Arrow's Impossibility Theorem | Infinite Series 

I feel these videos would be even better without colourblindness.

Supplementary explanation for anyone who might be feeling lost after watching: I think what they did a very bad job of is explaining where they got the ballots that they used. There was only one real set of "randomly generated" (made up) ballots. Everything else was determined from that. What they called "test cases" were not randomly generated. They were completely predetermined. They were an exhaustive list of permutations where some number of voters had purple first and the rest had purple last. The overall rankings of the test cases WERE made up. We knew that in the first test case when everyone rated purple last, purple would HAVE to be last (unanimity). Similarly, we knew that in the last test case, purple would HAVE to be first. Therefore, somewhere in the middle there, purple would have to switch from last to first. (This is of course assuming that they can't be anywhere in-between. They set this up as a homework assingment and said "figure that part out yourself" but given how essential it was to the proof, they REALLY should have explained it outright) But again, purple has to switch from last to first somewhere in the middle. In the video, we are told that the shift happens from round 2 to round 3. Again, this was made up. It could have just have easily been between any other rounds, it just would have implied a different voter as the dictator. This is what ensures that the proof is a general one. The "election results" that are presented right after this are also made up. Presumably, they were contrived to be convenient to use in the logical steps to follow. Just because it's convenient doesn't mean it's not a generalized proof, though. In this setup, voter 2 IS the dictator so any set of election results would ultimately show this conclusion. It just could be more difficult to prove with a different set of election results. From this, there's just a lot of logical inferences, all of which are difficult but valid. I had to listen to just about every sentence in the proof part of this video two times or more. The proof is all there, just not explained super clearly in my (and obviously a lot of other) opinions. Good luck!

i dont understand voting with colors.. maybe you could use animals like lions... just saying

We could vote for a voting system, then vote. But first, we need to vote on how to vote the voting system....

How did we determine that purple switched from last in round 2 to first in round 3? I know that the challenge was that it had to be either last or first, but my question is: why did the switch happen in round 3.

Let's say that the three candidates are red, green, and blue, and that green is the polarizing candidate. Suppose that, in some election, the outcome is R>G>B. By IIA, you can run the election on just red and green, and green loses. And you can run it on green and blue, with the same voters preferring green as before, and green wins. By unanimity, there are at least one voter who prefers red to blue, at least one voter who prefers green to blue (and also to red, since green is polarizing), and at least one voter who prefers red (and blue) to green. Switch the preferences of all voters who prefer red to blue so that they prefer blue to red, without affecting whether they rank green first or last. By unanimity, blue is now above red. But by IIA, red is still above green and green is still above blue. R>G>B>R>G>.... Contradiction.

If you plan on further discussing desirable voting system qualities, please consider Bayesian Regret. It doesn't get much attention, but you could make a strong argument that regret is more important than any collection of the standard criteria.

In my unimportant opinion, the example ballots might be easier to follow if each color was accompanied by its own shape.

Proof that a polarizing candidate must be ranked first or last: Suppose A is a polarizing candidate who is not ranked first, so that candidate B is ranked ahead of A. Let C be a third candidate. Our goal is to show that C is ranked ahead of A for any ballot. Consider a ballot which I will call Ballot 1. Since A is polarizing, for any given voter, this voter either prefers A over B and C (in the case that A is first), or she prefers B and C over A (if A is last). Equivalently, for every voter, A > C iff A > B, and A

What I absolutely LOVE about Infinite Series is the incredibly esoteric topics that come up as compared to many of the pop-sci channels. I.S. is definitely one of PBS' best short format video series.

4:30 - wow plot twist. I was under the impression that Arrow's Impossibility Theorem applied to all voting systems. I am happy to learn that it doesn't and that the Cardinal Voting System may be superior to ranked voting systems.

PBS doesn't get enough Thanks for their free content.

Orange was definitely the polarizing color.

So, for 1 Million dollars you'd get punched in the face? We all have our limits. I think my get punched in the face is more like -100,000

That was a lot to take in. Definitely going to rewatch like five more times!

