Push Fold Fundamentals: Winrate/Dealinrate
Shoutout to hue for recommending Japanese resources and proofreading this post.
I'm going to propose a heuristic that will help you answer push fold problems in a variety of circumstances. This is a concept that I borrowed from poker and modified for riichi mahjong. It’s a concept that is so important that I think about it in every game of mahjong that I play, and one that I wish I was formally taught as a beginner. It starts with a very simple idea: if I push, what percentage of the time will I win, and what percentage of the time will I dealin? If I win, how many points will I gain, and if I dealin, how many points will I lose?
Clarifying Comment: WR stands for Winrate - the outcome where we ron or tsumo. DR stands for Dealinrate - the outcome where we feed a winning tile. These values are measured as percents from 0 to 100. The safe tile / 10% tile columns indicate the dealinrate of the tile we need to push on the current turn. Bad shape refers to any tenpai waiting on 4 or less tiles, whereas good shape refers to any tenpai waiting on 5+ tiles.
As a beginner, I never understood why pushing bad wait tenpai against riichi was a good idea. Let's imagine that I am facing a riichi on T8 with a bad wait open hand that would win 4900 pts on any win - 3900 pts + 1K from the riichi stick. My opponent may have a bad or a good wait, but on average, they have a better wait than I do. Let's also assume that my opponent is the dealer. If I dealin to the dealer, I will lose 7500 pts on average, more pts than I stand to win. It seems like I'm outclassed in both wait and value - why should I push?
The reason why pushing is a good idea is the consequence of two related concepts. The first is that even if I fold succesfully, I will lose 1700 points on average to the opponent tsumo, or in exhaustive draw (a.k.a. Ryuukyoku, rkk) payments. Pushing can be negative EV, but if it's higher than -1700, it would be justified. For the rest of this post, when I say a push is +EV, I mean that it is positive EV compared to folding. The second concept is that my w:d ratio is positive. Despite the fact that my opponent has a better wait than me on average, I have a w:d ratio of 1.57 for a safe tile push, and 1.03 for a 10% dealin tile push. This is because I am comparing my own winrate (ron + tsumo) to a fraction of my opponent's winrate (ron on me). If I push, I can expect to win more than I dealin, even with a bad wait.
Let's make some conclusions based on the math above. From Eq. 2, we conclude that for pushing to be better than folding, our w:d ratio must be greater than 5800 / 6600 ~ 0.88. Our w:d ratio is 1.03, which means we should push. How did we get these numbers? For the numerator, we took the absolute value of our average dealin value and subtracted 1700 from it to get 5800. For the denominator, we took our average win value, and added 1700 to it, to get 6600.
Let me highlight the significance of these numbers. If we win, we will gain 6600 pts more than the folding EV. If we lose, we will lose 5800 pts below the folding EV. Relative to folding, we stand to win more than we stand to lose. In addition, with a w:d ratio of 1.03, we expect to win more than we lose, making this a positive EV push.
Now let's estimate our push EV, and compare it to the actual SMS EV chart, to see how accurate our simplifying assumption was. With a w:d ratio of ~1, we can estimate our EV using Eq. 1 to be (6600 - 5800) * .3 - 1700 ~ -1460. In SMS Table 2.11, this value is -1400. Our heuristic worked pretty well, estimating the EV within a hundred pts. For the rest of this post, we'll use Equations 1 and 2 to analyze if pushing is better than folding in a variety of borderline scenarios.
W:D estimate: Since we have 0 genbutsu tiles to fold with, our fold EV is lower than -1100, the fold EV vs non dealer riichi assuming 100% fold success rate, as well as -1400, the fold EV assuming 2 genbutsu (~4% round dealinrate). If we assume folding with 0 genbutsu has ~8% round dealinrate, the fold EV would be around -1700.
Let's first use Eq. 1 with values centered around -1100, and assume a 10% dealin push. If we win the riichi hand, we'll gain 3800 pts on average (SMS), or 4900 counting the 1100 pt offset. If we lose, we'll dealin to 5300 pts on average, or 4200 counting the offset. This corresponds to a push EV of (4900 - 4200) * .3 - 1100 = -910, before considering the riichi stick we have to wager. We lose our riichi stick around 70% of the time on a chase, so our EV estimate of pushing is -1610. This is greater than -1700, so we should push.
