1. W H Y
E V E R Y
E L E C T I O N
I S
R I G G E D
A B R I E F E X P L O R AT I O N I N T O
T H E G O A L S O F A N
E L E C T O R A L S Y S T E M , A N D
H O W T H E Y A L L F A I L
3. Motivation
• An electoral system only
functions if the electorate has
faith in it
• Without faith in an election
there will be political and civil
unrest
• By explaining that your vote
doesn’t matter we sow chaos
5. Election
Guiding
Principles
• All elections should be free and fair and open
to everyone.
• In 1951 Kenneth Arrow wrote a book entitled
“Social Choice and Individual Values” and
listed 3 “fairness” criteria:
- Unanimity: If every voter prefers candidate
A over B then the group prefers A to B.
- Independence of irrelevant alternatives: If
every voter’s preference between A and B
remains unchanged then the group’s
preference between A and B also remains
unchanged, even if preferences between A
and C or B and C change.
- Non-Dictatorship: There is no “dictator” ie.
No single voter possesses the power to
always determine the group’s preference.
7. Let’s formalise this theorem first
Let G be a set of outcomes, N a number of voters and we shall denote
the set of all full linear orderings of G by L(G).
A social welfare function (which is just an economic term for a function
that ranks social states by preference) is a function:
𝐹 ∶ 𝐿(𝐺)𝑁
⟶ 𝐿(𝐺)
Which aggregates voters’ preferences into a single preference order
on G. An N-Tuple
𝑅1, … , 𝑅𝑁 ∈ 𝐿(𝐺)𝑁
Is a called a preference profile. Arrow’s Impossibility Theorem states
that whenever G has more than 2 outcomes the 3 fairness criteria
become incompatible
8. Unanimity
If A is ranked strictly higher than B for every ordering 𝑅1, … , 𝑅𝑁
then A is ranked strictly higher than B by 𝐹(𝑅1, … , 𝑅𝑁)
9. Independence of Irrelevant Alternatives
For two preference profiles (𝑅1, … , 𝑅𝑁) and (𝑆1, … , 𝑆𝑁) if every
individual ‘i’ ranks A and B the same in Ri as in Si then A and B
will the same ranking in 𝐹(𝑅1, … , 𝑅𝑁) and 𝐹(𝑆1, … , 𝑆𝑁).
10. Non-dictatorship
There is no individual ‘i’ whose preference always prevails:
∄ i ∈ 1, … , 𝑁 such that ∀ 𝑅1, … , 𝑅𝑁 ∈ 𝐿 𝐴 𝑁
if voter i ranks A
above B by Ri then this implies A will rank above B by
F 𝑅1, … , 𝑅𝑁 .
11. Proof of the
Theorem
Step 3 Show that there exists a dictator
Step 2 Show that this pivotal voter for B
over A is a dictator for B over C
Step 1 Prove that there is a “pivotal” voter
for B over A
12.
13. 1. Prove that
there is a “pivotal”
voter for B over A
There are 3 candidates A,B,C but everyone
prefers A to B and C to B. So B just sort of
sucks. By unanimity, society prefer A and C to
B. Call this profile 0.
Arrange all voters in a random but fixed way.
For each ‘i’ let profile i be the same as profile 0
but move B to the top of the ballot for voters 1
through i.
After a certain number of votes are cast for
them B will eventually rank higher than A. The
person who puts them over the line is called
the “pivotal voter for B over A” and we will call
the profile where this happens profile k
14.
15. 2. This pivotal voter is
a dictator for B over C
To show this pivotal voter is a dictator for B over C we must
prove that it doesn’t matter how everyone else votes.
Call all voters 1 through k-1 segment one and voter k+1 through
N segment two.
Suppose:
Segment one ranks B>C>A
Pivotal voter ranks A>B>C
Segment two, ranks A>B>C
Because our pivotal voter ranks A>B then the societal outcome
must have A>B. Also, because B>C for every voter the societal
outcome must have B>C
16. 2. This pivotal voter is
a dictator for B over C
Now suppose:
Segment one ranks C>B>A
Pivotal voter ranks B>A>C
Segment two, rank A>C>B
In other words our pivotal voter has moved B above A and
everyone else has moved B below C. Well, this is the same set
of results that we had in profile k in Step 1 except C has been
changed. So, B must rank above A in the societal outcome.