Please bring this channel back

This is why ordinal (rank) voting is dumb. Cardinal systems are superior in every way

I think there's a mistake in the voting diagram at 6:30 - between rounds 4 & 5 voter 2 changes the order of R > G > B to G > B > R

please cover cardinal voting systems!

can you do a video on cardinal voting

What would happen if candidates were allowed to have the same ranking?

"You can focus on the important things like how to pack spheres in a 9 dimensional space" best Squarespace plug *ever*.

I can break this theorem - everybody ties! (and is a winner and loser at the same time) 1) no dictator - nobody's opinion matters 2) IIA - no candidate matters 3) unanimity - if A is ranked higher than B by everyone, B cannot be ranked higher than A in the end

It's worth noting that the theorem needs there to be more than two parties. If there are only two parties, say brexit or no brexit, then a democracy does work.

Here's a proof by induction on the number of candidates : First step : Two candidates, each is a polarizing candidate and each will be ranked first or last. General step : Suppose A is a polarizing candidate amongst n. By independance of irrelevant alternatives, we can remove one candidate, say B, without interfering with the overall ranking of the n-1 other candidates. By induction hypothesis, A is first or last in the result of the n-1 candidates voting. Now where does B stand in the actual voting ? the only problematic cases are when B is first and A second or when be is last and A second to last. Let's consider the first case (the second is identical) and let's change the ballots as follows. In every ballot , we move B to the least possible preffered option, without changing its position relative to A. Example : xxBxxA becomes xxxxBA and AxBxxx- becomes AxxxxB. Observe that nowany candidate but A is preffered to B by everyone but that the A vs B choice of anyone has been preserved. This is only possible because A is a polarizing candidate. According to the unanimity principle, B should now be ranked lower than any candidate but should still be ranked higher than A which is a contradiction. We discarded the two problematic cases and hence, A is ranked either first or last in the election.

Proof of challenge problem: Let A be a polarizing candidate and assume it is not ranked first or last. So there exist B,C such that B>A>C. We can swap B and C whenever B>C and get a new input. By unanimity, the new output has C>B. But by irrelevance of independent alternatives, the new output has B>A>C. This contradict the transitivity property of ordering. Therefore polarizing candidates must always be ranked first or last.

I proved the challenge problem using slightly different assumptions. Instead my assumptions are "independence of irrelevant solutions", and another property, which is that the ranking of a candidate may only be determined by the order of the votes, and not by which candidate the candidate is. We will assume candidate A is polarizing. If we only look at candidate A and candidate B, by independence of irrelevant solutions this should be enough to determine the ranking of A relative to B in the overall vote. A must be ranked < B, or > B. Example System (only looking at A and B): A B A B A B B A If we now add in the votes for one other candidate, who we will call candidate C, we will get something like this: Example System (only looking at A, B, and C): A C B A B C A C B C B A If we now only compare A and C (to determine the ranking between A and C) we get this: A C A C A C C A As you can see, this looks exactly like when we compared A and B. This is because A is a polarizing candidate, which means if any voter votes A > B for some candidate B. They must also have ranked A > every other candidate. Similarly if they ranked B < A for some candidate B, they must also have ranked every other candidate as < A. Because of the second assumption I made, C must be ranked the same way relative to A, as B was relative to A. This implies A must be ranked relative to B, the same way that A is ranked relative to any other voter. This means that if A > B, A must be > all other candidates in the overall vote, and will thus be placed first. Similarly if B > A, all other candidates must be > A, so A must be placed last in the vote. This means A is either > all other votes and first, or all other votes are > A and A is placed last. Interestingly enough, it also means that to determine if a polarizing candidate is placed first or last overall, you only need to compare that candidate to one other candidate.

I wrote a method last time, but here's another one I've come up with: 1. choose 2 candidates, grade all ballots based on which candidate is greater. 2. Give 1 point to all ballots that correlate with the determination of which candidate is greater. 3. Repeat for all possible combinations 4. The ballot with the greatest points is the "dictator," their score is the only one that matters. ?

