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The best example of this is the presidential election. For a lot of people, this might be the most important slide of the day, so I'm going to go into detail about this data.
I should mention that I got this idea from Mohit Shetty, who wrote a piece about the election in the Huffington Post.
For example, when the poll was taken in March , there were 14 Republican candidates. The population sample for these polls was drawn from Republican voters, so it's representative of the national population of Republican voters.
If we look at the plurality-voting contest, from July [onward], he's always in first place. But if you look at [whom voters are willing to support, i.
Keep in mind that in the very beginning of May , Trump won the nomination because he had won enough delegates. So the March and April polls were basically the last ones before that.
So plurality voting and approval voting are two different generalizations of majority rule, and they lead to two very opposite outcomes. This is why voting systems really matter.
Another thing I want to emphasize is that [approval voting is] ideologically neutral. In , approval voting probably would have helped the Republican candidate win.
I want to make a prediction that you can hold me accountable to: Given the plurality structure and the nomination process, I'm predicting that for the Democratic nomination for president, no candidate will receive a majority of regular delegates.
I think that the decision will be made either by non-elected super delegates or by a contested convention. If people are forced to vote in a particular way, it doesn't matter which aggregation function is used.
I'm going to talk about a lot of this with respect to intelligence explosion hypotheses, but it's not dependent on those.
Ray Kurzweil is a director of engineering at Google and probably the most famous proponent of intelligence explosion.
Among his several predictions, two are basically that 1 knowledge and technology growth are accelerating, and 2 that human lifespans can be indefinitely extended into the future.
What's also interesting is that Condorcet [came to that conclusion] in the s. What's even more interesting is that he mathematically modeled it with his Jury theorem.
So how does this theorem go? The Jury theorem says that as you increase the number of agents, the probability that the majority is correct approaches the numeral 1.
So basically, what Condorcet said was we want to make society into a multi-agent system that's accelerating towards judgment. Therefore, we need institutional change.
He believed that we need to improve honesty in society, and that we shouldn't [tolerate] dishonesty except when it improves honesty in the long run.
Condorcet wanted [more] independent thinking. One of my favorite examples [of the changes he advocated involves voting]. When masses of people voted, he didn't want them to meet in assembly.
He thought they should vote by mail, because he believed that if people meet in assembly, then they form parties.
And if they form parties, then they're going to stop thinking independently. That would have been more likely by mail, because back then, long-distance communication was difficult.
Another thing that Condorcet argued was that regardless of race, sex, or sexuality, everybody should be allowed to vote. This would increase the number of voters and let us get closer to the axiomatic approach [where the probability of a correct majority decision becomes 1].
The math isn't discussed in Sketch , and that's why it has taken so long for people to figure this out. But Sketch is probably his most famous philosophical work.
In it, he basically argues that societies that tend to fulfill the Jury theorem, or at least have institutions that promote the conditions of the Jury theorem, tend to be ones where knowledge and technology grow at a fast rate.
Sites that don't have this property, like late-antiquity European societies, tend to have reversed or slowed growth in knowledge and technology. There are several people who have generalized the Jury theorem.
I've given you a set of these people and they have very excellent papers, which I encourage you to look at if you get the chance.
But I think we should focus less on trying to generalize the Jury theorem, and more on [its application to] multi-agent systems. He developed several theorems in terms of information aggregation.
An example showing how the multi-agent framework is helpful, I think, is a preschool. Let's say you have a class of preschoolers — three-year-olds.
You also have an adult teacher. We'll also assume that among the three-year-olds, there's a four-year-old who bullies all of the other kids.
One question given these background conditions is: What is the best way to aggregate information? We can pretty clearly say that majority rule is not going to work, and we can say this because there's low information and the opinions are highly dependent on the opinions of the bully.
We want to implement institutions that decrease the dependence of the students on the bully's opinion and increase the students' knowledge of crossing the road.
This might take a few years, but [we can identify] the institutions we want in place so that the three-year-olds will start making this judgment independent of one another.
Maybe after five years, when they're eight, they can figure out when to cross the road and when not to cross the road.
You might say that the multi-agent framework is overkill when you're dealing with this preschool class, or when you're dealing with the classic Jury theorem conditions.
But that's because they're corner cases for simple illustration. When you deal with really complex, real-world situations, you're going to have a lot of interweaving interdependencies in judgments.
Here I lay out a very oversimplified model for what you might do. But unfortunately, the vast majority of cases are going to require approximation algorithms.
There [have been some applications] to other kinds of problems that require the epistemic aggregation of information, but not to problems that deal with people with dependencies and things of that nature.
This class is taught by Andrew Critch and Stuart Russell. There are a number of other equivalent definitions that can be used in the case that we are dealing with a finite rational group.
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If you see this placeholder for a long time, file an error report at the error reporting page. Categories : Semistandard definitions in group theory Semistandard terminology Group properties.
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Contents 1 Definition 2 Examples 3 Relation with other properties 3. An element with the property that it is conjugate to any other element that generates the same cyclic subgroup is termed a rational element.
A rational group can thus be defined as a group in which all elements are rational elements.
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