SMS, bad wait: -1300 EV T8 10% push
Nisi (2022), 456 wait: -1397 EV T9 2378 musuji push, -1610 EV T9 456 double musuji push
All of the sources place the push EV between -1300 and -1400 EV, meaning riichi is better than fold if we have 2 genbutsu or less. Unlike the last example, our heuristic was low by a few hundred pts of EV, but it's good enough to get the right answer. It turns out that given the assumptions we made, this heuristic generally underestimates push EV - we'll discuss the reason for this later.
W:D estimate: With 2 genbutsu tiles to fold with, our fold EV is roughly -2000. If we win the riichi + 1 hand, we'll gain 5900 pts on average (SMS), or 7600 from the -1700 pt offset. If we dealin, we'll lose 7500 pts on average, or 5800 from the offset. This corresponds to an EV gain over folding of (7600 - 5800) * .3 = 540, before considering the riichi stick, or -160 including it. The final EV estimate is -1700 -160 = -1860. This number is larger than -2000, so we push.
SMS: -1700 EV T8 10% push
Nisi (2022): -1529 EV T9 2378 musuji push, -1748 EV T9 456 double musuji push
The solution to this problem is simple: we just need to use a modified ratio: the winrate to (dealinrate - (fold dealinrate)) ratio, which I will refer to as the marginal w:d ratio. According to a chart from this website, if we attempt to fold, our round dealinrate on turn 9 with 2 genbutsu is ~4%. With 1 genbutsu, our round dealin is roughly 5%, and with 3 genbutsu, our round dealin is roughly 3%. With this in mind, let's update our bad wait w:d chart that we made before with marginal w:d ratios. We'll also interpolate the table for pushing tiles with dealinrates of 5% and 15%.
To understand the marginal w:d ratio, let's imagine we are pushing a 10% tile with a bad wait hand on turn 8, and we have 2 genbutsu. The regular w:d ratio says that we will win 30% of the time and lose 29% of the time. The marginal w:d ratio says that even if we try folding, we will dealin 4% of the time. So if we decide to push, we are only increasing our dealinrate by 25%, while gaining 30% in winrate.
Using equation 2, the required w:d ratio is 1.57 ((7500 - 1700) / (2000 + 1700)). With 1 genbutsu (5% round dealin), the marginal w:d or pushing a ~5% tile (1s push w/ 13 live suji) is 1.58, which passes the threshold.
SMS: -1700 EV T8 0% push , -2300 EV T8 10% push
Totsugeki: -1909 EV T9 Kanchan 3 Wait 2378 Musuji Push
Totsugeki: -1909 EV T9 Kanchan 3 Wait 2378 Musuji Push, -1816 EV T14 Kanchan 3 Wait 2378 Musuji Push
The stadard Naga version decides to fold on 3m draw. 3m is more dangerous than 7m by suji counting. In addition, 9p has become suji, and there are fewer rounds until rkk, making our fold success rate higher.
On the other hand, according to the SMS and Totsugeki charts, the EV of pushing has increased from T8 / T9 to T11 / T14. Since we gain tenpai points on rkk, and our rkk chance goes up later into the round, the charts suggest that we should be more inclined to push now, assuming that 3m and 7m have similar dealinrates. Although the SMS chart is more optimistic about this T12 push, it still suggests folding.
With 11 live suji, the dealinrate of 2p is 7.3%. We'll make a rough estimate of the 5% marginal w:d by taking the average of the T11 5% and 10% dealin values, roughly 1.45. Despite having 2 genbutsu, the 5% round dealin estimate is more accurate for a twice open hand, which has less folding options than a 14 tile hand. Let's take into account that the dealer cannot use the dora yakuhai in their hand. By the same calculation done in the last hand, the resulting w:d cutoff is ~ 1.3. We should push this hand.
