Vitalik has a really good post about collusion where he writes “The instability of majority games under cooperative game theory, is arguably highly underrated as a simplified general mathematical model of why there may well be no “end of history” in politics and no system that proves fully satisfactory; I personally believe it’s much more useful than the more famous Arrow’s theorem”.
This dynamic disequilibrium is a kind of strange idea and doesn’t typically conform with our notions of how things work. Surely if nothing underlying a system changes it can come to some stable point? While Vitalik gives a nice example of a majority game with no equilibrium I wanted to try to give an explicit example for the political setting.
This was mainly an exercise for myself but I wasn’t aware of a simpler model for this exact set up so I thought I’d put it up here. The problem we’re trying to represent is a system of voters who have fixed preferences over some polices and two political parties who get to pick a platform to run on each election cycle. The voters then all go to the party who is closest to them and that party then wins the election. We want to try to find a set up where even though voters never change their preferences every election cycle will see a different choice of platform being chosen.
The model is pretty simple, 3 voters, 2 policies and all voters rank where they sit on a scale of 1-10, and with that we can get the following set up:
Given this we can now get the following election cycles outcomes:
No matter what one party does the other one can always pick a different platform that will beat their rival. In this case we have a set of three non transitive platform options (5,2)>(7,7)>(4,6)>(5,2) and there is no platform that will be able to beat all three of these options and so no dominant strategy.
Science is the only news
Stewart Brand has a quote where he says “science is the only news”, the idea being that everything else moves around but doesn’t necessarily change in a fundamental way, science though has a direction of progress, forward. An advance in science is a real and lasting change in the world.
In the above example nothing about the preferences of the actual world is changing. Yet, you can imagine a wave of pundits and explanations every election cycle talking about how the electorate have been dissatisfied with the left wing policies of one government and are turning to the right. Or how the right wing has overdone it and people are craving a return to the centre. In the real world there will be more going on than in this small model but hopefully this shows that we can get seemingly radical cycles of behaviour that have far less actual meaning attached to them than one might think.
Proof
Above I kind of just said that the above system has no end to it but didn’t really prove it and it would be fair enough to question that claim. After all there are 102 potential platforms a party could take, how can we be sure that none will be stable?
To show this I’m going to use a graphical proof as I think it’s the most intuitive (and incidentally is how I generated the preferences in the first place). We can start off by plotting our voter preferences and the three platforms I put above on a grid. We then need to show that this triple of platforms are non transitive and that there is no other platform that will beat all of them head to head. We need the first bit so that if any of these platforms is chosen by one party the other always has a way to win and we need the second to make sure that regardless of what the other party is doing one of these policies will always beat it. Taken together regardless of what one party is doing one of these three platforms will always win and these platforms are in an unstable equilibrium.
Non-transitive platforms
To show this we will draw circles from each voter out to their most preferred option and then their second most preferred option. You can see that each platform is at the intersection of being one voters most preferred and a different voters second most preferred. This means regardless of what platform you chose there will be some other platform it will beat from our group (you find the voter for whom this policy is the most preferred and then you run it up against their second most preferred one). If you want to verify this you can just pick pairings of policies and use the circles to figure out who will vote for what.
No dominant platform
This relies on the fact that there is no point that is inside all of the circles for second most preferred option. We can see on the diagrams above that the three yellow segments have no point of overlap. Contrast this with the also non-transitive set up below where any party who took the green point as their platform would never be able to be beaten.
One last thing to add is that you might be thinking the model takes liberties a bit too much with allowing the parties to basically move from being left to right wing just to win elections. This is probably a fair criticism but I would just say that you can still achieve this effect without having to have such large changes if we allow for more policies and voters it just gets more complicated and also that actually flipping from one side of the isle to the other isn’t necessarily that far from reality.
Neil Hacker 