Instant Runoff Voting and How Safe Movies Win

[jedishrfu]

There are many ways to tally voting, one such scheme is Instant Runoff Voting where you are allowed to choose your top 3 favorites. However, this scheme can lead to mediocre choices when applied to Academy Award movie wins:

 

[Buzz Bloom]

There are many ways to tally voting, one such scheme is Instant Runoff Voting where you are allowed to choose your top 3 favorites. However, this scheme can lead to mediocre choices when applied to Academy Award movie wins:

Hi jedishrfu:

I think the word "mediocre" is a very misleading and biased word choice.

In the ordinary use of Instant Runoff Voting (IRV) the goal is to avoid the election of a candidate that does not represent a majority of the votes cast. I gather from the explanation given in the included video that IRV is not actually used, but that some modified form is used in which the voter casts only three ordered choices. It is not clear what the goal of this method of voting is intended to achieve, but if the number of candidates is much greater than three, a winner is again not likely to be the desired choice of a majority. I am guessing that the process achieves a winner with a larger base of support than the method used previously, rather than a more intense level of individual support by fewer supporters. If this is correct, then it makes sense that the film industry would prefer it's Oscar choices to better match what most movie goers would want to see.

Regards,
Buzz

 

[jedishrfu]

I quoting the view in the video where "safe" movies win over edgier movies.

 

[Buzz Bloom]

I quoting the view in the video where "safe" movies win over edgier movies.

Hi jedishrfu:

I must have missed the oral statement in the video that used "mediocre". If the word was used in the video, I still find it to be a misleading and biased word choice.

From
http://www.dictionary.com/browse/mediocre
adjective

1. of only ordinary or moderate quality; neither good nor bad; barely adequate:
The car gets only mediocre mileage, but it's fun to drive.
2. not satisfactory; poor; inferior:
Mediocre construction makes that building dangerous.​

The bias of "mediocre" is that the user of the word prefers edgier movies to safe movies, and "mediocre" is pejorative.

Regards,
Buzz

 

[jedishrfu]

Okay I changed the thread title to "Safe" to be safe.

 

[mfb]

The perfect voting system cannot exist - Arrow's impossibility theorem - at least not if you just ask for an order of preference. There are many systems and good arguments for most of them.

 

[Buzz Bloom]

Hi mfb:

Thank you for the link about Arrow's impossibility theorem. I will have to spend some time to fully understand its implications. In particular the reference to the Cardinal Voting exception will also need some study.

What is fascinating is that theorem also seems to (if I understand it) apply to an ordinary runoff system in which following an election in which there is no majority, the two candidates who had the highest number of votes have another opportunity to get a majority of votes from a second vote. I have not yet fully understood why this fails to work when the first election has only three candidates, except for the case where many voters whose first choice is eliminated have no interest in voting in the runoff. That means that there is no single candidate of the three who is acceptable to a majority of the voters. Perhaps a system might work in which it is determined by the first election with may candidates that none of the candidates in this election are acceptable to a majority of the voters, and that this result would require that a second election be run with entirely new candidates in which none of the candidates in the first election are included. I have not yet studied the theorem sufficiently to determine if this also violates the Arrow's theorem, except for he possibility when there is no possible candidate that is acceptable to a majority of voters.

ADDED
I am still not sure I understand the theorem, but the following quote from the Wikipedia article seems clear enough.

We are searching for a ranked voting electoral system, called a social welfare function (preference aggregation rule), which transforms the set of preferences (profile of preferences) into a single global societal preference order.​

This seems to assume that the purpose of an election is to determine "a single global societal preference order". But that is an unreasonable assumption about the goal of an election. Why is it not sufficient for an election to choose a candidate that is acceptable to a majority, and that among those candidates who are acceptable to a majority, it elects the one with the most votes when the fewest candidates with the least votes are eliminated?

Regards,
Buzz

 

[mfb]

Let 40% favor A>B>C, let 35% favor C>B>A, 25% B>A>C.
The first round gives 40% A, 35% C, 25% B and B is eliminated. The second round gives 65% A, 35% C and A wins.