The borderline +EV value cutoff for bad wait push vs dealer was ~2 han for riichi, and 1-2 han for open hands. It follows that pushing most good wait tenpai hands vs riichi is also +EV, since good wait hands have higher winrates and lower dealinrates than bad wait hands.The W:D Heuristic Underestimates EV Due To RKK
As mentioned previously, the w:d heuristic relies on the assumption that the EV of pushing and not winning or dealing in is equivalent to the successful fold EV. This is not true: in fact, the EV in these scenarios is greater than our fold EV because of rkk. If we push and reach rkk, we will gain points for being tenpai, whereas we would lose points if we had folded. The later the turn number, the higher the chance of rkk, and the more the w:d heuristic will underestimate our push EV. For the next post of the Push Fold series, we're going to analyze keiten pushes, where rkk payments will be the main incentive to push.
Reasons Not To Push
Being able to recognize if a push is +EV is a valuable skill, but may not apply to every situation. I want to end this post with a list of reasons why pushing may be suboptimal. Many of these concepts will be addressed in future posts in this Push Fold Series, but for now, I will provide a brief outline. Some reasons not to push may include:
Dama: With a low value closed hand, it might be better to push without declaring riichi, and potentially fold later. We may even upgrade our hand's value or wait. In this post we only compared chasing riichi and folding.
Mawashi: It's possible that we can cut a series of safe tiles in a way that gives us a chance of getting back into tenpai. So far we've only discussed whether pushing is better than folding, but it's possible that mawashi could be better than both options.
Positional Play: We may want to fold more the closer we are the South 4 in a hanchan, or the larger our point gap with our opponents is.
Reward System: We may want to fold more in a 4th avoidant reward system, or a negative sum reward system.
Exploitative Folding: If we think we are stronger than our opponents, we may be more selective with what risks we want to take.
Expected Lateral Movement: If someone else is pushing against a riichi, we may expect some lateral movement - one of the two fighting players may dealin to the other. If lateral movement is likely to happen, the EV of folding increases, since opponent tsumo rates and rkk rates are decreased. This is somewhat related to exploitative folding, where the presence of a high dealin player may increase the lateral movement rate, raising the fold EV for everyone else.
Introducing The Win Dealin Ratio
Your opponent declares riichi. Do you push a dangerous tile, or fold? Some players may refer to a flowchart (push good wait with X han), or a chart of expected values (EVs) to answer this question, but I think there’s a deeper understanding to be had. What is the math underlying these charts, and how can you modify your answer if parameters such as value, dealinrate, ability to fold/maneuver, or position change?I'm going to propose a heuristic that will help you answer push fold problems in a variety of circumstances. This is a concept that I borrowed from poker and modified for riichi mahjong. It’s a concept that is so important that I think about it in every game of mahjong that I play, and one that I wish I was formally taught as a beginner. It starts with a very simple idea: if I push, what percentage of the time will I win, and what percentage of the time will I dealin? If I win, how many points will I gain, and if I dealin, how many points will I lose?
Data Sources
In this post, we're going to use the win dealin ratio (w:d ratio) to estimate if a push is positive EV or not. We're going to compare our estimates to a variety of sources.- Statistical Mahjong Strategy By Miinin (2017)
- A book that provides nisi simulator data for push / fold. It presents round EV estimates for push / fold, as well as a more detailed breakdown of what outcomes (winrate, dealinrate, opponent tsumo rate, etc.) occur at what frequencies. Nisi is a riichi mahjong simulator built off of Houou data and is the cornerstone of every mahjong stats book.
- 2022 Nisi Chasing Riichi Round EV Charts
- Using a more recent version of the nisi simulator than SMS, this post presents (Push - Fold) EVs based on our wait and the tile to be pushed.
- Shin Kagaku-suru Mahjong by Totsugeki (2021)
- Open Hand Push Fold Charts from Pages 112 - 113. These charts present EVs for pushing every tile until the end of the round.
- Naga AI decisions
Why Is Pushing Bad Wait Positive EV Over Folding?
