Become a Creator today!Start creating today - Share your story with the world!
Start for free
00:00:00
00:00:01
63: Broken Voting Systems (Voting Systems and Paradoxes)
527 Plays3 years ago
Voting systems are, in modern times, essential to the way that large-scale decisions are made. The concept of voicing an opinion to be, hopefully, considered fairly is as ancient and well-established as the human concept of society in general. But, as time goes on, the recent massive influx of voting systems in the last 150 years have shown us that there are as many ways to vote as there are flaws in the way that the vote is tallied. So what problems exist with voting? Are there any intrinsic weaknesses in group decision-making systems? And what can we learn by examining these systems? All of this, and more, on this episode of Breaking Math.
Licensed under Creative Commons Attribution-ShareAlike-NonCommercial 4.0 International License. For more information, visit CreativeCommons.org.
Recommended
Transcript
The Importance and Flaws of Voting Systems
00:00:00
Speaker
Voting systems are, in modern times, essential to the way that large-scale decisions are made. The concept of voicing an opinion to be, hopefully, considered fairly, is an ancient and well-established as the human concept of society in general, but there are as many ways to vote as there are flaws in the way that the vote is tallied. So what problems exist with voting? Are there any intrinsic weaknesses in group decision-making systems? And what can we learn by examining these systems?
Introduction with Sophia and Gabriel
00:00:24
Speaker
All this and more on this episode of Breaking Math. Episode 62,
00:00:28
Speaker
broken voting systems.
00:00:36
Speaker
I'm Sophia. And I'm Gabriel. And you're listening to Breaking Math. Before we begin our episode talking about the math behind voting systems, we're going to do a few plugs. First, tensors are the mathematical object used in Einstein's general theory of relativity. And now they're in poster form. We have a lovely 24 inch by 36 inch poster where we talk about tensors. And it's available on facebook.com slash breaking math podcast. Just click on shop. The posters are matte, full color, and make a perfect addition to any office.
00:01:05
Speaker
So for $15 and 15 cents plus $4 and 50 cents for shipping and handling, totaling $19 and 65 cents, you can get this poster for you or someone you know. Check it out at facebook.com slash breaking math podcast when you click on store and see if it's right for you.
00:01:21
Speaker
Also, we're on Patreon at patreon.com slash breakingmath. You can go there if you want to support the show. Even a $1 donation, and I'm not kidding with this, makes a huge donation. We will send you a thank you message. With $1 or more, you can gain access to episodes slightly early and without ads. We also include the outlines that we use to produce the show. We really appreciate your patronage at patreon.com slash breakingmath.
00:01:43
Speaker
Now for news about the show, we're on Twitter at Breaking Math Pod. We are also on Facebook at facebook.com slash Breaking Math Podcast. And we also have an interactive website at breakingmathpodcast.app. And also you can get in touch with us at breakingmathpodcast.gmail.com with questions, comments, suggestions, corrections, or anything else. All right, so what's our topic today?
00:02:08
Speaker
Today we are talking about, as the subject says, voting systems and the flaws in voting
Understanding Gerrymandering
00:02:14
Speaker
systems. And I'm excited to say today we are going to solve all politics and all problems in politics that have ever existed forever. Yep. We're going to do it with Matt. Oh, absolutely.
00:02:25
Speaker
Now, Gabriel, tell me in the audience why voting is pointless. Voting is pointless. Where do I begin? You're going to have to listen to this entire episode, but at the end, we will prove you. We'll prove to you. No, no, I'm just kidding. No, no, it is not. It is not pointless. And that is absolutely not what we're saying, even though at certain points of the episode, it'll feel like that. Correct. Because voting is something that we all hold sacred pretty much across the world. Yeah. Sweden has some interesting elections the way they elect their parliament.
00:02:52
Speaker
Yeah, yeah, yeah, absolutely. But one thing that I wanted to go into too is that the same way that we consider logic and mathematical stuff kind of sacred as a society, but then we have the Godel's impossibility paradox, you know, saying that there's some limits to logic and formal systems. That's kind of the way that I view the paradoxes that we're going to be talking about today. Because at the end of the day, I don't think dictatorships are the best way to do things. I think they're dumb. But I also acknowledge that logic is slippery.