If C wouldn't have participated, B would have gotten 60% in the first round and would have won.
A third candidate turned the outcome between A and B.

While my numbers are not very realistic, the situation itself is very realistic. The recent French election was an example. There were two different right-wing candidates, one moderate candidate and one left-wing candidate, all with roughly similar poll results. The moderate candidate was the only one many would have preferred over other candidates, but he could have gotten kicked out in the first round. He stayed in, and won the second round with a clear victory over one of the right-wing candidates.

Edit: Simplified the example a bit.

 

[Buzz Bloom]

Hi mfb:

Thanks for the excellent example. It shows that the way IRV is normally used is flawed. One has to assume that a voter may give no preference at all to a candidate that is unacceptable. Therefore, the votes should show acceptability is: 100% B, 65% A. 35% C. Instead of eliminating B, C should be eliminated, and B would be the winner with 60% of the runoff vote.. Note also, if the original votes are:

40% A=1, B=2, C=blank
35% C=1, B= blank, A=blank
25% B=1, A = 2, C = blank​

65% find A acceptable, 65% find B acceptable, and 60% find C acceptable, so C is eliminated again, but A wins the runoff . This seems to me to be the "right" result for this election. This might be made clearer it we give party associations for A, B, and C as follows:

A = extreme liberal, B = moderate liberal, and C = conservative.​

Thus there are 35% who are conservatives and who find either form of liberal unacceptable, but among the liberal voters who find both liberals acceptable, the majority prefer the extreme liberal.

Regards,
Buzz.

 

[mfb]

You can find counterexamples for that as well. You don't even have to find numbers (although they are useful to see what is going on). The theorem guarantees that there is always a counterexample.

If we always eliminate the candidate who is ranked third by most:
Let 40% favor A>B>C, let 35% favor C>B>A, 25% C>A>B.
While C gets 60% of the votes we eliminate C, afterwards A wins. Without B C would just win.

If we only do that if no one gets an absolute majority:
Let 40% favor A>B>C, let 20% favor C>B>A, 25% C>A>B, 15% B>C>A
Again C is eliminated (45% of the first choice votes, but 40% hate C, compared to 35% who hate A), afterwards A wins. Without B, C would have won 60% of the votes and would have won.

 

[Buzz Bloom]

You can find counterexamples for that as well.

Hi mfb:

Another collection of good examples. You seem to have a knack for them.

You have convinced my that my suggestion does not work either.

Regards,
Buzz

 

[jedishrfu]

To expand on the fairness in voting, here's an article on redistricting:

https://priceonomics.com/algorithm-the-unfairness-of-gerrymandering/

and how it can skew elections in favor of one party over another.

I remember there was some work done on making this into a mathematical framework suitable for courts to decide if a redistricting plan would take away voting rights from some groups.

https://www.quantamagazine.org/the-mathematics-behind-gerrymandering-20170404/

I know in Austin Texas, they redistricted several congressional districts to favor Republican candidates and to unseat Congressman Lloyd Doggett.

https://www.texastribune.org/2012/02/28/court-delivers-election-maps-texas-house-congress/

https://www.texastribune.org/2017/08/15/federal-court-invalidates-part-texas-congressional-map/

 

[Buzz Bloom]

I am hoping someone can offer me some advice.

I would like to program a Monti-Carlo simulation to evaluate different rules for choosing an IRV candidate to eliminate on each round. To do this I need to define a rule that identifies a simulation run as OK or a failure. I find the rules of the Arrow assumptions too strict, so I will need to weaken them somewhat.

There is also an issue of whether some of the permutations of the ordering of preferences by voters can be eliminated as generally unrealistic. I am thinking that it is reasonable to assume that the candidates may be structured into distinct political groups, and there may be multiple candidates within each group ordered between extreme and moderate. This will place some restrictions on plausible voting ordering.

The final step of assigning a random distribution of number of voters to the remaining permutations will be straight forward.

Regards,
Buzz

 

[Buzz Bloom]

Hi @mfb:
I have taken a closer look at your first example in post # 8, and the "odd result" now doesn't look to me so odd.