Statistical Mahjong Strategy (SMS) Table 2.10 + W:D Ratio
Clarifying Comment: WR stands for Winrate - the outcome where we ron or tsumo. DR stands for Dealinrate - the outcome where we feed a winning tile. These values are measured as percents from 0 to 100. The safe tile / 10% tile columns indicate the dealinrate of the tile we need to push on the current turn. Bad shape refers to any tenpai waiting on 4 or less tiles, whereas good shape refers to any tenpai waiting on 5+ tiles.
As a beginner, I never understood why pushing bad wait tenpai against riichi was a good idea. Let's imagine that I am facing a riichi on T8 with a bad wait open hand that would win 4900 pts on any win - 3900 pts + 1K from the riichi stick. My opponent may have a bad or a good wait, but on average, they have a better wait than I do. Let's also assume that my opponent is the dealer. If I dealin to the dealer, I will lose 7500 pts on average, more pts than I stand to win. It seems like I'm outclassed in both wait and value - why should I push?
The reason why pushing is a good idea is the consequence of two related concepts. The first is that even if I fold succesfully, I will lose 1700 points on average to the opponent tsumo, or in exhaustive draw (a.k.a. Ryuukyoku, rkk) payments. Pushing can be negative EV, but if it's higher than -1700, it would be justified. For the rest of this post, when I say a push is +EV, I mean that it is positive EV compared to folding. The second concept is that my w:d ratio is positive. Despite the fact that my opponent has a better wait than me on average, I have a w:d ratio of 1.57 for a safe tile push, and 1.03 for a 10% dealin tile push. This is because I am comparing my own winrate (ron + tsumo) to a fraction of my opponent's winrate (ron on me). If I push, I can expect to win more than I dealin, even with a bad wait.
W:D Math
Let's do a rough estimate of the EV of pushing this 4900 pt hand vs folding on T8, using the w:d heuristic. We're going to make the flawed assumption that if we push and don't win or dealin, our EV is -1700, the same as if we had folded. For non dealer vs non dealer situations, we will use -1100 instead. We will revisit the accuracy of this assumption multiple times throughout this post by comparing our EV estimate to the EVs from various charts.Let's make some conclusions based on the math above. From Eq. 2, we conclude that for pushing to be better than folding, our w:d ratio must be greater than 5800 / 6600 ~ 0.88. Our w:d ratio is 1.03, which means we should push. How did we get these numbers? For the numerator, we took the absolute value of our average dealin value and subtracted 1700 from it to get 5800. For the denominator, we took our average win value, and added 1700 to it, to get 6600.
Let me highlight the significance of these numbers. If we win, we will gain 6600 pts more than the folding EV. If we lose, we will lose 5800 pts below the folding EV. Relative to folding, we stand to win more than we stand to lose. In addition, with a w:d ratio of 1.03, we expect to win more than we lose, making this a positive EV push.
Now let's estimate our push EV, and compare it to the actual SMS EV chart, to see how accurate our simplifying assumption was. With a w:d ratio of ~1, we can estimate our EV using Eq. 1 to be (6600 - 5800) * .3 - 1700 ~ -1460. In SMS Table 2.11, this value is -1400. Our heuristic worked pretty well, estimating the EV within a hundred pts. For the rest of this post, we'll use Equations 1 and 2 to analyze if pushing is better than folding in a variety of borderline scenarios.
Borderline Bad Wait Riichi
Fighting Non Dealer as Non Dealer
Let's first use Eq. 1 with values centered around -1100, and assume a 10% dealin push. If we win the riichi hand, we'll gain 3800 pts on average (SMS), or 4900 counting the 1100 pt offset. If we lose, we'll dealin to 5300 pts on average, or 4200 counting the offset. This corresponds to a push EV of (4900 - 4200) * .3 - 1100 = -910, before considering the riichi stick we have to wager. We lose our riichi stick around 70% of the time on a chase, so our EV estimate of pushing is -1610. This is greater than -1700, so we should push.
