentage of wins that are riichi goes down as the starting shanten goes up, which makes sense and explains why the total value is going down. The dama rates remaining relatively constant is interesting.
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