Challenge problem solution: Let P be a polarising candidate and suppose P is neither first nor last in the result. So there must be candidates A and B such that A > P > B in the result. Consider all possible relative positions of A, B, P a ballot might have: 1) P > A > B 2) P > B > A 3) A > B > P 4) B > A > P Take all the ballots with A > B (i.e. 1s and 3s) and swap A and B (so there are only 2s and 4s left over). None of the A > P and P > B rankings were removed so by IIA final result still has A > P and P > B, and hence A > B. But by unanimity the final result must have B > A, contradiction.

Challenge problem: Proof by contradiction: Let's assume we have a voting V where A is a polarizing candidate but neither first nor last in the final result. We'll derive a contradiction from this. Because A is not first, there is candidate B winning against A. Because A is not last there is a candidate C losing against A. Now add a candidate X to V just above B on every ballot. Because of unanimity, X must win against B and therefore against A. Now remove all candidates other than X and A and call the results V1. Because of Independence of Irrelevant Alternatives, X also beats A in V1. Now, instead, add X to V just below C on every ballot. Because of unanimity, X must lose from C and therefore from A. Now remove all candidates other than X and A and call the results V2. Because of Independence of Irrelevant Alternatives, X also loses from A in V2. Because A is polarizing, on every ballot its position relative to every other candidate is the same and so V1 and V2 must be identical. So X can't lose from A in V1 and win against A in V2. And there is our contradiction. QED

Challenge question: Let P be a polarising candidate. Suppose for contradiction that P is not in first and not in last. This means there are candidates A and B such that the group ranks A above P and P above B. Now imagine changing all the votes as follows: --If a voter preferred B over A, leave their vote unchanged. --If a voter preferred A over B, swap around the positions of A and B in their vote, leaving all their other preferences unchanged. Now observe that the voting system will continue to rank A above P: Changing individual preferences between A and B does not change individual preferences between A and P (because P is polarising), and so by Independence of Irrelevant Alternatives, A is still ranked above P. Similarly, the voting system will continue to rank P above B. Hence, A is ranked above B. But every voter prefers candidate B over candidate A, contradicting Unanimity.

For the challenge lemma, the independence of irrelevant alternatives is all we need: For each candidate (named A) other than the polarizing candidate (named P), we can construct a corresponding sequence of 1's and 0's such that if the kth voter ranked A above P, the kth digit is a 1, and otherwise it's a 0. For example in the round 2 of the example (around 7:00), green is given the sequence 01111, with purple being the polarizing candidate. We find that blue and red also given this sequence: in fact, the sequences corresponding to *all* candidates other than P are the same because P is polarizing (and so the other candidates are always grouped together on one side or another of P). The independence of irrelevant alternatives implies that the relative ranking of A and P in the global ranking is only a function of their relative rankings on the individual ballots; that is, it's a function of A's corresponding binary sequence. Therefore, since all the sequences are the same, P's ranking relative to any other candidate (in the global ranking) is the same: thus P is either ranked first or last. ◼︎ I didn't use unanimity in this proof because the lemma actually holds for any voting system and its opposite, the "opposite" being the voting system that returns the opposite ranking that the original voting system would give. If one of these systems verifies the unanimity property, the other one doesn't, but they both put polarizing candidates first or last.

im having some trouble understanding how purple is ranked first overall in round 3. i get that purple is polarizing, but in round 3 it's still ranked last 3 out of 5 times, which just doesn't appear sufficient enough to warrant first overall to me. round 4 is legit, i can totally see that. it just makes no sense lol at 11:00, the election results, green was ranked higher than blue 4 out of 5 times. i 100% fail to see how that means blue is greater than green. that makes no sense to me.

Another election-related topic of mathematical interest is apportionment/proportional representation. The methods discussed/used in the US for distributing seats in the House of Representatives among the states are often the same as those used in Europe to distribute seats among parties in party list-PR systems. There are some nice and surprising algorithms to do that, and some surprising paradoxes involing intuitively appealing methods. I suggest a video on that.

In your last video you stated that the pairwise comparison system that led to the Condorcet Paradox failed. But I would argue that it didn't. It revealed that there was no group preference.

