The Mathematical Peril of Democratic Voting
Let’s explore how it is entirely possible for those who are in a position to set agendas to pass a policy that is detrimental and far removed from what the current policy is.
By drawing attention to a cognitive bias, we can visually explore how agenda-setters can change policies over time.
Transitivity
Let’s start with the principle of transitivity. In the following number line, we can see 3 values. We’ll arbitrarily label them as a, b, and, c.
The principle intuitively states that if a>b and b>c, then a>c. That’s pretty simple, right? But there are some things that it doesn’t apply to, like voting in a democratic society, for example.
Take a look at the following example:
The first person wants Policy A, likes Policy B, and dislikes Policy C. Second person wants Policy B, likes Policy C, and dislikes Policy A. Third person wants Policy C, likes Policy A, and dislikes Policy B.
If there were a vote between Policy A and Policy B, Policy A would win, since two voters like it.
Likewise in B vs. C, B wins. Now if you were to hold a vote against A and C, you might think transitively that A wins over C.
But if a vote was held, C would actually win against A.
Voter preferences in general don’t follow the rules of transitivity. This creates a very interesting problem.
The Agenda Setters
Imagine if Policy C were the current one, and there was a vote to switch to Policy A. In the current setting, it wouldn’t change, as C is preferred to A.
But what if we first voted Policy C vs. Policy B? Since B is preferred to C, it would shift to B, then a vote can be held between A and B, moving the policy ultimately to A.
Manipulation of the agenda can completely change what policy we end with. With one agenda, we stayed with Policy C, but in another, we first switched to Policy B, then to Policy A.
We can further explore these implications with mathematics.
Representing Policy on a Graph
We first need a way to represent policy visually. On any given policy, you will have x amount of spending on a certain topic, and y amount on another. You can then plot the policy on a graph, with each axis being some spending amount. (Graphs can go beyond two variables, but here we will keep it simple with two.)
Now we can also plot a voter on this space that occupies an ideal policy. This would be their preferred policy.
With this method, we now have a way to measure a voter’s preference. We will assume that a voter will always prefer a policy that is the shortest geometric distance from them.
Given two policies, a voter can then measure the distance between the two, and choose the one that is shortest. That policy is the closest to their true ideal policy, after all.
So let’s now explore how a series of well-chosen votes can lead to a particularly undesirable outcome.
In the following image, we have a society of 5 voters, each of which is at their ideal policy. The current policy is labeled in blue. As you can see, the current policy is at a reasonable distance from everyone, leading to a decent compromise.
Now imagine we wanted to switch to a new policy, colored in red. As you can see, it’s quite far away from everybody’s ideal policy, and thus very unlikely to pass. But using the power of agenda setting, we can actually do it.
The goal is to propose a series of policies that the citizens will vote for, one after another until we arrive at the policy we want. Just like our example from above, where switching from Policy C to A was not possible at first, but by switching to Policy B first, then to Policy A. Let’s see how we ultimately convince voters to support a policy that is far worse.
Voter Manipulation
The idea is simple. In this five-person society, we make a policy that is slightly better for three of them, the majority, and worse for two of them. Remember that a person will vote for the policy that is geometrically closest to them. So we can draw a line from a person’s ideal policy to the current policy, and draw a circle. In doing so, we can say that a person will always vote for any policy inside that circle.
By picking three voters, and drawing their respective circles, we can find the area that all three would vote in. The overlapping area is the region inside all three circles. We then pick a point in that region that is slightly better for the three voters but much worse for the other two.
Now we rinse and repeat. By picking three different voters and drawing their circles to find their overlap. Again, we find a policy that is slightly better for these three, and much worse for the other two.
Each time we propose a new alternative, the majority will always find it better than the current policy. So the alternative will win. Because voters who oppose the policy are always hurt more than those who support the policy helped, by repeating this over and over again, we can get a policy further and further away from the voter’s true preference.
By repeating the process enough, we eventually find a policy so far removed from the voter’s policies, that we arrive at the red policy. By pitting the voters against each other, we, as agenda-setters managed to manipulate voters to vote for a policy that is far worse than what they started at.
Conclusion
Note that this is a two-dimensional space and only with 5 voters, but this works with more voters and with more dimensions too. In rare cases, there may be a perfect equilibrium and a policy will not be voted against. But any slight deviation and the agenda-setters can come in.
Now obviously, the world doesn’t quite work this way. People don’t always blindly vote for whatever policy is closest to their ideals, but this model gives us a few insights, particularly on the power of agenda-setters.
However, there is a way to beat the agenda-setters: by cooperation. If a voter chooses to consider the needs of other voters before voting in just their circle, they can avoid the circle that is the Tragedy of the Commons. By taking into consideration needs other their own, cooperative voters can arrive at a policy that still benefits them, while not punishing other voters as well.
The great nuance of voting isn’t black and white as in this example. For further reading, I recommend looking into the Median Voter and McKelvey Chaos Theorem.
Now that you’ve seen this as a possibility, can you find some examples of agenda-setters in modern-day politics? Let me know.