What happens in an ordinary election with 3 candidates and no runoffs? Frequently two candidates have somewhat similar politics, say A and B, but both want to run, and by so doing they split the majority who would have voted for either A or B if the other was absent. In this case the minority candidate C wins. In such an election, the majority of voters, which are those who don't like C, have a dilemma. They have to guess whether voting for the A or B candidate, which they much prefer to C, will allow C to win. The runoff result where A wins will always be a better result to the majority than that.

In your example, if C is absent B wins, which shows that C's running has influenced the election such that a majority of the voters preferring C are prevented from getting their second choice B. That doesn't seem to me like a very terrible result, as compared with the no runoff election. Also note that if in the example the 35% relationship were modified to C>A>B, then A would win whether C ran or not.

Regards,
Buzz

 

[mfb]

Consider the French election as real life example how it could have gone wrong (lead to a candidate the majority hates).

An alternative is a single voting round and then a coalition between parties, as it is done in various countries.

 

[Buzz Bloom]

Consider the French election as real life example how it could have gone wrong (lead to a candidate the majority hates).

There were two different right-wing candidates, one moderate candidate and one left-wing candidate, all with roughly similar poll results. The moderate candidate was the only one many would have preferred over other candidates, but he could have gotten kicked out in the first round. He stayed in, and won the second round with a clear victory over one of the right-wing candidates.

Hi mfb:

Could you make up some plausible 4 candidate ordered preferences that would demonstrate what might have happened if the French elections used IRV?

Regards,
Buzz

 

[mfb]

The French election does use it. Here is its Wikipedia page.

The relevant candidates:
A) Emmanuel Macron, moderate
B) Marine Le Pen, right-wing
C) François Fillon, conservative/right-wing
D) Jean-Luc Mélenchon, left

All of them had about 20% in polls (the rest was distributed over many minor candidates). The voters for B didn't like C and hated D, the voters for C didn't like B and hated D, and the voters for D hated B and C. Macron (A) was the only one seen as acceptable by the majority.

What actually happened: In the first round A got 24%, B 21%, C 20%, D 20%. A and B went into the second round, A won clearly with 66% and large support from the population.

What could have happened with slightly changed numbers: Distribute 3% of the votes from A over the other candidates, now A has 21%, B 22%, C 22%, D 20%. Now the second round is between two right-wing candidates. 44% of the voters can keep voting for their favorite candidate, the rest hates both option. The president will be elected with something like 1/3 support from the population and 2/3 will hate the president.

An alternative: Distribute 4% of the votes from A over the other candidates, now A has 20%, B 21%, C 22%, D 22%. Instead of A and B, we now have C and D in the second round. Again 44% of the voters can keep voting for their candidate, some voters for B will vote for C, and the rest has no candidate they feel confident to vote for. Again the president will be elected with something like 1/3 support from the population and 2/3 will hate the president.

 

[jedishrfu]

A lot of this analysis reminds me of a game theory problem where a husband and wife go camping at some national park. The husband likes to camp on the mountains and the wife likes to camp in the valleys.

The game is played with husband choosing the latitude and the wife choosing the longitude. As you can imagine neither gets their favorite choice and it’s hard to figure who will really win the game. Basically, a two-person election with multiple candidates along the lines selected.

 

[Buzz Bloom]

Hi @mfb:

The French election did not use IRV. The ordinary runoff the of election the French used is generally an improvement of no runoff, since a no runoff election, which is used in the US, frequently results in a minority winner, while the ordinary runoff more rarely does so.

Based on the data in your post and the Wikipedia article, it is likely that using IRV would have been elected Macron even with the voting modifications you gave in your example. Based on your modified results example data, the IRV would be something like the following. (I had to make up some numbers, based on preferences described your post, since neither you nor Wikipedia provided them.) I omitted Cheminade since he had only 0.18%.
(A) Macron, (B) Le Pen, (C) Fillon, (D) Mélenchon, (E) Hamon, (F) Dupont-Aigan, (G) Lasalle, (H) PouTou, (I) Asselineau, (J) Arthau
20% A>B
21% B>C
11% C>A>B
11% C>B>A
22% D>A
6% E>B>A
5% F>B>A
1% G>C>A
1% H>C>A
1% I>C>A
1% J>C>A

The rules for eliminating candidates are as follows.