On the Condorcet thumbnail the cycle is a mobius loop

It is perhaps worth noting one reason that Arrow focused on ranked voting systems was philosophical. He thought that we can't compare the intensity of preference between different people so the maximal information we can have is everone's list of preferences.

so we're doing cardinal voting next, right? this video made too much of a buildup for you guys to do otherwise :P

Yay! so glad you finally did one on Arrow's impossibility theorem! please do more on social choice theory and fair division

I was so confused after watching this the first three times, but I rewatched it, thought about it, read more, wrote it out myself, and now it makes sense! Thank you!

So glad you tackled this topic! Better voting systems could potentially solve a bazillion political problems we have right now, in the US and around the world.

My attempt at the Challenge question, prove"If a voting system has Unanimity and Independence of Irrelevant Alternatives, then there must be a polarizing candidate who is ranked either first or last." We should start by assuming the voting system has both Unanimity and Independence of Irrelevant Alternatives. Unanimity implies that everyone who has voted will rank one color over another color. Lets call the color of preference "Color A" and the other color "Color B." Independence of Irrelevant Alternatives implies that changing the ranking of some color C will never change the relative ranking of Color A to Color B. Use Unanimity to find the most preferred color in the voting system. Since our voting system has unanimity we may search for the most preferred color. If Color A is the most preferred color, namely no other color is ranked higher pairwise to all other colors, then we are done and we found the polarizing candidate who is ranked only first. If not, then there must exist another color everyone prefers to A. Since A is always preferred to B this new color must also be preferred to B. Repeat this process until the most preferred color is found. We may use Unanimity to find the least preferred color in the similar way. We either say B is the least preferred color or there must be another color which is ranked under B. We have thus constructed an ordered ranking of colors with Unanimity. Stating that our voting system has Independence of Irrelevant Alternatives means that the relative ranking of two colors will stay the same if the third is adjusted. This implies that our original ordered ranking could have been something like this C>.....>A>B>......>D where C was the most preferred and D was the least preferred. However we can change the ranking of A to be such that A>C>......>B>......>D Where C is still greater than all previous pairwise combinations (specifically C>B>D) Given this fact, we know that there must always be a polarizing candidate who is either ranked first or last. In this case I suppose there is two polarizing candidates, one ranked first and one ranked last. Let me know what you think.

can anyone explain whay purple above green at 10:41 😅

Can we even democracy?

The condition which holds that - if orange drops out of the race, that should not change if green is preferred to red in the overall ranking, is not Independence of Irrelevant Alternatives (IIA). Its is Sen's condition alpha. IIA holds that - whether or not x is preferred (or indifferent) to y in our social choice ranking should not depend on where any other candidate, z, falls within the individual preference rankings. This is not the same as condition alpha, which holds that - if x is preferred to y in candidate set S, then we remove some elements of S and end up with S’, x ought to be preferred to y in S’ so long as neither was removed.

I get this paranoid feeling Arrow's Impossibility Theorem corresponds directly with Gödel's Incompleteness Theorems...

2 continuous videos on topics discussed by undefined behaviour!

my brain has a lot fun while i watch this,but after some time it gets tired and makes a decision to pause for a little bit :D *that's a compliment

The distinction between "Arrow's idealized world" and "the real world" (resp. Satterwhite et al) is spurious. Arrow shows that for 2 or more choices, social choice mechanisms cannot satisfy all necessary rationality criteria (the ideal world). If you relax the rationality criteria (the real world), rationality criteria are still not satisfied. So, the statement "Arrow's Theorem does not apply to the real world" is false.

This series hasn't yet touched on the purpose and value of votation and systematicization (but neither did probability-and-statistics textbooks 50 years ago)-Single-objective voting itself is nearly impossible to realize, so we'll count it roguish nation (e.g. war vs. snooze)... more-realistic votation systems should probably serve up choices of decision-graphing....