00:03:21
Speaker
Yes, absolutely. All that to say, in any system that we have, there are inherent flaws, should we say, or exploitable characteristics in those systems, and we hope to talk about a few of them today.
Corncabia: A Gerrymandering Analogy
00:03:34
Speaker
For our very first topic, we are going to talk about gerrymandering, something you may have heard about recently in the news. We have a wonderful analogy involving a magical kingdom where we illustrate what gerrymandering is. Before we do that, however, we'd like to talk about the term gerrymandering and where it came from.
00:03:51
Speaker
All right. So the word gerrymander actually started off as a portmanteau. A portmanteau is something like brunch, which means halfway between breakfast and lunch. You take the BR from breakfast and the unch from lunch. So, uh, Jerry and salamander is what it means. So why salamander? Well, basically there's this dude, uh, governor Elbridge Jerry, who was changing the outlines of districts in a way that looked like a giant salamander surrounding the state. But, um, so the type of election we're going to be talking about in this case is one where you have different districts that all have the same vote, right?
00:04:21
Speaker
Yep, absolutely. And of course, what you said earlier is you specifically divide a region in a certain way that may affect the outcome of the vote one way or the other. Yeah, and you might wonder why people set up systems like this in the first place. The answer is because most of these systems were come up with in the 18th century.
00:04:40
Speaker
They're old systems. They have not been revised in a lot of time and they have a lot of flaws, especially systems in Jerry. So for example, congressional districts are prone to gerrymandering, right? Yes. Where each congressional district gets a vote depending on who wins the majority and then they all vote together. I mean, this is just such a flawed method for so many ways. I mean, I will say that any system that can be gerrymandered probably deserves to be gerrymandered.
00:05:05
Speaker
Ah, I see. Interesting. But we're going to go a little bit into this whole system. Yeah, we've got an analogy here that is a fictional analogy. And without further ado, we have here an analogy or a story to illustrate what gerrymandering is. Imagine a large and full city of magic and whimsy.
00:05:25
Speaker
We'll call this land Corncabia. Here's a large mountain range at the north end of the city. Snowy mountains, right? Yes, exactly. Next to the mountains, just south of them, are hills. There's a business district in the southwest next to the slums.
00:05:41
Speaker
Also, there's an orchard to the northwest and farmland due west. On the east side of the city, there's a beach just south of where the forest starts. And south of that, there's a large living area, mostly condominiums. South of all of this is a large desert. Dividing east from west, from the mountains down to the desert, is a large river spanning the town. And I just want to announce right now, we will give a free poster to anybody who draws that map for us.
00:06:07
Speaker
Actually, let's say the first five people who sent us a map. Yes, yes. Send us a map of Corncabia. And I don't even care how good you draw this, right? You can draw with crayons. Send it to us and we'll do something with it. We love receiving stuff from y'all. Exactly. And you'll get a free post for the first five of you.
00:06:27
Speaker
Yeah. Okay. So we have this, this area that we call corn copy. All right. And let's say that the following 15 areas, that's a, that's a lot of areas, but I'd say the following 15 areas have roughly equal populations. Right. So the East mountains, the West mountains, the East Hills, the West Hills,
The Impact of Redistricting
00:06:46
Speaker
the East forest, the West forest, you see a pattern here, the forest Hills, the farmland, the orchard and the business district and the slums, the beach, the condominiums, finally the West desert, East desert,
00:06:55
Speaker
Did you catch all of that? Did you get all of that there? Yeah, I'm sure everyone remembered all of that. So what we're trying to say here is in the town of Corncobia, you've got 15 distinct little regions here, right? Yep. So there are elections in the land of Corncobia. Trouble has been brewing for the residents of Corncobia. There have been dragons and whimsical magic worms in all manner of irritating magical vermin. Plus issues of access and elitism which have dominated the talk. There's a peasant, Habenrud, who is running for Lord Mayor.