Furthermore A must rank above C because of IIA.
So, the societal outcome will rank B>C even though our pivotal
voter is the only one to rank B>C. This makes him a dictator for
B over C
17.
18. 3. There
exists a
dictator
What we can do is apply the first 2 steps to find the
pivotal voters for B over C and C over B.
If we apply step 1 to find the pivotal voter of B over C by successively
moving B to the top of more ballots, the pivot point where society will
rank B>C must come before our dictator.
Reversing the roles of B and C we find the pivotal voter of C over B
must come after dictator. We have shown then that:
𝑘𝐵
𝐶
≤ 𝑑𝑖𝑐𝑡𝑎𝑡𝑜𝑟 ≤ 𝑘𝐶
𝐵
𝑘𝐵
𝐶
≤ 𝑘𝐵
𝐴
≤ 𝑘𝐶
𝐵
If we repeat this entire argument with B and C switched we also have:
𝑘𝐶
𝐵
≤ 𝑘𝐵
𝐶
Putting this all together we get:
𝑘𝐵
𝐶
= 𝑘𝐵
𝐴
= 𝑘𝐶
𝐵
Lastly, we can repeat all 3 steps for different pairings to show that all the pivotal voters
occur at the same position and so this voter is the dictator for the whole election
“Why Every Election is Rigged”: A brief exploration into the goals of an electoral system, and how they all fail. Hi my name is Stephen. I’m a 3rd Year Applied Maths and Physics student and I’m hoping that 2nd part won’t disadvantage me here. I’m also a massive politics nerd and I’m going to talk today about how your vote doesn’t count.
First let’s go through the plan of action on what I hope to discuss in the next 9 to 9.5 minutes or so. A campaign plan if you will. That was an election joke. I can’t promise it’s the only one but I can promise they likely won’t be funny. First I’m going to discuss the Fairness Criteria. These are the guiding principles for any election. I’ll go on to explain Arrow’s Impossibility Theorem and why our Fairness Criteria never work. We’ll formalise it and then go onto to prove it.
Before beginning any project/presentation I like to ask what is our motivation? What are we hoping to set out to achieve? Well, an electoral system only functions if the electorate has faith in it. Without faith in an electoral system there will be political and civil unrest. So, by explaining that your vote doesn’t matter we sow chaos and there is no aim more virtuous than chaos.
So our goal is Civil and Political Unrest
What are the guiding principles for any election? That all elections should be free and fair and open to everyone. Not quite. An economist by the name of Kenneth Arrow wrote a book and described the follow 3 fairness criteria. Unanimity, where if every individual voter prefers one candidate to another then society prefers that candidate. Independence of Irrelevant Alternatives which states that if each voter’s preference between 2 candidates stays the same then societies preference between them should stay the same regardless of how well or poorly another candidate does. And Non-Dictatorship, so no one person gets to decide the election. Arrow was nice enough to include a theorem in this book as well…
No ranked voting system satisfies all 3 criteria. This is Kenneth Arrow, a strapping fellow who I’m sure was all the rage with the ladies.
Let’s start by formalising the theorem. Let G be a set of outcomes, N a number of voters and we shall denote the set of all full linear orderings of G by L(G). This is basically just ordering the outcomes. A social welfare function (which is just an economic term for a function that ranks social states by preference, so most to least desirable) is a function: f maps collection of all votes to a single society outcome. It aggregates all the preferences to tell you who society prefers. This N-tuple is called a preference profile and is a list of every voters ballot. Arrow’s Impossibility Theorem states that whenever G has more than 2 outcomes the 3 fairness criteria become incompatible.