"Independence of irrelevant alternatives" reminds me of the psychological pricing trick that I've heard called "spoilers". I forget the particulars, but it goes something like this: At a movie theater, they sell three sizes of drink, small, medium (about the size of 2 smalls) and large (about 3 smalls). The prices are $1.00, $1.95 and $2.15, respectively. Now, the medium seems like a terrible choice - and virtually no-one chooses it - but the very existence of the medium size at that price, makes the large look like great value. The movie theater across the street has the same small size, and the same large size, at the same prices, but they don't sell as many large drinks because the "spoiler" medium size - the "irrelevant alternative" - makes a difference. This result is shown in real psychological tests, it's something that really happens in the world.

Could you do a video on Gödel's incompleteness theorems? That would be amazingly interesting...

3:10 someone please explain to me how orange ranks first here? Seems to me it should be green, blue, then orange 1-3

If we're going to be pedantic, then the challenge problem is wrong unless we define being ranked first and last to include ties. For example, what if we have two people voting for blue or red, and person 1 ranks blue above red, while person 2 ranks red above blue?

this theorem is always cited as some important fact about voting, and it assumes ordinal voting. if we use cardinal voting we can solve this all kerfuffle. sure, cardinal voting doesn't guarantee a definite answer in all cases, but if our system is fair (whatever that means) and has concluded that two candidates are identically good in the eyes of the populous - than just flip a coin, who cares. they're both literally exactly as good. is there another more substantial downside to cardinal voting? or did arrow's theorem just came about because we purposelessly imposed the restriction of ordinality upon ourselves?

Good job of breaking the proof into parts and preparing us for what was coming. You are very good at your explanations and the parts that you leave for us to prove are well thought out and fun to try.

I wonder if this is why so many civilizations start out with chiefs and kings; in pursuit of perfect fairness, you unintentionally create autocracy.

Challenge problem solution: Lemma: If a voting system satisfies Unanimity (U) and Independence of irrelevant Alternatives (IIA). Then a polarizing candidate must be ranked first or last. Proof: We prove this by induction. When there are 2 or less candidates any candidate will either be ranked first or last, so the induction start is trivial. Assume the lemma to hold for n-1 candidates, and consider a voting result with n>2 candidates with a polarizing candidate p. Let c be a candidate that is not p. If we remove c from the ballot it will not change the fact that p is a polarizing candidate, and since there are then n-1 candidates (in this case) we know by the induction assumption that p will come out first or last. We are going to assume here that p comes out last in the election with c removed, the proof for when p come out first is simmilar. By IIA the final order of all the candidates that are not c will be the same as the order of them when c was removed - that means all we have to prove is that c is ranked higher (ranked before) p in the final rank. If we change the current casting of the votes such that all the voters that placed p first places c second, and the rest (the ones that placed p last - since p is polarizing) places c first - all of this without changing the order of the other candidates - then c will be placed before all candidates that are not p on all the ballots, and therefore by IIA and U c will end up before all the n-2>0 candidates that are niether c nor p, but since the relative position of these n-2 candidates and p will be unchanges, they will still end up before p. That is c is ranked higher than n-2 candidates (at least 1) that are all/is ranked higher than p thereby c is ranked higher than p. Notice now that if we swap back to the orriginal voting, the relative order of c and p will not change, so since c won over p before the swap, it will again after the swap. Thereby p will end up last. As mentioned before you can make a very similar argument for when p ends up first in the voting with n-1 candidates then p will end up first in the voting with n candidates, which concludes the induction step and thus the proof! QED.