00:07:25
Speaker
There's also a wizard, Scrooge McGandalf, who's also running. And blood is boiling. That's actually a magical thing. It doesn't mean people are angry, right? Oh, no, no. They're both. Their blood is spurting out of their arms. They're just like willy-nilly. And they're mad. Okay. Just to clarify, because I know it's a magical kingdom. Oh, yeah. It's very magical. Everybody has teeth on their shoes and crap. Very good.
00:07:52
Speaker
It's whimsical. So now I know that in this big area of Corncobia, I imagine that there are different areas that support different candidates and different interests. So I would say that the peasants of the land, there's a lot of peasant support in areas like the orchard or in the mountains west and east, the hills west, the farmland, the forest west, desert west, and the slums. We just chose all the west areas.
00:08:19
Speaker
Yeah. And a lot of cities are divided like this. That's kind of why I chose it, right? Yeah. East and West. Yeah. But the business area, which is in the West, I'm assuming would be for the wizard. Okay. I mean, his name is Scrooge. Yeah. Yeah, exactly. So of course, yeah. You know, in all these regions, you'd find support for the wizard in the Hills East, the Forest Hills, Forest East. Yeah. You're right. All the East side places, the beach, businesses, desert East, condominiums.
00:08:44
Speaker
Now, is there any trends here in these places? Like would you say it's sort of like in America where like say in Oregon, you've got the, um, the, uh, populous, uh, the populated areas that are typically a little more blue and then the more, uh, rural areas that are more red. Would you say that that trend? Well, in this town, it's more like, uh, old style Kentucky, right? Okay. Where the workers are, um, more leftist and the elitists are more right. Not to get too political, but.
00:09:10
Speaker
Yeah. Okay. Okay. I see.
Gerrymandering and Democratic Fairness
00:09:13
Speaker
Yeah. There's a whole lot of ways of slicing up this cake here. Yeah. And what we see basically most of the support in the West, there are, um, so there are eight districts in the West, the orchard, the mountains, East and West, the Hills West, the farmland, the forest West, the desert West, and the slums.
00:09:31
Speaker
all with about equal populations support Hoppenrud, the peasant, versus the wizard Scrooge McGandalf is supported by the forest hills, the forest east, the beach, the businesses, the desert east, and the condominiums. And also we will point out right now that
00:09:48
Speaker
If you just count the amount of people, more people support the peasant. That reminds us of elections in the United States, for example, when you have more people actually supporting one candidate, even though the way our system is set up, the electoral college may elect somebody else. So that's happened a few times in our past.
00:10:05
Speaker
Oh, yeah. I mean, it hasn't happened forever, though, not since 2016. And before that, not since the year 2000. Yes, yeah. It happens way too frequently here. Yes, it does. Left or right, you must acknowledge this. Yeah, absolutely. Absolutely. So now let's divide these places, these populations into voting blocks. So why would we do that? Just off the bat, why would you ever do this? The answer is it's a very old system thing. But basically, we're dividing these populations into how many voting blocks?
00:10:31
Speaker
Three, three voting blocks. Of five districts, right? Yes. And we're just dividing it into the districts to simplify it for y'all. I mean, a way that cities, cities are divided into districts by a lot of ways. I mean, there's counties, zip codes. There's actually, you can even look at it on the, there's a census map from 2010 that gives a racial data of everyone. And you will, if you look at your town on there, you'll most likely see actually racial lines within your own town. Yep.
00:10:58
Speaker
If you're not aware of this, I would check it out at demographics.virginia.edu slash dot map dot map or capitalized. Yeah, it's amazing. And also, if you want to see a few places, if you know any place like New York City or San Francisco, you can see how actually still fairly segregated they are in terms of where different ethnicities are located. So if you're curious how homogeneous your region is or your city is, check out that website. And again, what's the website?
