Mah Jongg Statistics (part 2)

In our first Mah Jongg Statistics article, we discussed win rates, think times, hand popularity, number of tiles remaining in the wall, and the influence of jokers. The article was extremely well received, so we decided to dig a little deeper down the Mah Jongg statistics rabbit hole!

This time, we were particularly interested in exploring the following:

• The impact of being East on the outcome of the game
• Whether the second Charleston is worthwhile
• Which numbers are the most successful at achieving Mah Jongg, for given hands or categories

Our sample data was gathered from games played on ilovemahj.com over the last few months (so an extremely large sample size), using the NMJL 2022 card and from every skill level. As I Love Mahj offers many tools for new players (Exercise Room, Practice Mode with hand suggestions, tutorials, etc), our demographic includes many beginners. We also have a significant number of players at the opposite end of the spectrum, who come to challenge themselves against our advanced, level 3 bots, and every skill level in between.

So overall, we believe the data gathered on our platform is reflective of the general Mah Jongg population and that the numbers presented here are likely realistic. We included data from both human and bot players, so we could further analyze trends between these two groups.

The Impact of Being East

Intuitively, I'm sure we all believe there is an advantage in being East, since East starts with an extra tile and makes the first move. However, we wanted to quantify this advantage. We began by measuring the win/loss/wall rate of players when they were East vs when they were not. Here are the results:

When Player is Not East

When Player is East

Interestingly, the percentage of wall games remains the same. But, the chance of winning when a player is East increases by 10.82%!

This result confirms that being East has a significant impact on the outcome of the game. Of course, since East rotates around the table, this advantage passes to each player in turn. As long as a multiple of four games are played, the advantage evens out.

What is not clear from the data is how much of this advantage is based on starting with one extra tile vs making the first move, and therefore being one step ahead of other players at the table. We suspect the former has much more impact but, in order to prove that theory, it would be necessary to gather data from games where all players start with the same number of tiles, which isn't possible. So, for now, we can only speculate.

Is the Second Charleston Worthwhile?

Stopping the Charleston is often considered to be a controversial decision. Even though the rules clearly state that a player may stop the Charleston for any reason, there are players that have a tendency to become extremely upset when a player does. Some players rarely stop the Charleston, some stop it every time, others stop it only when they have fewer than 3 tiles to pass and some stop it because they can't make up their mind between two hands, and just can't bring themselves to give up any of their tiles.

We were keen to study the influence of stopping the Charleston on the game. In particular, we wanted to know:

• How often is the Charleston stopped?
• How many tiles does a player have towards Mah Jongg when they stop the Charleston?
• What are the win/loss/wall ratios for the player that stopped the Charleston?
• Does East stop the Charleston more often?
• What is the influence on game duration?
• How much does a player benefit during the second Charleston?

How Often is the Charleston Stopped?

Given how opposed many players seem to be to stopping the Charleston, we were surprised to see that it occurred in 11.45% of games. We then broke this figure down between human and bot players:

• Humans stop the Charleston 9.35% of the time
• Bots stop the Charleston 2.1% of the time

This is a striking, but not totally unexpected, difference. Bots are programmed to only stop the Charleston when they have a significant number of tiles towards Mah Jongg. By contrast, we often see people stop the Charleston because they can't make a decision between two hands.

We decided to dive deeper into these numbers to take a peek at distribution (ie, are there players that skew the data one way or the other?). We noticed that 46% of players didn't stop the Charleston at all (at least during the time period studied), and 3.4% of players stopped the Charleston during every single game they played. The rest of the distribution was pretty spread out with a clustering between 0 and 8%.

Number of Tiles Towards Mah Jongg

Next, we looked at how close a player was to Mah Jongg when they decided to stop the Charleston. From a strategic perspective, players are generally advised to continue the Charleston if they have at least three tiles to pass. Of course, there are additional factors that may prompt someone to stop the Charleston: perhaps they do have 3 tiles to pass, but would be required to pass a pair, or all winds, or flowers, and some may decide it's wiser to stop the Charleston in these cases.

Here are the results. On average:

• Bots stopped the Charleston with ~10 tiles towards Mah Jongg
• Humans stopped the Charleston with ~8 tiles towards Mah Jongg

This reinforces our earlier conclusion that human players are more likely to stop the Charleston when they're not close to Mah Jongg (likely due to indecisiveness).

How Stopping the Charleston Influences the Outcome of the Game

When a player stops the Charleston, the expectation is that they will win more often, since it's presumed they have a plethora of tiles towards Mah Jongg. Of course, this hypothesis assumes players stop the Charleston for the correct reason. Let's see what the data reveals!

Below we show the percentage of games that ended in a loss/wall/win for players that did not stop the Charleston and for those players that did:

Did not Stop

Stopped

The data highlights that stopping the Charleston results in more wall games, but also a significantly higher win rate (an increase of 27.58%) for the person who stopped it. But, let's be careful with the conclusions we draw here. The data is not saying that stopping the Charleston magically increases one's chances of winning a particular game. All it's showing is that when a player is in a position to stop the Charleston (presumably because they have many tiles towards Mah Jongg - more on that in a minute!), they have a higher likelihood of winning. This makes perfect sense.