The challenge problem is mostly about working out mathematical notation to me. So let's say that we have input vectors x_j which represents how a voter "j" ranked all candidates by giving them numbers from 1 to n (the first coordinate x_1j encodes the rank given to the first candidate, x_2j encodes the rank given to the second and so on). Now for convenience let's pack all of these vectors together into a matrix [x_ij] and run it through a function f([x_ij])=y, where y is results of the elections and the final rank of a candidate "k" is y_k=f_k([x_ij])=f(x_11,...,x_ij,...x_nN) with N being the number of voters. Now let's translate conditions of unanimity and independence of irrelevant alternatives into this notation. Unanimity would be: if for every voter "j" candidate "m" has a lower rank than candidate "l" i.e. x_li>x_mi for every i, it means y_l>y_m. And independence means that for the result of a candidate "k" it is irrelevant, if we swap ranks of two other candidates for a specific voter or that f_k(x_li,x_mi)=f_k(x_mi, x_li) for every l,m≠k. I'm going to assume for simplicity that the polarizing candidate is candidate "1", i.e. x_1i= 1 or n for every i. Finally the proof goes as follows: first assume that every voter chooses x_1j=n, this automatically means from the condition of unanimity that y_1=n. The opposite (x_1j=1 for every j) by the same logic gives the result y_1=1, but first let's concentrate on y_1=n. If now any random voter "j" changes his mind about candidate "1", it means he must swap him with candidate he ranked first "l", since 1 is a polarizing candidate and can be either first or last (it also shouldn't undermine the generality of taking into account all possible voter choices since l and j are chosen completely arbitrarily at this point). So in this notation x_1j=1 and x_lj=n, but because of the independence condition f_k(x_1j,x_lj)=f_k(x_lj,x_1j) for every k, or in other words the rankings of all other candidates aside from "1" and "l" should remain unchanged. After the swap it's either that the candidate "1" is still the loser with rank n or now he has swapped places with candidate "l" and is now the winner. If the candidate "1" is still loosing we can continue the process with voters changing their opinions about him until the voter is found, whose change of hearts determines the result of the election, and we know this is guaranteed to happen due to the fact that, if all of voters rank the candidate 1 first, he's going to win.

Solution for challenge question: Proof by contradiction - assume the result is A>P>B where P is a polarizing candidate. Now, if we allow A to be an irrelevant variable, then for all voters that selected P to be in first place, due to independence of irrelevant variables, we can place A in first. This will cause the placement of P to now be last (due to the rule of a polarized candidate) and that causes all voters to think that B>P. Due to unanimity, this contradicts the result. Similar argument can be used to show that P>A. Please note that this argument can be used to show that P must be in both first AND last place at the same time within the results. Thus any election with more than two candidates cannot have both unanimity and independence of irrelevant variables while using this election type (even if there is only one voter)... sigh.

I feel like there should be some close relationship between independence of irrelevant alternatives and preclusion of strategic voting. Is there one? And if so, what is it? Thanks.

My Plurality Criterion says "A Condorcet winner who is also the plurality winner should never lose." With that requirement, top-two runoff passes, while IRV fails. In IRV, the plurality candidate may fall to 3rd place and be eliminated and never get a chance for a head-to-head comparison to his rivals. It's important because REAL WORLD voting includes compromise before the election, and so using a voting method with this criterion encourages voters to "agree to agree" before the election, and that prevents the threat of elections like 100 candidates and 100 voters where rational ranking is impossible. If you know "I have to be first or second most popular", weaker candidates may choose to endorse someone else before the election even occurs.

It's important to state that this only works if there are 3 or more candidates.

Awesome video

Do you think you will be able to explain cardinal voting systems too? Or do you think it is too far off topic from the series? I was always curious about what is the best option to treat the non valid votes (people who don't give a complete information about their preferences)

What about the property of unrestricted domain for proving the Arrow's theorem? Or only unanimity and IIA are enough to show dictatorship?

The dictatorship kinda happen in majority between as well, if you have a tie. The next vote will be the "dictator".

There is a question that I have never really understood: does plurality voting (candidate with most votes in First round wins) count as an example of Ranked choice voting? I believe it does, but I would like to know how the properties discussed here apply for it.

Thank you SO MUCH, this is an excellent video!

Note that the pairwise comparisons system described at the start of the first video meets both unanimity and a weaker (but still pretty strong) version of Independence from irrelevant alternatives. Adding a candidate never reverses a group preference but can only turn a clear group preference into a paradox. So A>B can't be changed to B>A by the introduction of option C. Instead it can only be turned into an ABC cycle or paradox. If you, as I do, consider a paradox to be a tie then adding a candidate can turn A>B into A=B. The worst effect of the way other systems turn A>B into B>A by the introduction of C is how that leads to tactical voting which denies the system true data to analyze.

