Polling Paradox: the Crowded Bus Paradox

 I've found some mention of a "Crowded Bus Paradox" online, but it's always been poorly explained. Additionally, its impact on polling data is seemingly non-existent. I want to outline an example to help explain why I feel this is important. 

Lets imagine a fictional city. In this city is a train line; perhaps a subway, perhaps some form of LRT, or perhaps commuter rail. Regardless, on this train line, is a station. The one we are using in our little example. At this station is a bus line. On this bus line, a bus departs every 5 minutes, right on schedule. 

People walk to the station to catch the bus. One every minute, to be exact. The train also uses this station, it arrives and departs at 10 minute intervals, and on each train, are 5 people who use the bus line we are looking at.

Now. What will happen. Lets assume its 2:00pm.

At 1:56pm, a passenger walks up. Another at 1:57, 1:58, and 1:59. At 2:00pm a 5th passenger walks up. At the same time, a train arrives, and 5 people get off the train. Also at the same time, the bus is ready to depart, and so 10 people board the bus. 

What happens next?

Well we continue to get 5 people walk in, so the next bus leave with 5 people on board. Remember, no train arrives at 2:05pm, the next one arrives at 2:10pm. 

Next we have another bus with 10 passengers departing, followed by one with 5, and one with 10, and one with 5, and so on to infinity. 

So, a statistics agency hires a polling firm to get some data. They ask the bus drivers. "How many people are on your bus". Averaging the answer, they get 7.5

Now, they ask the passengers. "How many people are on your bus". Averaging the answer, they get 8.3

But wait! How? Where did these additional people come from?

Up to now you've probably been looking at this from a driver's point of view. 10, then 5, then 10, then 5. But look at this from a passenger's point of view. 

You can split this 10-5-10-5-etc thing into two groups. One group with 5 passengers, and one group with 10. This is a total of 15 people. Of those 15, 10 of them get on buses with 10 people, and 5 on buses with 5 people. This means that 10 people see 10 people on the bus, and 5 see 5 people on the bus.

 10+10+10+10+10+10+10+10+10+10+5+5+5+5+5=125

125/15 = 8.3333333

Lets assume for the sake of argument that 8 people is the most that can sit on our tiny little bus. 

If you were to look at the total provided by the drivers, you'd find that only half the buses are overcrowded. However, if you ask the passengers, two third of them say they are on an overcrowded bus.

Now we have 50.0% vs 66.7%. 

So, we have radically different conclusions from the exact same data, simply changing based on how you look at it.

So. Why am I posting this? It is a bit unusual, given what I usually post about. There are a few reasons. One, is that from time to time I see polls asking people to estimate things about others, and the results of those polls are always far off from base. Part of the reason is related to this paradox. It also explains why people can experience things (like an overcrowded bus) that the data (how many buses are overcrowded) suggests are uncommon, at rates that seem to imply the data is faulty. Lastly, I have a few posts I hope to make in the future that may require referencing back to this post, and this paradox. As such, I felt it was good to 'get it out of the way' now. 

I'm still going to post a regular weekly update on monday; but I also hope to make additional posts based on some stuff I've been working on that are not directly related to the blog; in particular, looking at historic and past elections; in particular, in germany. 

26JUL2021 updates

 In Bulgaria, ITN has come to its senses, and is trying to put together a coalition of other reformists. As outlined, that would have 112 of the 121 they'd need for a majority. A reformist coalition that's in a minority situation could still govern based on getting issue by issue support from other parties. Additionally, it would be fairly easy for such a government - if it provides good governance - to be re-elected to a coalition majority should the traditional parties in opposition block reformist legislation. 

In Germany, I've run a quick poll-based prediction. it is as follows.

188 C/U (142 CDU  / 46 CSU )

130 Grn 

117 SPD 

91 FDP 

78 AfD 

46 Lnk

As you can see the CDU (along with the CSU) is far from a majority, and even a CDU-GRN coalition would not have a majority. This means a three party coalition is likely at this time, with the only question being if that will be CDU-SPD-FDP, which is likely, or the less likely, but still possible CDU-SPD-GRN "kenya coalition". There is always a tiny chance of a GRN-SPD-FDP "traffic light" coalition, but I'm, not seeing it at this point. I note that while a CDU-SPD-FDP coalition does not have an official name, it is very similar to the "Weimar Coalition" of parts of the 1920s, and also contains the colours of the German Flag.

I continue to monitor other countries. In Italy, we are getting close to the FDI and PD both overtaking the League. I'll let you know when that finally occurs. I've also been working on comparative election results from germany going back over 100 years. I'm hoping to have a post about that up somewhat soon. 