However, things get even more interesting as we dig deeper and break down the results between bots and human players.

Bots Only

When stopping the Charleston, bots end up winning ~58% of the time (a whopping increase of 157%) and it also reduces the percentage of wall games.

Humans Only

When stopping the Charleston, the win rate for human players actually decreased! Interestingly, the percentage of wall games increased dramatically (by 80%!) when there was no second Charleston.

We interpret this as a confirmation that a significant portion of human players stop the Charleston due to indecisiveness, rather than having a spectacular hand. In doing so, they deprive themselves (and their tablemates) of potentially nine additional exchanged tiles, and it's likely that the player opposite them will decide to forgo the Optional Pass, so potentially three additional tiles are never received. The result being fewer wins and more draws.

East Stopping the Charleston

We were also interested to discover if East stops the Charleston more frequently than other players at the table.

As mentioned previously, the Charleston was stopped in 11.45% of games, 3.73% were stopped by East and 7.72% by other players who were not East. Since there are three players who are not East, that works out at 2.57% per player. In conclusion, East stops the Charleston 45% more often than other players at the table.

Since East has an additional tile from the very start of the game, they have an advantage when it comes to identifying and making progress towards a particular hand. East is also more likely to be closer to Mah Jongg, or perhaps more likely to be stuck between two hands (see previous discussion) by the end of the first Charleston. So, it makes sense that East would stop the Charleston more often than other players, though the difference (45% more often) is more pronounced than we would have expected.

Game Duration

We moved on to analyzing the average duration for games where only the first Charleston was completed. Intuitively, one may assume the game would be shorter since players save time by skipping the passing related to the second Charleston. As an aside, an I Love Mahj member recently commented that she'd noticed several players in a tournament agreeing to skip the second Charleston during the last game in their round, in the hopes that they could catch up and finish before the clock ran out.

However, the data shows the opposite to be true! Games where the Charleston was stopped took, on average, 18% longer to complete than those where the Charleston continued.

This does make sense, actually. As discussed earlier, skipping the second Charleston deprives all players of tile exchanges, which could have helped players reinforce their hand. This leads to more wall games and more difficulty completing a hand, and therefore more time elapsing for the game.

How Much is Gained During the Second Charleston?

We often hear players say something along the lines of "what's the point of the second Charleston? I never get anything new." Is this true or is it an old Mah Jongg myth? Let's see if we can find out!

Looking at all games where the Charleston was continued, a player gains an additional 0.6 tiles, on average, towards Mah Jongg during the second Charleston.

Below we set out the distribution of gains (and losses!) achieved during the second Charleston:

We see that in ~47% of cases, no tiles are gained, though there's a possibility that the player switched hands and they now have a stronger starting point from which to work from. For example, perhaps they now have a needed pair for a hand they were keeping an eye on, or maybe their tiles are more evenly distributed in the hand they've switched to, making it easier to call discards during the game.

36.6% of the time, the player is 1 tile closer to Mah Jongg and in 11.6% of cases, they are 2 tiles closer. These are pretty significant improvements to a player's hand.

Interestingly, you will notice that there are also instances where players ended up with fewer tiles towards Mah Jongg at the end of the second Charleston! This is likely due to the fact that the player had a pretty good hand at the end of the first Charleston, but did not realize it, and they ended up passing some of these tiles.

There are a few outliers in the data, including instances where a player gained 6 tiles during the second Charleston, while another lost 4!

If we break down the results by bots vs humans, we end up with the following:

Bots gain more from the second Charleston, on average, than humans (presumably because they are better at recognizing the hands they have).

Finally, we took a peek at an internal proprietary metric that computes how close a player is to Mah Jongg, while also taking into account the difficulty factor of the chosen hand (ie, whether it contains pairs, is concealed, etc). Using this metric, we calculated that a player achieves, on average, a 10% improvement to this score during the second Charleston.

Conclusion

Overall, it is generally to everyone's advantage to continue the Charleston, especially if you’re very familiar with the card and can easily recognize the best hand options. If you have a strong hand and are less than three tiles away from Mah Jongg, then stopping the Charleston makes perfect sense. It also makes sense to stop the Charleston if you're close to Mah Jongg, but don't have suitable tiles to pass (ie, pairs and flowers). However, not being able to make a decision between two mediocre hands is not a good enough reason to stop the Charleston. In this case, you will only be harming your ability to win and the game has a far higher likelihood of ending in a draw.

Which Numbers are the Most Successful?

Let's turn our attention now to analyzing the popularity of certain number tiles. In our previous article, we revealed which hands on the NMJL 2022 card were the most popular. In this article, we wanted to dig a little deeper into those hands that have flexible number options (mostly found in the Any Like Numbers and Consecutive Run categories), to discover which numbers are used most often in winning hands. Please note that by "popular" we don't mean which hands people enjoy playing the most. We are only looking at winning hands, so "popular" here means that these number choices resulted in a successful outcome.