You are wrong on one crucial point, The arrow impossibility theorem doesn't just apply to Ranked choice ordinal voting, Borda system which provides cardinal numbers based on ordinal collective preferences doesn't satisfy the independence from irrelevant alternatives, as the number of options and rank has direct implications for votes. The theorem is general in nature and applies to all voting systems

Suppose we have pink be a polarizing color, (as it tends to be). Then for every other color we choose, getting rid of irrelevant alternatives produces the same 2 color election. If for voter A, pink was last, then no matter what color we compare it to, that other color will be always be ranked higher, and so on for every other voter. So if the election decides that pink is ranked higher than any given other color, it must rank pink higher than all other colors, due to the rule of irrelevant alternatives. Hence pink is first, if it wins any 2 color election, and last otherwise.

Dictator and ties don't matter. Neither does it matter if there is a circular condorcet mapping. What matters is that plurality is an extremely poor voting method because in a real situation with hundreds of candidates and millions of voters, a condorcet vote could have opposite results than plurality. It doesn't matter if there's a tie because it's not a lie, whereas a plurality vote gives results based on the candidate space. That's why they tried to hack the math by adding parties.

the arrow's theorem i know of splits unanimity into two parts. One part requires that there exists a set of votes that outputs any given outcome, and the other part requires that when one candidate's rank is increased by some voters and decreased by none, then that candidate's rank cannot fall. it feels like unanimity is simpler, but is there some mathematical advantage of stating it in this way instead?

What about a "reverse" borda count? every candidate wins as many points as his rank (first candidare gets 1 point, second gets 2...) and the winner is the one with least points?

The pairwise system includes only one opportunity for tactical voting and that would be an attempt to create a paradox or cycle that includes your preferred candidate. But you would only be interested in doing so if your first choice candidate were likely to come second. But engineering a paradox with a single vote is very difficult because you can't vote for the entire paradox but only two sides of the cycle (engineering a more than three sided cycle would be even harder so I am assuming three sides). The third side of the cycle you have to vote against. So you would have to determine from polling results which side of the cycle is most easily turned and make certain your vote reinforces the cyclw on that side. But too many people tactically vote and they will turn another side undoing the cycle and potentially helping their least preferred candidate to win. Thus the system deters what little tactical voting is possible under it by making it very likely to backfire.

Maybe it should be noted why independence of irrelevant alternatives matters: it's another way of saying that every voter should be able to express their true preference, without any strategic voting according to how they expect others to vote.

We proved Arrow's Theorem in graduate math class in 1992 at Rutgers University. I find nothing special or important about one of those axioms. The one that forbids candidate X from being group-ranked 1st place in spite of no voter ranking X 1st place. I see no particular reason why one "needs" or "should have" that assumption. As long as everybody agrees on the particular algorithm to generate a group ranking from individual voter rankings, there's no logical reason to rule out X being group ranked highest.

Some interpretations argue that using cardinal methods do not actually avoid Arrow's Impossibility Theorem because cardinal information from one person is not directly comparable to the information obtained from another person. Plus, the Gibbard-Satterthwaite theorem still applies, as well as various extensions and related theorems. This means that gaming the system will always be possible, making some people want to vote against their own actual preferences in order to obtain a more desirable result (which also applies to Ordinal Systems). Thus, a Cardinal System is not necessarily an obvious choice against an Ordinal System. Unless you only have one voter and/or one candidate, there will always be some way the system is flawed. Even a system as simple when there are two parties, there can be the chance of a tie if there are an even number of voters. Plus, not everyone in a population does vote, and so questions have to be raised as to what counts as a win. If only one person out of an arbitrarily large voting population decides to vote, then that person alone decides the election, against the wishes of who knows how many people.