00:11:26
Speaker
demographics.virginia.edu slash capital D lowercase O T capital M lowercase AP dot map. And I wonder where they pull their data from the U S census or census. Yeah. Okay. Okay. Very good. Yeah. Fascinating site. So anyways, uh, yeah, this fictional place here, uh, I wonder, would it be more segregated or more, I guess, at least in terms of the reasons that we, I guess it could be segregated pretty well between economic classes, right? Yes, I think you're right.
00:11:54
Speaker
All right, so now let's say let's divide it into districts, right? Let's do it. Let's say the orchards and farm right to the west, the entirety of the mountains, and the west hills are a block. This would vote entirely for the peasant, right? Yes. Oh, and one thing that we forgot to mention before is that the east mountains are supporting the peasant as well.
00:12:15
Speaker
Okay, sure, so there's one east label place that is supporting the peasant. Interesting. Yeah, and one less west label place, too. I mean, it's not labeled it, but the business area is on the west, right? Okay, sure thing. In support of the wizard. But yeah, we're going to say that the mountains, the east and west mountains, the orchards, the farms, and the west hills are a block. This block votes entirely for the peasant. Yes.
00:12:38
Speaker
So we got one big vote for the peasant. Another block will be west desert, slums, and all the forest areas, west, east, and hills. So that would vote three and two a favor of the peasant, right? Yes. And the rest are voting entirely for the wizard.
00:12:53
Speaker
Yep. So that's two votes for the peasant and one for the wizard. Yes. So the peasant would win here, right? Yep, absolutely. Now, suppose that we redistrict. The hills in the forest will be one block. This block will vote three to two in favor of the wizard. The beaches, the businesses, the condos in the desert will be another block. This block will also vote three to two in favor of the wizard. Finally, the orchard, farmlands, slums, and mountains will be the
Ranked Choice Voting and the Condorcet Paradox
00:13:18
Speaker
last district, which votes entirely for the peasant.
00:13:21
Speaker
So, now that two votes for the wizard and one for the peasant, the wizard wins. Yeah, see, even though there was not a majority vote for the wizard, the wizard still won. Yes, yeah. And what we should have mentioned as well is the blocks, you know, vote collectively.
00:13:37
Speaker
Oh, yeah, absolutely. Yeah. Which is like you might wonder why would anybody do this here from somewhere that's not the United States or the countries that do this. And we actually have even a worse situation not to editorialize too much in the United States called the electoral college, where your vote can be worth as much as three times as somebody else's vote, depending on what part of the country you're from. Yep. Yep. We'll talk about that on a problem episode.
00:13:57
Speaker
Yes. So as we can see, as we said earlier, depending on how you district things, you can have different outcomes in your election, which is a huge incentive to figure out how districts are made. So in this town, I wonder who decides, in this fictional town, who decides the districts?
00:14:13
Speaker
Yeah, and that's always the thing, right? Is that the people who are deciding the district have a lot of power that's not accounted for, right? Yes. I mean, we have the balances between legislative, executive, and judicial branches. In theory, yes. In theory, yes. But in practice, we have a lot more moving parts that we really have to consider. Yep.
00:14:35
Speaker
Hey, Breaking Math fans. First, I want to thank you for listening. I have an important message for everyone. You can start your own podcast right now with Anchor. Anchor lets you create and distribute your own podcast. Just get an idea, record, and upload. It's just that easy. Anyone can do it. I'm on my way to accomplishing my dream, and you can too. Just get on your device's app store and download Anchor. It contains everything you need to make a podcast. With Anchor, you can put your podcast on all the big platforms.
00:15:03
Speaker
Apple Podcast, Spotify, Google Podcast, Amazon, and more. Reach the whole world with Anchor. Best of all, Anchor is free. You have nothing to lose with a free platform. Download the Anchor app or go to anchor.fm to get started.
00:15:23
Speaker
There are other flaws in voting systems besides gerrymandering. For example, in what's called a Ranked Choice Voting System, there's a flaw that emerges known as the Condorcet Paradox. A Ranked Choice Voting System is one in which all voters can indicate their preferences in order. Australia has a version of this, as does New Hampshire and many other places throughout the world. We came up with an example to illustrate a Ranked Choice Voting System and how the Condorcet Paradox may emerge.