19JUL2021 updates

 In Bulgaria, the victorious ITN party has decided to form a minority government, with no coalition partners. They only have 65 seats (ISMV only ended up with 13), but with the other reformists, they do have 112, which is still short of 121. Such a government may be very weak, assuming it can even get into office by passing any needed confidence votes. It is likely this is a political ploy as it will say to the voters that "everyone else" is out to stop them, and therefore, they need a majority. 

Russia closes in on its own elections. My current thinking is as follows

270 - UR (110/160) [pro-putin] {-73}

72 - CPRF (45/27) [communist] {+30}

61 - LDPR (40/21) [hard nationalist] {+22}

42 - SRZP (30/12) [moderate] {+19}

5 - Oth (0/5) [various] {+2}

The first/last number indicates the number of proportional/constituency, seats won.  While this is a loss of many seats for United Russia (Putin's party), they've been this low before in the legislature, taking only 223 seats in 2003, and 238 in 2011. 

For comparison, the two parties that would form United Russia took a combined 141 seats in the 1999 legislative elections (held a few weeks before Putin became President) The communists took 113 in that election, and, 52 in 2003, 27 in 2007, 92 in 2011, and 42 in 2016. The Nationalists took 17 in 1999, 36 in 2003, 40 in 2007, 56 in 2011, and 39 in 2016. There was no clear predecessor to Just Russia (SRZP) in 1999, but the Rodina party took 37 seats in 2003. Just Russia then took 38 in 2007, 64 in 2011, and 23 in 2016.

The TL:DR of that number-filled paragraph is "nothing unprecedent is expected" 

I'm keeping an eye on things in Germany, Iceland, and Norway; all of whom are holding elections this fall, but as of today, nothing to post. 

Bulgarian election

With a small number of votes left; the results are as follows:

64 - People [ITN] (+13) - Reformist

63 - GERB (-12) - Conservative
36 - BSP (-7) - Socialist
34 - Democratic [DB] (+7) - Reformist
29 - Rights [DPS] (-1) - Liberal/Minority

14 - Stand Up [ISMV] (+-0) - Reformist

This is a big win for the reformists, who have gained 20 seats from the traditional parties. Regardless, they have not reached the 121 seats they need for a majority, standing at 112. What the lead party; "There is such a People" or, ITN, will need is what we in Canada would deem winning a confidence vote. In short, they would need a majority of the assembly to either back them, or, agree not to block them. From what I can tell, DP seems willing to form a coalition government with ITN. I'm not clear on ISMV, but, previous reports suggested they'd be willing to either support it from outside the government, or, join such a government. 

What's important is that with 112 seats, the reformists are close enough to the 121 mark to attempt to govern as a minority. They could, on an issue by issue basis, rely on the support of one of the other three parties. They could also try to peel off MPs from the other parties, and, with time in government, these outsiders can prove trustworthy enough to increase their vote in the next election if they are defeated on a reformist issue. 

As such, this is generally good news for Bulgaria, but it remains to be seen what kind of coalition, if any, will form. 

10JUL2021 updates

 First, I did a quick rough projection of Manitoba using current polls that show the NDP and Tories might swap popvote totals when compared to the previous election. That resulted in the map here; which was quickly drawn on top of the base map from election-atlas. 

Bulgaria 

The election is still shaping up similar to the previous one. Neither the pro-reform, nor the anti-reform camps will likely be able to form a majority coalition. The "difference" this time is that the leading reformist party, ITN is seen as open to a coalition either with the socialists, or the pro-turkish minority liberal party. 

Elsewhere

As often happens for the summer, elections simply die down. Nothing else big/important is coming up until September, when we have Norway, Russia, Iceland, and Germany! 

01JUL2021 updates

 I've decided to do a single-poll projection for Italy, to show what the 3 largest parties being "tied" actually looks like:

84 - 20.70% - FdI (hard nationalist)
83 - 20.30% - Lega (neo nationalist)

76 - 18.80% - PD (moderate progressive)
67 - 16.60% - M5S (left populist)
28 - 7.00% - FI (conservative)
16 - 3.90% - A (progressive)
11 - 2.70% - SI (socialist)
9 - 2.30% - Art.1 (left progressive)
9 - 2.10% - IV (liberal)
7 - 1.80% - +Eu (liberal pro-EU)
6 - 1.50% - EV (green)
4 - 1.00% - CI (liberal conservative)

As you can see, they are not on exactly the same number of seats, but rather, are close to it. 

I am also following other upcoming elections, but none of them are 'good news'; Bulgaria for example, appears set to mostly re-elect the same parliament that could not cobble together a coalition, except with more hard nationalists. Moldova, which has had deadlocked parliaments in the past decade, is possibly headed back to one. Germany has seen the Greens fall back to former levels due to the weakness of their lead candidate, meaning another long and drawn out coalition negotiation post-election, is again possible. 

So, sadly, there is no good news to offer this week. Everywhere pretty much remains deadlocked for the time being.