Any Like Numbers

We first looked at the Any Like Numbers category and calculated how often winning hands were completed for each of the nine numbers. Here are the results:

For those that prefer to see things graphically, these numbers are plotted in the bar chart below:

The number one appeared in significantly more Mah Jongg hands for the Any Like Numbers category than any other number. The next most popular was the number nine. Six, on the other hand, was the least successful number in this category, with the number three being not too far behind. The results surprised us, as we expected them to be more evenly distributed. We're not entirely sure of the reasoning behind the results, other than the number one is likely being used less often by others at the table, and so there is more opportunity to collect tiles of that particular number. If you have an interesting theory to explain these results, we'd love to hear it!

We also analyzed Line 4 of the Winds & Dragons category, which uses two kongs of like numbers. Here are the results:

Once again, we observed a similar shape to the distribution, though the significance of the number one was not quite as dramatic. Six was the least successful number, with the number three a close second, for this particular hand, as was the case for the Any Like Number category.

Our conclusion is that if you're aiming for an "Any Like Number" hand, you're more likely to be successful if you use ones (and nines to a lesser extent). The least likely to succeed are the numbers six and three. This information could aid in making crucial decisions during the Charleston: for instance, if you have several ones and several sixes and you're trying to decide which number to continue to hold, focusing on ones could result in a more successful outcome!

Consecutive Runs

We performed the same analysis for hands in the Consecutive Run category. We looked at each of the most played hands to see which run of numbers was most successful in achieving Mah Jongg.

Consecutive Run - Line 1

This hand has 2 versions 1-5 and 5-9. The results were split fairly evenly: 50.8% and 49.2% respectively. So, building a hand using the lower numbers results in a marginally more successful outcome.

Consecutive Run - Line 2

The eight versions of this hand are listed below, along with the percentage of times each number combination achieved a successful result:

Once again, numbers at each end of the spectrum achieved Mah Jongg more often. The numbers six (when combined with the number 5) and three (when combined with the number 2) remain on the lower end of popularity.

Consecutive Run - Line 3

Below are the results for the 1-suit version of this hand:

Below are the results for the 3-suit version of this hand:

The shape of the distribution for these two versions are very similar, with the lowest and highest-numbered runs resulting in Mah Jongg more frequently than runs using other numbers. As with previous analysis, runs that include the number six are the least likely to result in a successful outcome..

Consecutive Run - Line 4

There are eight versions of this particular hand. The results are shown below:

Once again, runs that include the numbers one or nine are the most successful, and those that include the numbers six or three are the least successful.

Consecutive Run - Line 5

Below are the results for the 1-suit version of this hand:

2-suit version

In both versions, runs that include the numbers one or nine are the most successful, and the run that includes both the numbers six and three is the least successful.

Winds and Dragons - Lines 5 & 6

As these hands contain a mini run, we have included them in our analysis.

Once again, the runs that include the number one or nine have been the most successful at achieving Mah Jongg. The runs that include the number six have been the least successful.

Conclusion

Though there are some differences in the distributions for each of the Consecutive Run hands analyzed above, overall the shape for each is very similar. Low and high-end runs ended up being the most successful at achieving Mah Jongg. This may be counter-intuitive, since it would seem that building a run somewhere in the middle, with the opportunity to build on either end, would be more likely to be successful. Perhaps, this is actually why these hands end up winning less often, as there is just too much competition for the same tiles. Whereas the extreme low and high end numbers are less in demand. We'd love to know how you interpret these results!

13579 Category

Several of the hands in the 13579 section of the card offer 2 options (1-3-5 or 5-7-9). We delved into the data to discover which version (low or high) achieved Mah Jongg more often.

Interestingly, we discovered that the low numbers (1-3-5) achieved Mah Jongg more often than the high numbers (5-7-9). We're not totally sure why this would be, but suspect that there is just a tad less competition for numbers on the lower end than those at the higher end.

Other "Flexible" Hands

Finally, we took a peek at several additional hands that offer flexibility in the numbers that can be used as part of the hand.

2468 - Line 2

This hand has the option of using 2, 4, 6 or 8 for the two sets of kongs. Here is the distribution:

Interestingly, we observe once again that the number six is, by far, the least likely to result in a successful Mah Jongg. It seems that there is just too much competition for this particular number with other sections on the card.

369 - Line 6

This hand has the option of using 3, 6 or 9 for the kong. Here is the distribution:

Here we see that the number three is the most popular and six is, once again, the least popular.

Conclusions

Well, it's been a very interesting dive into the data and our discovery of the numbers that are most likely to achieve Mah Jongg. Our journey has highlighted the numbers one and nine as the frontrunners, with six and to a lesser extent three being the least successful, on average. That being said, the hands we’ve analyzed have been won with every single number, just less often in some cases. Our advice would be to keep our analysis in mind, especially when making decision between number options, but to let the tiles lead the way.

We hope you've enjoyed geeking out with us again, and that these insights may help inform some of your decisions during the game (and result in more wins!). As usual, we welcome your feedback and suggestions for future articles (please email us at info@ilovemahj.com).

Mahj on, everyone!

Philippe & Julie
ilovemahj.com