SMS, bad wait: -1300 EV T8 10% push
Nisi (2022), 456 wait: -1397 EV T9 2378 musuji push, -1610 EV T9 456 double musuji push
All of the sources place the push EV between -1300 and -1400 EV, meaning riichi is better than fold if we have 2 genbutsu or less. Unlike the last example, our heuristic was low by a few hundred pts of EV, but it's good enough to get the right answer. It turns out that given the assumptions we made, this heuristic generally underestimates push EV - we'll discuss the reason for this later.
Fighting Dealer as Non Dealer
SMS: -1700 EV T8 10% push
Nisi (2022): -1529 EV T9 2378 musuji push, -1748 EV T9 456 double musuji push
Marginal W:D
Let's revisit Equation 2 that was presented above. When we derived this equation, we assumed that folding and non win / dealin outcomes were equivalent to -1100 or -1700 EV, the fold EV assuming we fold succesfully. If we can't guarantee that we are able to fold 100% succesfully, we can't use Equation 2 to make our push/fold decision. In previous problems, we accounted for this by using Equation 1 for an EV estimate, and then comparing to an alternate fold EV number that changed with how many genbutsu we had. However, doing this kind of math is infeasible during a match - we'd rather use Equation 2 to calculate the w:d ratio we need to push, and then use our memorized w:d ratio to make the decision.The solution to this problem is simple: we just need to use a modified ratio: the winrate to (dealinrate - (fold dealinrate)) ratio, which I will refer to as the marginal w:d ratio. According to a chart from this website, if we attempt to fold, our round dealinrate on turn 9 with 2 genbutsu is ~4%. With 1 genbutsu, our round dealin is roughly 5%, and with 3 genbutsu, our round dealin is roughly 3%. With this in mind, let's update our bad wait w:d chart that we made before with marginal w:d ratios. We'll also interpolate the table for pushing tiles with dealinrates of 5% and 15%.
SMS Table 2.10 + Interpolated Dealin + Marginal W:D Ratio
To understand the marginal w:d ratio, let's imagine we are pushing a 10% tile with a bad wait hand on turn 8, and we have 2 genbutsu. The regular w:d ratio says that we will win 30% of the time and lose 29% of the time. The marginal w:d ratio says that even if we try folding, we will dealin 4% of the time. So if we decide to push, we are only increasing our dealinrate by 25%, while gaining 30% in winrate.
Borderline Open Bad Wait
Fighting Dealer as Non Dealer
Earlier we concluded that with 3 han bad wait, fighting dealer riichi was positive EV over a 100% fold. Now let's take a look at 1 han push with varying amounts of genbutsu.SMS: -1700 EV T8 0% push , -2300 EV T8 10% push
Totsugeki: -1909 EV T9 Kanchan 3 Wait 2378 Musuji Push
We now have 2 genbutsu. With 11 live suji, 7m has a dealinrate of ~8.4%. Using the 10% dealin column 4% marginal w:d ratio for T8, we get a ratio of 1.20, which is less than 1.57. We should fold by the w:d heuristic. According to SMS, the EV of pushing a 10% tile on T8 is -2300, worse than the -2000 EV for folding from 2 genbutsu. Our heuristic is consistent with SMS (which the chart is based on), but not Naga or Totsugeki. Totsugeki reports an EV of -1909 for T9 Kanchan 3 Wait 2378 Musuji Push.
One possible explanation as to why Naga pushes is that the w:d ratios used in our chart are based off of chasing riichi data. When we declare riichi, we commit ourselves to pushing every tile until the end of the round, but a strategy where we push, and then fold on particularly dangerous tiles will have a better w:d than those presented in the chart. Another possible explanation is that the opponent's turn 1 dora yakuhai discard decreases the expected value of their hand. If we assume that we dealin to an average of 6500 pts rather than 7500, our new w:d cutoff is 4800 / 3700 ~ 1.3. We should still fold by our heuristic, but it's a lot closer now.
Comparing SMS to Totsugeki, it seems like the newer nisi simulator data is more inclined to push open hands vs dealer riichi by a few hundred pts of EV.