Step 1: Add the votes for each candidate for all ballots that include the candidate as an acceptable choice. The results:
100% A
63% B
58% C
22% D
6% E
5% F
1% G
1% H
1% I
1% J
Eliminate all candidates with less than 50+%. That leaves only A, B, and C. Distribute the highest acceptable choice among remaining candidates. This gives the result:
42% A>B
32% B>C
13% C>A>B
13% C>B>A
Note that it is possible with IRV that no candidate has greater than 50% acceptability. I have not seen any clear rules about what should happen in such cases. One possibility I find reasonable is to elect the candidate with the greatest acceptability, rather than the most top preference votes, but many people might reject that as not reasonable. Another possibility is that an entirely new collection of candidates should be selected for a new election.

Step 2: Eliminate the candidate with the fewest votes, C, and redistribute the ballots for that candidate to the remaining candidate with the highest preference. (This step is repeated until a candidate has greater than 50% of the votes.) This results in the following:
55% A>B
45% B>C
A wins. Note however, if all voters who preferred C had B as the second choice, B would have won. This shows that a candidate which fewer voters find acceptable can still win if they have at least 50+% of acceptability.

Although you have shown in your examples that it is possible for IRV to have a "bad" result, at least the results are not as bad as no runoff or ordinary runoff which methods can elect a candidate who has less than 50% acceptability.

Regards,
Buzz

 

[StoneTemplePython]

The ordinary runoff the of election the French used is generally an improvement of no runoff, since a no runoff election, which is used in the US, frequently results in a minority winner, while the ordinary runoff more rarely does so.

This seems to be going off topic, and is wrong. In the US if no candidate has a majority of electoral college votes, then congress votes in the next president. It is simply a different non-comparable system.

- - - -
edit: I poked around and it seems that I was focusing on majority win voting while there are plurality win voting structures as well, and the run-off structuring can vary quite a bit. Most run-offs that I've read about in international politics, e.g. Peru's 2016 election, seem to be designed to deal with the case when there is no majority winner in round one (i.e. something different than the US setup), but it seems this musn't always be the case.

The issue that I tried to highlight is that you cannot have a minority winner in the US by construction. (The fact that it is done via electoral college and not popular vote further muddies the waters and makes the comparison inappropriate in my view.)

Stepping back a bit, I am getting a feeling that there could be an interesting insights piece on Arrow's impossibility theorem lurking under here...

 

[Buzz Bloom]

This seems to be going off topic, and is wrong. In the US if no candidate has a majority of electoral college votes, then congress votes in the next president.

Hi Stone:

I have to agree that much of this thread has been about voting methods rather than just as applied to the Oscar awards. Perhaps this discussion should be moved to a separate thread in the General Discussion forum.

Regarding the point in the quote above:

(1) There are other US elections than for president. For example, governor or senator, or congressman.
(2) Voters in the US do not directly vote to elect the president. They vote for a slate of electors. It is only when the chosen electors do not have a majority vote for a presidential candidate that the house of representatives gets into the process.​

The issue that I tried to highlight is that you cannot have a minority winner in the US by construction. (The fact that it is done via electoral college and not popular vote further muddies the waters and makes the comparison inappropriate in my view.)

There is a process in the works whereby the electoral college may someday become irrelevant.
https://en.wikipedia.org/wiki/National_Popular_Vote_Interstate_Compact

The National Popular Vote Interstate Compact (NPVIC) is an agreement among a group of U.S. states and the District of Columbia to award all their respective electoral votes to whichever presidential candidate wins the overall popular vote in the 50 states and the District of Columbia. The compact is designed to ensure that the candidate who wins the most popular votes is elected president, and it will come into effect only when it will guarantee that outcome.[2][3] As of December 2017, it has been adopted by ten states and the District of Columbia. Together, they have 165 electoral votes, which is 30.7% of the total Electoral College and 61.1% of the votes needed to give the compact legal force.​

Regards,
Buzz

 

[StoneTemplePython]

Hi Stone:

I have to agree that much of this thread has been about voting methods rather than just as applied to the Oscar awards. Perhaps this discussion should be moved to a separate thread.

