Chapter 23 More Nontransitive Paradoxes

When I first looked at the title of this chapter I had not a clue what it was about, but it was really not that hard of a concept to understand. For something to be transitive it must be true the signs that you use when you were in math class such as less than, greater than, or equal to signs are examples of this. So in easy way to put this is a=b, b=c, then a=c. and you could switch the signs with greater than or less than and it still be the same. These are examples of transitive things so then what are examples of nontransitive things then? Lets assume Person A loves Person B, but Person B loves Person C. This does not mean that Person A will also love Person C. Another example if the simple game of rock paper scissors. One is always dominate over the other, but not one has complete dominance over them all.
             A nontransitive paradox is when we may think that something is transitive but it is actually not. An easy way to grasp the concept of a nontransitive paradox is when we vote. Lets assume we have three people to vote for and 40% of the population chose one, but that leaves 60% to split between the other to individuals. If the other two individuals receive 30% a piece then the person with 40% wins. This is a nontransitive paradox because even though that 60% of people didn't vote for him he still won. One thing that is talks about in the chapter is the five conditions, but it never actually stated what these five conditions were which kind of confused me. There were multiple other nontransitive paradoxes in this chapters that dealt with cards and coins that was very interesting to read about. When they were talking about the coins it was easy to understand because they broke it down first to just HT or TH, but ended up going all the way to TTTT and HHHH and all the possibilities in between which was quite an interesting read.
          
            In Social Choice and Individual Values, by K. J. Arrow goes on to explain in further detail on this topic. He explains more about how the voting aspect is nontransitive. In the passage above I talk about the Five Conditions which were Arrows, the author. This also goes into more detail about his "impossibility theorem" which was an interesting read, but didn't have much to do with the subject at hand.
            In Theory of  Voting by R. Farquharson it goes on in further depth about voting. Unlike Arrow however he did his work based on ordinal preferences. It was neat how the same topic of voting was able to be discussed on different views. I liked how though they just kept it simple for us to use in this chapter.
         
            In The Theory of Committees and Elections by D. Black it also goes into more detail about voting. He used the social choice theory like Arrow did, but differently. He used it so that it would cover everything for a resulted outcome. Like I mentioned above I liked how they took just had the main point in this chapter rather than all the different views they all had.

As you may have seen all three of these go into more detail about the voting topic, but this is no coincidence. All three of these books were around the same time frame between the 50's and 60's. Arrow was first and that was why there are Arrows Five Conditions, but these other people had different views than he did.