Just as we formalised the theorem, let’s do the same for the criteria. If A is ranked strictly – not the dancing - higher than B for every ordering of votes then A is ranked strictly higher than B by our aggregation of votes. ie Society prefers to A to B
Independence of Irrelevant Alternatives. For two collection of votes R1-Rn and S1-Sn. If everyone ranks A and B the same in each then that should be the same for society’s preference. It doesn’t matter if candidate C beats them both in Ri but loses to them both in Si.
Finally, Non-Dictatorship. There does not exists an ‘i’ element of 1-N, so a voter, such that for every vote cast if they vote one way then that determines the outcome of society’s preference.
Now, let’s turn out attention to proving this theorem. This proof isn’t exhaustive. It doesn’t account for ties but gives a good concept of why the criteria fail. It’s a 3 step proof. Step 1 - Prove that there is a “pivotal” voter for B over A. Step 2 - Show that this pivotal voter for B over A is a dictator for B over C. Step 3 - Show that there exists a dictator. Actually, we can skip straight to Step 3 and show there’s a dictator.
Woops, sorry. That’s a different kind of dictator. Let’s go back to Step 1 shall we?
First, prove that there is a “pivotal” voter for B over A. Imagine there are 3 candidates A,B,C but everyone prefers A to B and C to B. So B just sort of sucks. By unanimity, society prefer A and C to B. Call this profile 0. This is our starting point. Now we arrange all voters in a random but fixed way. For each ‘i’ let profile i be the same as profile 0 but move B to the top of the ballot for voters 1 through i. I’ll explain this more in a second. After a certain number of votes are cast for them B will eventually rank higher than A The person who puts them over the line is called the “pivotal voter for B over A” and we will call the profile where this happens profile k
We start with everyone disliking B. For Profile 1, Voter 1 puts B on top but everyone else has him last. For Profile 2, the first 2 voters put B on top but everyone else has him last. Eventually we will reach voter k where if k voter for candidate B then B will rank above A. Voter k is our pivotal voter. Obviously profile N has everyone voting for B so B will win the election.
Step 2 we need to show this pivotal voter is a dictator for B over C. To show this we have to prove that it doesn’t matter how everyone else votes. Let’s split the electorate into segment. I won’t take any accusation of gerrymandering. Call all voters 1 through k-1 segment one and voter k+1 through N segment two. Now, suppose we have this set of result where our pivotal voter ranks A>B. This is essentially profile k-1. Because our pivotal voter ranks A>B the societal outcome will be A>B by how we’ve defined our pivotal voter. Also, every voter ranks B>C so the societal outcome must have the same. Now say we switch some of these votes. We’re going to move B down a ranking in segment 1 and 2 but move B up in our pivotal voters ranking.
This is the set of ballots we will receive. Note that we have only moved B in our rankings. If you look, our pivotal voter has now ranked B>A so by definition B must rank higher than A in the societal outcome. However, we only moved B on the ranking so our comparison between A and C must stay the same because of IIA. Therefore the result will be B>A>C. So society is ranking B>C even though our pivotal voter is the only one to rank B>C. This makes him a dictator for B over C.
Switching A and B on the ballot of voter k causes that change in the outcome of the election. It is irrelevant how every other person votes.
Finally, we need to show that a dictator for the whole election exists. We apply the first 2 steps to find the pivotal voter for B over C and vice versa. If we gradually switch voters’ ballots like in Step 1 then the pivot point for B>C will come before the dictator and by reversing the roles of B and C the pivot point for C>B will come after the dictator. We have shown that the position of the pivotal voter of B over C comes before the dictator which comes before the position of the pivotal voter of C over B. We also know our dictator is the pivotal voter of B over A. We can repeat this entire argument with B and C switched and come up with the opposite result. The only way for these both to be true is if they are all equal. The final step is to repeat the everything for different pairings to show that all the pivotal voters occur at the same positions. So this voter is the dictator for the whole election.
And they seem quite happy about that
This is an alternative proof but I wanted the opportunity to get some stick people in there.
So the final question is, “What is our result?”. Well if your vote doesn’t count then the solution is to make sure you are the dictator. That or be the person setting the rules. If it is all rigged there was really only one predetermined outcome, victory. Thank you