00:15:51
Speaker
Imagine a place that has three candidates for mayor. Candidate one, Aoki. Candidate two, Bronson. And candidate three, Chavez. Aoki is a candidate who wants wide sweeping financial reform for local banks. No banks can take advantage of the common folk. Yes, very good. Bronson is a candidate who wants to repair infrastructure.
00:16:19
Speaker
We need to drive on bridges not made of rubble! Chavez is a candidate who wants to invest in small businesses. Save small business!
00:16:29
Speaker
Wow, they all sound the same. The clones don't tell their constituents. Okay, very good. But let's say that the voters, let's say 98% of them can be divided about equally into the following three groups. Voters in the first block would prefer Aoki to Bronson and then Bronson to Chavez. Since they prefer Aoki to Bronson, they probably want financially poor
00:16:52
Speaker
They probably want financial reform more than infrastructure repair, right? Yes. And Bronson de Chavez means they want small business investment less than infrastructure repair. Yep. So we can imagine that this person, typical of this group, doesn't care much for or about small businesses, sees a need for infrastructure repair, but would benefit the most from financial reform of banks. A person typical of this group might be someone who works as a server at a restaurant without a bad time with their bank.
00:17:17
Speaker
Suppose that voting block 2 prefers Bronson to Chavez and Chavez to Aoki. This voting block's main interest is infrastructure. They also have an interest in small business, but have no interest in financial reform. Suppose that a person in this block would be, say, a driver who works for a small business and has had no issues with the bank.
00:17:39
Speaker
And finally, the voting group three that we're considering prefers Chavez to Aoki and Aoki to Ronson. This person's main focus is small business. They also want financial reform of local banks, but don't care as much about infrastructure. We can assume that this person is some, let's say a person typical of this group to discuss later is a florist, right? Who has their own small business who has struggled with their business. Yes.
00:18:02
Speaker
Now, so imagine that these three groups of voters cast their ranked votes. Now let's see who prefers what. We can see that both the florist group, the third group that we talked about, and the servers group prefer Aoki to Bronson. That is because they prefer financial reform to investment in infrastructure.
00:18:24
Speaker
We can also see that both the servers group and the drivers group prefer Bronson to Chavez. That is to say they prefer infrastructure to small business. So we know these two things, right? That the majority of the group prefers Oki to Bronson and the majority of the group prefers Bronson to Chavez. Yes. So we might conclude from just this data alone that the three groups in general prefer Oki to Bronson and Bronson to Chavez, right?
00:18:46
Speaker
Yes. So we could think that at this point that we might be able to rank them Aoki, Bronze and Chavez. Yes. However, there are also two times as many people who prefer Chavez to Aoki, the florists and the drivers, that is those who prefer investment in small businesses versus those who prefer bank reform.
00:19:03
Speaker
as those who do not. Therefore, the voting preferences are cyclical. Yeah, because most people want Aoki more than Bronson. Most people want Bronson more than Chavez. And most people want Chavez more than Aoki. Yes, absolutely. So again, this is what it's called, the Condorcet Paradox. So how likely is it? You might think this might not happen.
00:19:27
Speaker
Yes. One estimate based on something called the impartial cultural model estimates that the preferences are distributed evenly amongst the candidates 8.77% of the time for three candidates.
00:19:41
Speaker
Yeah, that's to say that 8.77% of the time, about one time in 12, you'll have an election that has a cyclical kind of preference. Yes. Which might seem like a problem, right? Because it means that there's no way of respecting everybody's rankings. Yes. Yeah, yeah, absolutely. Yeah. And actually, I was curious if in your research, if you came across any solutions for these kinds of paradoxes that come up.