SMS: -2300 EV T8 10% push, -2100 EV T11 10% push
One possible explanation as to why Naga pushes is that the w:d ratios used in our chart are based off of chasing riichi data. When we declare riichi, we commit ourselves to pushing every tile until the end of the round, but a strategy where we push, and then fold on particularly dangerous tiles will have a better w:d than those presented in the chart. Another possible explanation is that the opponent's turn 1 dora yakuhai discard decreases the expected value of their hand. If we assume that we dealin to an average of 6500 pts rather than 7500, our new w:d cutoff is 4800 / 3700 ~ 1.3. We should still fold by our heuristic, but it's a lot closer now.
Comparing SMS to Totsugeki, it seems like the newer nisi simulator data is more inclined to push open hands vs dealer riichi by a few hundred pts of EV.
Totsugeki: -1909 EV T9 Kanchan 3 Wait 2378 Musuji Push, -1816 EV T14 Kanchan 3 Wait 2378 Musuji Push
The stadard Naga version decides to fold on 3m draw. 3m is more dangerous than 7m by suji counting. In addition, 9p has become suji, and there are fewer rounds until rkk, making our fold success rate higher.
On the other hand, according to the SMS and Totsugeki charts, the EV of pushing has increased from T8 / T9 to T11 / T14. Since we gain tenpai points on rkk, and our rkk chance goes up later into the round, the charts suggest that we should be more inclined to push now, assuming that 3m and 7m have similar dealinrates. Although the SMS chart is more optimistic about this T12 push, it still suggests folding.
Good Wait Chart
SMS Table 2.4 + Interpolated Dealin + Marginal W:D Ratio
The borderline +EV value cutoff for bad wait push vs dealer was ~2 han for riichi, and 1-2 han for open hands. It follows that pushing most good wait tenpai hands vs riichi is also +EV, since good wait hands have higher winrates and lower dealinrates than bad wait hands.
Pushing Dora vs Dealer
As one final example, let's imagine we have a 1 han good wait hand, it's T8, and we're pushing a 10% tile vs dealer riichi. The w:d cutoff would be (7500 -1700) / (2000 + 1700) = 1.57, which is far below our marginal w:d of 2+. Now imagine we have to push a dora vs the dealer. We'll estimate the average dora dealin value by the average ippatsu dealin of 10900, making the new w:d cutoff 2.49. We can push a 5% tile, but not a 10% one.The W:D Heuristic Underestimates EV Due To RKK
As mentioned previously, the w:d heuristic relies on the assumption that the EV of pushing and not winning or dealing in is equivalent to the successful fold EV. This is not true: in fact, the EV in these scenarios is greater than our fold EV because of rkk. If we push and reach rkk, we will gain points for being tenpai, whereas we would lose points if we had folded. The later the turn number, the higher the chance of rkk, and the more the w:d heuristic will underestimate our push EV. For the next post of the Push Fold series, we're going to analyze keiten pushes, where rkk payments will be the main incentive to push.
Reasons Not To Push
Being able to recognize if a push is +EV is a valuable skill, but may not apply to every situation. I want to end this post with a list of reasons why pushing may be suboptimal. Many of these concepts will be addressed in future posts in this Push Fold Series, but for now, I will provide a brief outline. Some reasons not to push may include:
Dama: With a low value closed hand, it might be better to push without declaring riichi, and potentially fold later. We may even upgrade our hand's value or wait. In this post we only compared chasing riichi and folding.
Mawashi: It's possible that we can cut a series of safe tiles in a way that gives us a chance of getting back into tenpai. So far we've only discussed whether pushing is better than folding, but it's possible that mawashi could be better than both options.
Positional Play: We may want to fold more the closer we are the South 4 in a hanchan, or the larger our point gap with our opponents is.
Reward System: We may want to fold more in a 4th avoidant reward system, or a negative sum reward system.
Exploitative Folding: If we think we are stronger than our opponents, we may be more selective with what risks we want to take.
Expected Lateral Movement: If someone else is pushing against a riichi, we may expect some lateral movement - one of the two fighting players may dealin to the other. If lateral movement is likely to happen, the EV of folding increases, since opponent tsumo rates and rkk rates are decreased. This is somewhat related to exploitative folding, where the presence of a high dealin player may increase the lateral movement rate, raising the fold EV for everyone else.
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