Analysis - The Threat of Pao

In this analysis, we'll look at how Houou players react when faced with the threat of Pao. In other words, when one player has called two dragons, and the third is live. We'll also look at what happens when the third is discarded, and see how dangerous it actually is. Is it worth holding the tile so you don't deal into a yakuman, or is a yakuman so rare that you shouldn't worry about it?

If an image is too small to read, click on it for a larger version. Also, if you don't know what pao is, it's when a player feeds another player their third dragon pon. In a situation like this, the player who fed the last pon is responsible for the daisangen, and has to pay alone if they tsumo, or split the payment on a ron. You could also just deal into the yakuman, so it's doubly scary.

First, the sample size. We're using the same sample as last week, which is 1,246,147 games from 2011 to the middle of 2019. Of course, situations like these don't come up in every game, so let's look at how many rounds actually had a potential pao.

Of the thirteen million rounds examined, not even a hundred thousand had the chance for a pao to occur. It's not a huge sample, but it should be enough to get a general idea. So, what happened in those 96k cases?

To give a general idea of how often players are asked to make such a decision, I tracked the number of times at least one player drew the third dragon. They could still have one in their hand from before the second call, though.

The next five rows are events that remove the pao threat. Either the opponent shows they don't want the tile, a player has two of the third dragons, or daisangen is impossible. The next three after that show the decisions. When players discarded the tile with none visible, one visible, or never discarded it at all. Of the times the threat doesn't go away, the third dragon only gets discarded around a third of the time.

The "Third Dragon Never Discarded" case could have a few explanations. Either people were too scared to discard it, it wasn't worth the risk, or they were stuck deep in the wall where they were never reached, or people used it as part of their hand. It's hard to know exactly, and might be something I look at if I revisit this.

Note that these numbers are, very slightly, inflated. Since the replays don't allow us to know what the initial dora is when the game isn't won, some games that went to ryuukyoku would have had the initial dora indicator being the third dragon. From the last post, we know that around 15% of games go to an exhaustive draw. Combined with the 1/34 chance of the indicator to be the third dragon, this means the discard numbers are about 0.44% higher than they should be, or around 50-60 instances. We can't know for sure the number, so this isn't represented in the above table.

Anyway, what happens when a player discards the third dragon? We'll first look at the more dangerous case, where the tile is discarded when no others are visible.

Ten times out of eleven, the tile passes the scary player. But the remaining time, it either gets called, or deals in. And when it deals in, it's an average of nearly 20,000 points! Weighing the chance to deal in against that average value means you'd, on average, lose 872 points for discarding the tile. But boy is it a big swing between the two outcomes.

Still, I believe these numbers indicate it's worth it to push if you're tenpai or close to it with a good hand. At least, from a simple EV perspective. The placement EV might be something to think harder about.

Before we talk about that average point loss including pao line, we should look at what happens after pao is applied.

Even if the player gets all three calls, they still win less than half of the time. Of course, when they do win, it's a yakuman, and you have to pay. So, if you get pao applied to yourself, you'll be losing over 14k on average. This is factored into the "Incl Pao" line in addition to the chance of dealing in. You could think of the "Average Point Loss" line as being "Average Immediate Point Loss," and the "Incl Pao" line as being "Average Round Point Loss."

I made a chart, too. Look at it.

Ah, yes. What a chart. Let's move on to looking at the less dangerous option, where the tile gets discarded while one is already visible somewhere.

As you might have expected, the chance of pao or dealing into daisangen drops dramatically. Judging by the point loss numbers, it's about half as dangerous. It seems very easy to justify pushing when you can already see one.

That "dealt into a different yakuman" instance is pretty unique. You can find the replay here. Of course, there's only one other yakuman it could be.

If you watched that replay, you'd notice that the player who discarded the tile was in riichi. Whether the discarder was in riichi or not doesn't really affect the danger, but it does mean that some of the discards were not intentionally done. Let's look at some riichi statistics.

People are still willing to riichi in the face of two dragon calls. Given the average point loss values above, this seems sensible to me. If your hand is already ready, it's likely good EV to push, though perhaps not the best placement EV. Plus, the chance of drawing the dragon isn't that high, so you could justify it way easier than actually discarding the dragon itself.

Looking at the number of times the third dragon was discarded by a riichi player, it's around 10% of the time. We can put this into a pie chart to have a more visual idea:

With three-quarters of the riichis happening after the two calls, this implies to me that the two calls usually happen pretty early in the round. Let's take a look.

Yeah, there's a pretty big clump between turns 3 and 6. We can represent this in a graph, too, as all these percents are a bit hard to read.

Now, more goes into pushing than just EV considerations. Next, we'll look at how players in different seats and placements act when faced with these situations.

People are more likely to push while they're in fourth, which makes sense. Every seat is around the same likelihood to push, except for dealer when there's none visible. That's kind of interesting. I can't think of an explanation for why that is.

If the player with two calls is the dealer, the hand is a lot scarier. Do the pushing habits change when pushing against a dealer?

Using these two tables, we can generate a similar table for pushing against a non-dealer, for easier comparison.

One thing that's notable is that the dealer is inordinately the one with two dragon pons. This could just be because the dealer has a better incentive to call them even when their hand has nothing else, due to the dealer scores being higher and repeats being possible. Perhaps this means the danger of two dragon pons is less when the dealer does it. Either way, I'm not really seeing any huge difference between pushing the third dragon vs the dealer and a non-dealer.

If you look back to the first seat/placement table, you might notice a diagonal line of inflation. East pushes in First more, South pushes in Second more, West pushes in Third more, and North pushes in Fourth more. I think this is due to the influence of East 1, as thoseare the placements at the beginning of the game. Let's remove all the discards that happened in East 1 and see how the table looks.

These numbers seem more logical, but are probably still a bit inflated in the same way. I'm not sure if they're more useful, but let's look at them anyway. Again, dealer pushes the most when 0 are visible, but the seats are pretty equal outside of that. Fourth pushes most, as expected.

You can also see the pushes from fourth place rising as we go around the table. This might just be an effect of that inflation I mentioned, or it could be people wanting to push more the further they are from their dealer turn (or if their dealer turn has already passed).

While we're at it, let's look at the push count by round. From last week's post, we know that most games make it to South 4, so there should only be a slight bias towards earlier rounds.

People get less likely to take the risk the closer the game gets to South 4. I imagine this is especially true of people with good placements. There could also be a reduction in the number of "empty" double calls, ie, ones without a chance of daisangen or shousangen, as people start needing bigger hands to come back.

Finally, since I was looking at the wins anyway, I compiled a table of the non-pao daisangen wins, so we can see how many calls are in an average daisangen hand.

The ones with three dragon calls are ankans of the third dragon, so nobody is in pao. Even with the ~270 daisangens that were won with pao (3 calls, 3 dragons), it's still more common for a daisangen to be won with only two calls.

You can find the spreadsheet with all of this data here. Some of the numbers might be slightly different than the ones in this post, as I did a lot of minor refining after prepping the screenshots the first time. You can find the scripts used here.

I did look briefly at daisuushii, however, there has never been an instance of pao being applied for daisuushii in Houou in the dataset I have. When searching, I did find an interesting replay. The player gets in tenpai for daisuushii/shousuushii and tsuuiisou, double yakuman. There are three kans on the field, and he draws the fourth of one of his winds. If he calls the kan and doesn't rinshan, the round ends in a draw. There are only two draws left, though, so it's one guaranteed draw versus two maybe draws. Would you risk it?

Next time, we'll look at how Houou players deal with another risky situation: Having a lone dora dragon in their hand.