00:20:01
Speaker
Um, honestly, these paradoxes, as we'll see in the next couple of sections are sort of intrinsic to the way that we do decision-making. Interesting. Fascinating. Fascinating. So, so the point of this section is that even in ranked elections, there is sometimes no way to come up with a general order. This is almost, as you said earlier, similar to Godel's incompleteness theorem or, you know, any theorem that talks about the, uh, limits of our knowledge.
Arrow's Impossibility Theorem
00:20:27
Speaker
Yeah, absolutely. Or the limits of, uh, that are possible with mathematics and with human knowledge.
00:20:31
Speaker
So I don't mean to be too cynical here, but that's still, that's just, you know, that's part of voting and part of human behavior. Also, yeah, because we kind of are a part of this larger organism than ourselves. And I don't think it's really that surprising that we have to give up maybe even, maybe we can even consider it as some individuality when we're making group decisions. Yep. Yep. Absolutely.
00:20:53
Speaker
Because if we had absolute individuality then everyone's things would be respected But I mean we've learned we learn as children right that you can't always have a group decision that makes everyone happy Yep, pretty much pretty much. Yeah, and in fact there's even a saying that I think is often attributed to Churchill and that's saying is that Is it democracy is the worst of all forms of government except for all the other ones? Oh, yeah something like that. Yeah So yeah, it's hard for everyone to get to what they want. There's also some sort of song from I think the 70s
00:21:23
Speaker
You can't always get what you want. That's it. You can't always get what you want by the Rolling Stones. So much fun to deconstruct these voting systems and make everyone skeptical of their own systems. Let's talk about one more paradox while we're at it. Arrow's impossibility theorem. So this shows that even ranked voting systems, no matter what, will have limitations.
00:21:47
Speaker
Sure. Absolutely. So first let's talk about some, we need some background to talk about arrows and possibility theorem. Um, so we need to talk about the idea of choice, right? Yes. So let's say that we have two choices for a candidate of something, right? Yes. X and Y. Yep. We can either say three things at any given time, right? Yes. We can say X is preferred to Y, right? Yep. We can say Y is preferred to X. Yes. But we can also say that X is indifferent to Y.
00:22:13
Speaker
Yeah, to boil it down mathematically, those are the only things that you can have when you have two choices in front of you. Yeah, and so Eero, Kenneth Eero in the 1950s came up with a few axioms, right? So he defined two relations, P and R, on decisions. So X, R, Y means X is preferred to, or at least indifferent to, Y.
00:22:35
Speaker
So for any two, all x and y, x are y or y are x, right? Yes. Because either x is preferred to, indifferent to y or y is preferred to or indifferent to x. Okay. That doesn't strike me as bad, right? Not yet, no. So x came to. For all x, y, and z, x is preferred to or indifferent to y and y is preferred to or indifferent to z means x is preferred to or indifferent to z. Okay. So let's say x is preferred to y and y is indifferent to z, right? Yes. Then x would be indifferent to z. Yes.
00:23:03
Speaker
So it makes like, if you do it with any combination of these things, it'll make sense.
00:23:08
Speaker
Okay, okay, sure. So then we define the relation P and I. P means not X, Y, or X, meaning that Y is neither indifferent to nor preferred to X. Okay. Meaning X is preferred to Y if X, P, Y. Yes. So X, I, Y means X and Y are indifferent to one another, basically by saying that X, I, Y, if and only if X, R, Y, and Y, R, X. Okay. And remember R in these situations is like an operator, like plus or minus or times are divided. Yes.
00:23:37
Speaker
So now we gotta talk about the concept of social welfare. So the concept of social welfare can be defined as a function that takes in two or more rankings and spits out a ranking. And you can think about this as any way of voting, right? So we vote and we rank who won the most votes. Basically any voting system, like pretty much any voting system except for voting systems that we'll talk about later, which rely specifically on giving like star ratings or one to 10 ratings are a part of this.
00:24:07
Speaker
So we want the social order function to meet a few conditions, right? We want it to be defined on any sets of orderings, meaning that there's no vote that's illegal. If for an individual, X rises in their ordering and X is preferred to Y with respect to the social ordering before X rises. So what that means is, let's say we have a certain vote that'll get X more than Y, right? Or Y more than X. Is it preferred to Y or Y preferred to X?