Regarding the point in the quote above:

(1) There are other US elections than president. For example, governor or senator, or congressman.
(2) Voters in the US do not directly vote to elect the president. They vote for a slate of electors. It is only when the chosen electors do not have a majority vote for a presidential candidate that the house of representatives gets into the process.​

1.) You were talking about Macron -- i.e. head of state elections. You now seem to be further departing from the the thread.
2.) I know this and said as much in my post.

There is a process in the works whereby the electoral college may someday become irrelevant.
https://en.wikipedia.org/wiki/National_Popular_Vote_Interstate_Compact

This is completely irrelevant to the thread.

 

[Buzz Bloom]

Hi Stone:

I apologize for for misunderstanding the intent of that part of your post #20 discussing the electoral college.

Regards,
Buzz

 

[StoneTemplePython]

It's ok.

I happen to really like Arrow's Impossibility Theorem but I've only read a bit about it and that was some time ago... I still have my fingers crossed I can learn more from this thread.

 

[jedishrfu]

I started the thread to discuss the math behind voting methods and used the Oscar article as a starting point .

Hence any discussion related to the math of voting including apparent voting fails or redistricting is okay with me.

My uncle once had to write a program for a court case in upstate NY around 1974 or so to show if voting rights were lost due to differing populations in counties and voting districts.

 

[jedishrfu]

Heres a more recent Supreme Court case testing the notion of One Person One Vote.

Ie whether redistricting be based on everyone who lives in the district or only on those who are registered voters.

https://www.nytimes.com/2016/04/05/us/politics/supreme-court-one-person-one-vote.html

 

[Buzz Bloom]

I happen to really like Arrow's Impossibility Theorem but I've only read a bit about it and that was some time ago

Hi Stone:

I find the Arrow theorem interesting mathematics, but after giving it a more careful read, I find that it does not apply to practical election processes.
https://en.wikipedia.org/wiki/Arrow's_impossibility_theorem

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

1) If every voter prefers alternative X over alternative Y, then the group prefers X over Y.The purpose of eleciong someone
2) If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
3) There is no "dictator": no single voter possesses the power to always determine the group's preference.​

The purpose of most elections is to choose one person as a winner who becomes the elected person. It is NOT to determine an ordering of candidates, but just one single candidate. (There are some elections intended to choose several winners, but that topic should be discussed separately.) I confess that I do not understand what (2) and (3) mean.

The purpose of the IRV method is to chose a single candidate who is acceptable to a majority of the voters, and among those candidates that meet this criterion, choose one as the winner. The method first eliminates those who do not have 50+% acceptability, and then to iteratively eliminate from the collection of remaining candidates the single one who has the least number of highest choices among the redistributed ballots. @mfb has by examples demonstrated that this method can sometimes choose someone who would not be chosen if a candidate removed him/her self from consideration before the voting takes place, but after the ballots had been prepared. In that case it is assumed this candidate would be removed from the acceptable choices made by voters. (Note that dealing with this case is not a required criterion related to Arrow's theorem.) It is not clear to me that it is an important criterion to try to satisfy. If someone can come up with some reasonable additional criteria, perhaps the method used to eliminate candidates one at a time could be improved.

Regards,
Buzz

 

[StoneTemplePython]

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

1) If every voter prefers alternative X over alternative Y, then the group prefers X over Y.The purpose of eleciong someone
2) If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
3) There is no "dictator": no single voter possesses the power to always determine the group's preference.​

The purpose of most elections is to choose one person as a winner who becomes the elected person. It is NOT to determine an ordering of candidates, but just one single candidate. (There are some elections intended to choose several winners, but that topic should be discussed separately.) I confess that I do not understand what (2) and (3) mean.

I think you have hit on something with 2.) in particular, which is closely tied in with a total ordering of candidates ("rationality").

There's a nice little writeup from Terence Tao, here:
https://www.math.ucla.edu/~tao/arrow.pdf

I think point number 2 seems to be a key underpinning of the voter system rationality axiom.