What makes voter 2 specifically the dictator? What are the conditions of the distribution for a specific voter to be the dictator? And is there any way to manipulate your way into dictatorship?

wouldn't a mean value ranking system satisfy most of these? Sort of the inverse of a point system. any time a color is ranked 1st it recieves 1 point, 2nd place is 2 points, etc, and the total is averaged, so the mean value comes very close to the rank that color recieves most often. the colors are then ranked by this new mean value. in the case of a tie, a second round of just those two colors is calculated. this satisfies unanimity and independence of irrelevant alternatives. it's also not dictatorial. using this system the switch for purple occurs evenly across all rounds, as it should, and does not affect the rankings of any of the other colors, other than bumping one to achieve its rank

I´m confused, does election in america work by ranking the candidates? in my country works by choosing only one preference for a candidate and counting the total votes. Does this still works then?

I have a question regarding what you mean by voting system. I would assume that in a voting system all voters are indistinguishable, meaning that if we permute the votes, the outcome would be the same. Therefore if all voters cast different votes and there is a dictator, permuting the votes may result in the dictator making a different vote, but the outcome should remain the same. How does this exactly work?

I don't understand, why did you said overall it's purple above green in modified bollots and vote test 3; when it's actually green above purple

[Cries in colorblindness]

Is there any way you could avoid using blue and purple together in future videos? I'm colorblind and I couldn't distinguish between them.

The weekly problem is trivial given Arrow's Impossibility Theorem. If there is only one ballot, then the only polarizing candidates are ranked first and last respectively by this singular ballot. These candidates rank first and last in the dictatorial vote, respectively. If there is more than one ballot, then by Arrow's Impossibility Theorem, our assumptions are violated. Therefore in every case that we have those two properties, every polarizing candidate must rank first or last. QED.

How does proportional representation figure into the problem?

I don't see why the challenge problem should be true: say one person votes A>B>C, and one person votes C>A>B, then C is a polarizing candidate, but what's wrong with the ranking A>C>B?

Shouldn't the Condorcet Criterion be that if a candidate wins a head-to-head election with *every* other candidate they should be the overall winner? If it's any overall winner then everyone could vote A-B-C and the criterion would say that B should win because they beat C, even though A clearly beat both B and C.

All I understood is........ dictatorship is where one main ballot acts as a bottleneck of the whole election

Meanwhile , ARROW - what are you trying to teach ??

This theorem is not merely a problem for democracy, but for _all_ human affairs with more than two possible outcomes: _nothing_ is immune; sports are particularly susceptible (round-robin or pair-wise comparison, brackets etc.); inheritance and succession are as well.

Not trying to start something because i know you all worked hard, but around the 7 min point whats up with box lining the boxes together?? Its slightly off. Where the layers not in the same program? Its driving me bananas. But good vid overall. Rant over. Thanks. Good job

Why is purple first in round three? Shouldn't it be last?

at 7:10 how does purple end up at the top of the rankings in round 3? Clearly, more people prefer blue to purple, red to blue and red to purple, green to red, green to blue and green to purple, so the order should be Green, Red, Blue, Purple?

Only -1000000 for punching in the face? I would expect -infinity on this channel...

Challenge Problem Solution: (EDIT: When I originally wrote this, I accidentally assumed that all candidates are treated equally and I didn't notice, but luckily this is still provable using unanimity!) If we have an election with a polarizing candidate, consider all groups of two candidates where one of the candidates is polarizing. Every group of two will have exactly the same voting pattern, with each voter either preferring the polarizing candidate or not preferring them. So however our voting procedure decides a relative ranking between the polarizing candidate and another candidate, (assuming all candidates are treated equally, [1]) the relative ranking will be the same regardless of the other candidate. This means that the procedure prefers the polarizing candidate to every other candidate or prefers every other candidate more than the polarizing candidate. So by of the Independence of Irrelevant Alternatives, the polarizing candidate will come in either first or last place. Q.E.D. References: [1] David de Kloet, "I can prove your extra assumption using unanimity", Reply to Cantor's Cat's Solution, 24 June 2017.

At 7:29, I don't know why purple wins round three, because overall, green is ranked higher than purple, thus it cannot be first, right?

I missed the point about what differentiated voter 2 from the others. What made him the dictator? I guess there must be certain circumstances for that to happen.