00:24:34
Speaker
So what we're saying is if X rises in the orderings, then X being preferred to Y will still remain true no matter what. And that's something that we would like to have. We also want to say that the social welfare function is imposed if X is X or Y is true. That is to say X is better than or indifferent to Y, no matter what the individuals say. Yes.
00:24:57
Speaker
An example of this is dictatorship, where one person determines what the entire group thinks. So any range voting system will not follow all these conditions. So namely, just to make it simple, the three following conditions, only two at most can be followed. That's why the first one is, if X is favored over Y for everyone, then so it is for the group. Meaning that if people support Aoki over
00:25:24
Speaker
Zamora, then Oki will always be more than Zamora in the final results. Okay. Then if X and Y do not change in order for everyone, then if other preferences do change, the order between X and Y do not change.
00:25:40
Speaker
meaning that if we have Aoki, Zamora, and Feinberg, then if we remove Feinberg from the equation, let's say Feinberg dies, right? It's not gonna change the ordering between X and Y. There are certain voting systems such as voting systems where you sign points like one, two, three, and four to everyone, and tally them up, where let's say you have votes one, two, three, and four, right? Assigned to A, B, C, and D with half of the candidates, and half of them consigned to D, B, C, and A.
00:26:06
Speaker
X will win most of the time there. But if you remove Y from the equation, then X and Z will be tied, which is against the system. Yes. And then finally, there is no dictator. One person decides what's best and what everyone else wants. For the group, yeah. For the group.
00:26:24
Speaker
Now, it seems a little hopeless when you think about this, right? Yes. Now, yeah, you're right. You're right, which is why I'm glad that there is a quote by Arrow that he leaves us with. And the quote is, most systems are not going to work badly all of the time. All I've proved is that all can work badly at times.
00:26:46
Speaker
Finally, we'd like to talk about one more thing called Gibbard's theorem. Now, Arrow's theorem doesn't hold for cardinal voting systems.
00:26:55
Speaker
Now those would be like stars, right? So in ranked voting systems, if you like candidate A way more than candidate B, that tallies in the same way that if you like A marginally more than B, right? Yes. But there are systems where you could assign like stars or ratings from zero to one to people, right? Yep. Or like one to 10. Yeah. So like in that system, saying that somebody is like a nine versus somebody else being a three is completely different than saying that the first person is a three and the second is a one.
00:27:22
Speaker
Yes. Yeah. Yeah. Absolutely. Now there are still limits to collective decision making. Gibberts theorem states that for any deterministic voting process, meaning we're not throwing dice into the voting system here. And remember voting process means anything here, right? It means going for a sandwich with your friends and deciding what place you go to. Yes. Anything can be considered a voting system under this theorem. Yes. Like we're not considering formal ballots. We're just considering group decision making processes that are deterministic.
00:27:52
Speaker
Yes, yes. Um, yes. So one of the following must be true. First, the choice in the process is restricted to two outcomes. So that means that we only have two choices. Let's say we have Sparos versus, uh, McDonald's. Yes.
00:28:08
Speaker
Secondly, the process is dictatorial, meaning that one agent influences the outcome. Yeah, which would suck. It kind of defeats the purpose of voting, right? Yes, perhaps it does. So maybe it's the leader of the pack who's the most charismatic, or if there's consequences for going against what the leader wants. Third, the process is subject to strategic voting. That is, strategic voting is voting to stop an undesirable outcome.
00:28:32
Speaker
And this happens a lot in the United States, right? A lot of people support third candidates, but will vote for the main two candidates because they consider voting for the third party candidate throwing their vote away. Which is something that ranked voting systems, ranked choice voting systems can actually help with. Yes. Yeah, absolutely. And again, once you've got ranked choice voting systems, you've got other problems as well. So, implications. Essentially, the world and systems in the world are not perfect.