A couple interesting things that come up in here:

1.) If we allow randomness to enter the picture we could make things ex ante fair. This comes up at the bottom of page 1. This is an idea that comes up in a lot of different domains and in general this is a nice way to get better and more fair results. However for this particular problem, the simple case of having one voter chosen at random effect his /her preferences on everyone... seems quite ill advised.

2.) On the last page he mentions that a simple majority vote with 3 candidates yields a rational outcome, asymptotically, 91% of the time. Though again, I kind of think you're right -- if you're talking about elections for a specific head of state office, it's winner takes all and ordering beneath the winner seems not so relevant -- though figuring out the progression to a run-off could be tricky if there is no majority winner in round one.

 

[Buzz Bloom]

There's a nice little writeup from Terence Tao, here:

Hi stone.

Thanks for the link. Tao's presentation is much clearer than Wikipedia's.

Regards,
Buzz

 

[mfb]

The purpose of most elections is to choose one person as a winner who becomes the elected person. It is NOT to determine an ordering of candidates, but just one single candidate.

The theorem applies to the case of finding a single winner.

@mfb has by examples demonstrated that this method can sometimes choose someone who would not be chosen if a candidate removed him/her self from consideration before the voting takes place, but after the ballots had been prepared. In that case it is assumed this candidate would be removed from the acceptable choices made by voters. (Note that dealing with this case is not a required criterion related to Arrow's theorem.)

This is exactly what condition 2 of the theorem states. Adding or removing another candidate should not change the winner (unless the new candidate wins).

 

[Buzz Bloom]

The theorem applies to the case of finding a single winner.

Hi mfb:

I am confused about this quote above, and I hope you can clarify it for me.

As I understand Axiom 1 it says that there must be a match between the final ordering of preferences among any sub-collections of candidates for which all of the ballots indicate the same ordering of this sub-collection. Is this correct? If so, how does this axiom apply to choosing a single winner who has among all the ballots an indication that this winner has at least 50% of all the voters indicating they find the winner acceptable? If the axiom does not apply, then how does the theorem prove without this axiom that this objective is impossible?

I confess I do not understand what "If every voter's preference between X and Y remains unchanged," means. In order to understand this I think I need to understand what it means for a voter to change one or more of his/her preferences. Is this intended to mean a hypothetical change? I just don't get it.

In Tao's presentation of Arrow's theorem, there is the following Axiom:

(Independence of a third alternative) The relative ranking of X and Y is independent of the voters preferences for a third candidate Z.​

Is this intended to mean the same thing as Axiom 2? If so, then perhaps your example shows this is not guaranteed even if Axiom 1 is removed. What do you think about this?

I would also much appreciate your giving me some advice about the Monte-Carlo trials I am working on. I am making progress and nearing a point when I expect to be able to (under some assumptions about the distribution of random numbers) estimate the probability that an outcome like the one in your example would occur. If I understand correctly about the rules of the PF, I am not allowed to discuss the result of this personal project in a thread. However, would it be OK for me to discuss this in an inbox conversation? Would you be interested?

Regards,
Buzz

 

[mfb]

I confess I do not understand what "If every voter's preference between X and Y remains unchanged," means. In order to understand this I think I need to understand what it means for a voter to change one or more of his/her preferences. Is this intended to mean a hypothetical change? I just don't get it.

It is a hypothetical change (we require that it doesn't happen: If you prefer X over Y, then you should do this no matter which other candidates exist).

It doesn't matter if you look at a single winner or a full ranking. If we remove the winner, the group's preference between the second and third (and second and fourth, ...) should not change. If you could construct a voting system that can determine a winner while keeping the other conditions, you could use the same system to create a full ranking by removing the winners step by step.

Is this intended to mean the same thing as Axiom 2?

Yes.

I would also much appreciate your giving me some advice about the Monte-Carlo trials I am working on. I am making progress and nearing a point when I expect to be able to (under some assumptions about the distribution of random numbers) estimate the probability that an outcome like the one in your example would occur. If I understand correctly about the rules of the PF, I am not allowed to discuss the result of this personal project in a thread. However, would it be OK for me to discuss this in an inbox conversation? Would you be interested?