00:29:00
Speaker
No, yeah. And we have to think about decision making processes because like there's decision making processes where everyone is equal. There's decision making processes where you vote for someone to decide for you in certain situations. It just shows how much flexibility there is and how much we actually have to pay attention to our voting systems. We can't say that anyone voting system is a panacea, right? Correct. Yes.
00:29:22
Speaker
because we really do have to analyze our democracy mathematically, because gerrymandering is being taken advantage of in many different ways. Absolutely. So no matter what, there's ways of slicing the cake to take advantage of certain things and be unfair.
00:29:41
Speaker
Speaking of slicing the cake, that kind of reminds me of something. So let's say I have a big cake, right? Yes. And I say I have one other person I'm splitting it with. Yes. A really fair way to split it would be to have one person divide the slice or the cake into two pieces. Yes. And have the other person choose which slice they want, right? Yes. I like that. But if you have three people, how do you do that?
00:30:02
Speaker
Ooh, great question. How do you do it? There's no way. It's impossible. There's no fair system that works on the first try with this. Okay, interesting. Just draw straws, right? Well, not even draw straws, but it's like one of those things where you always have flaws. It reminds me of the two-body system versus the three-body system in astrophysics.
00:30:23
Speaker
Yes. You want to talk about it real quick? Oh, absolutely. Yes. Yes. So the two body system we know with with mathematical precision, perfect precision, where orbits where two planets will move, assuming that we know their their masses and the distance between them and the additional relative velocity.
00:30:42
Speaker
Yes, and their relative velocity, assuming that they are an isolated system. But once you throw a third body into the equation, it's chaos. You no longer have perfect predictability. Yeah, in fact, I think past is something like 10 to the 15th years or 10 to the, some huge number of years. We don't know the fate of the solar system, right? Yes, that's correct. Yeah, for example, in 3.3 billionaires, there's a one in a hundred chance that Jupiter's gravity can make Mercury's orbit so it's eccentric that it collides with Venus.
00:31:12
Speaker
Wow. Goodness. So there's a lot of things that could happen. Yeah, absolutely. So it's an imperfect world. So goodness, I guess with that, where do we go from here? How would we advise a society in the future how they do things?
00:31:26
Speaker
Look at how you already do things, right? Look at how your society, your culture makes decisions and maybe base it on that. Because a boating system is like any other system, right? It's subject to flies and we have to recognize the flies. And also I'd say we should make systems as flexible as possible sometimes, right? To have a certain amount of flexibility. And also we need to recognize that we cannot make perfect systems on the first try. We're not some kind of God, right? Correct.
00:31:52
Speaker
Yes, I'm kind of reminded of the way Douglas Adams writes in his book. What was the book? Hitchhiker's Guide to the Galaxy, where you have the Improbability Drive. Oh yeah, the Infinite Improbability Drive, which is generated. It was created because the Infinite Improbability Drive was determined to have a finite improbability in the books. And so they used the finite improbability drive to create the Infinite Improbability Drive.
00:32:19
Speaker
by figuring out how improbable the infinite improbability drive was and putting that value into the finite improbability. Oh wow. Okay. Which is, it's, it's just a bunch of sci-fi fun, but honestly you kind of have to get into that weird mindset to really talk about voting systems. Yeah. And in order to, to, to, uh, figure out, I don't know, it'd be interesting to apply that to a voting system.
00:32:42
Speaker
Voting systems have many difficulties. These arise not from how much effort was put into the voting systems, but, in the existential sense, as a result of how voting systems work. Voting systems are a way of formalizing the way the decisions are made, but, like all deterministic systems, have a way of trending towards intrinsic but meaningful limits. All this to say, we need to be mindful not only when we vote, but when we decide how to count votes.
00:33:08
Speaker
I'm Sofia. And I'm Gabriel. And you've been listening to Breaking Math. All right, so poster, you could get that at Facebook.com slash Breaking Math podcast. Also, please send us a picture of corn cobbia. I don't care how rough it is, it would be fun and you get a free poster, right? Absolutely. And without further ado, or I don't know, I guess that's a wrap.