Should be fine here. I wonder how you choose voter preferences.

 

[Buzz Bloom]

The current version of my IRV Monte-Carlo tool is a spreadsheet which calculates results for just one trial, and it is limited to one particuar set of assumptions (see below). I next plan to extend it to simultaneous run a large number of trials. For the present I hit F9 to run a new trial and manually keep track of the results.

The assumptions are:
1. There are three candidates, A, B, and C.
2. Each ballot selects a 1st choice and optionally also a 2nd choice.
3. The candidates have an ordering with respect to politics, for example: A=Liberal, B = Moderate, C=Conservative.
Therefore a voter's 2nd choice will never be C if the first choice is A, and vice versa. This allows for 7 ballot possibilities: A, A>B, B, B>A, B>C, C, C>B.
4. Seven random numbers are generated from a flat distribution [ =rand() ]: T, U, V, W, X, Y, Z. The fraction of ballots for each of the 7 ballot possibilities are calculated as follows:

P(A) = (T/(T+U+V) ) * W
P(A>B) = (T/(T+U+V) ) * (1-W)
P(B) = (U/(T+U+V) ) * X
P(B>A) = (U/(T+U+V) ) * (1-X) * Y
P(B>C) = (U/(T+U+V) ) * (1-X) * (1-Y)
P(C) = (V/(T+U+V) ) * Z
P(C>B) = (V/(T+U+V) ) * (1-Z)​

Note that the sum of these seven probabilities is 1.
5. The "acceptability" E for each candidate is calculated as follows:

E(A) = P(A) + P(A>B) + P(B>A)
E(B) = P(B) + P(B>A) + P(B>C) + P(A>B) + P(C>B)
E(C) = P(C) + P(C>B) + P(B >C)​

6. If the “acceptability” for a candidate is not greater than 50%, then that candidate is removed from consideration.
Note that it is possible for all three candidates to be removed for this reason. At the present time I have no follow-up for choosing a winner if this happens. Early manual trials show that this occurs very roughly about 2% of the time. These results also show that at least one candidate is always eliminated for this reason. (I think I might be able to prove this as a theorem, but my proving skills are much diminished from what they were when I was much younger.) This result makes it unnecessary to eliminate one candidate of three for having the least number of first round votes.
7. It there is only one “acceptable” candidate, then that candidate is the winner. If there are two “acceptable” candidates, the final count of votes is made. This adds the ballots which had the eliminated candidate as 1st choice to the 2nd choice of these ballots. Then the candidate with the resulting greater number of votes is the winner.

The following are some early results based on 120 trials:

A: 28, B: 75, C: 15, NONE: 2.​

Since A and C are symmetrical, it is reasonable to adjust for this as follows

A: 21.5, B: 75, C: 21.5, NONE : 2.​

Some thoughts:
The very high frequency for B is unexpected and suggests that the assumptions are biased away from reality. Perhaps the flat distribution for W, X, and Z, the probabilities for having a 2nd choice, should be adjusted. I have in mind trying a triangular distribution and possibly a mean other than 50%. However, I intend to postpone these changes until I add the feature for running a large number of trials simultaneously.

 

[StoneTemplePython]

The current version of my IRV Monte-Carlo tool is a spreadsheet which calculates results for just one trial, and it is limited to one particuar set of assumptions (see below). I next plan to extend it to simultaneous run a large number of trials. For the present I hit F9 to run a new trial and manually keep track of the results.

1.) I think a couple of us on this thread were hoping you were going to run the simulation in Julia...

2.) If you do choose to stick with spreadsheets, I'm inferring Excel. If so, you may consider using the free module "PopTools" so at a minimum you don't need to keep manually pressing F9. http://www.poptools.org/

 

[Buzz Bloom]

If you do choose to stick with spreadsheets, I'm inferring Excel.

Hi Stone:

I am using LibreOffice Calc. This is more-or-less compatible with Excel. It can input an Excel spreadsheet, and also convert its format to an Excel format. These conversions more-or-less work, although here are a few incompatibilities.

Regards,
Buzz