June 9, 2015

# Politics & Mathematics | Legislative cooperation with Nash Equilibriums

Brace yourselves, because this is one long and detailed blog post. You have been warned!

Why would opposing parties work together to pass bills that obviously favor the ideology of one over the other? What forces, seemingly more important than the voters that got them into those positions, persuade political actors to act ostensibly against their interest?

There are a multitude of factors that affect the result in the previous paragraph, but by far, one of the most important is cooperation. Let’s start with a counter example (and please, disregard the unfortunate parallels between the former and latter actors mentioned).

Imagine, for instance, that two robbers decide to rob a bank together, and set up ground rules for this job, perhaps like this:

1. This is a one time job.
2. We split the loot equally.

The robbery starts well, and they get the loot, and stash it away. The police catch them afterwards, and decide to interrogate them separately (this story is just a restatement of the classic ‘prisoner’s dilemma’), as the police know that there is insufficient evidence to convict any of them with just the evidence that they collected.

Therefore, every robber in this situation has two options: To Confess or to Not Confess. If both of them, independently (as they are being interrogated separately and do not know what the proceedings are with the other robber) decide to Not Confess, the police have to set them free, and they can split the loot as agreed (we could simplify this loot and say it has a value of “10”).

If however, one confesses and the other one doesn’t, the former gets free for helping out the police (and therefore, can collect the total loot), while the latter gets the harshest penalty available, 10 years (expressed negatively as -10). The last option, is that both of them confess. If that were to happen, both of them would get a slightly more lenient penalty of 2 years in jail each.

We could attempt to simply this scenario (a “game” of sorts) with a visual guide:

1. 2 players (Robber A and Robber B), decide on 2 different…
2. Actions: Whether to “Confess” or to “Not Confess”. They do so attempting to get the following…
3. Payoffs: 2 years in jail if both confess (a “payoff” of -2 each), the full loot if they confess but the other party does (10, positive in the former case, and -10 for the latter) and finally, 5 when they split the loot.

Visually, it could look something like this. This is what’s called a normal form representation of a game. The game in question is known as the Prisoner’s Dilemma (for obvious reasons).

In the previous game, Robber A gets payoffs from the first number in each cell, while robber B gets payoffs from the second number in each cell. With this in mind, a couple of things should be readily apparent.

First, if any robber were to maximize their payoff, the best result they could possibly strive for, would be to confess while their partner in crime does not, therefore taking the maximum loot.

Second, the scenario where any of them is at a maximum without making the other actor worse off is where they both avoid confessing. This result is called a Pareto Efficient allocation.

And third, perhaps most importantly and jarring: by going only through their own payoffs (that means not taking into account what the other person is doing), any robber in this situation would choose to confess rather than to not confess. This is of particular importance as this is not the best result they could both get if they had some way of knowing what the other party is doing. This is called having a dominant strategy (in this case, to confess), and the resulting scenario, where they both get a payoff of 2 years in jail, is called a Non-cooperative Equilibrium, also known as a Nash Equilibrium (from John Nash Jr., mathematician and game theorist extraordinaire, also made famous by Russell Crowe in his portrayal of Nash in ‘A Beautiful Mind’).

The next figure shows these dominant strategies with arrows. Robber A, going just by their own actions, and considering the actions of the other robber as constant (caeteris paribus), would maximize their payoffs by confessing: When Robber B confesses (first column), Robber A’s best response is to confess as well, and move from -10 to -2 (as shown by the red arrow point upwards). When Robber B does not confess, again Robber A’s best response is to confess! Thus sending their partner in crime to jail!

The same occurs for Robber B, whose dominant strategy is shown in blue arrows in this case: Regardless of what Robber A decides, the actions of Robber B always lead them to confess, which again, is not the best case scenario for both of them.

This is all well and good, but what does this have to do with politics? As seen, the incentives to avoid cooperation lead both players to the worst outcome possible (the non-cooperative or Nash Equilibrium in this case), where they could benefit tremendously if they were just to cooperate.

Imagine however, that instead of robbers we have legislators, trying to pass amendments, laws or acquiring benefits for their constituents. They can’t possibly do it alone, can they? They usually require some other politicians, perhaps from different parties or different regions to vote according to the wishes of this first politician.

If legislative bodies had a beginning and an end for terms, it would be like repeating the Prisoner’s Dilemma over and over again with an end in sight. With a process called backwards induction, one can demonstrate that any number N of repeated games will lead to the same outcome if the N is finite (if we play Prisoner’s Dilemma 1, 10, or 1,000,000 times).

One of the most elegant solutions for this problem, that allows for cooperation, is removing term limits in legislative bodies: By allowing a (theoretically) infinite repetition of games, not only do the players (the politicians or legislators) can get accustomed to each other, but also can work to avoid the Nash Equilibrium and foster cooperation. But again, the question arises.

Why would they do that? And the answer is the following: Because, the payoff for the politician comes in many ways. While the robber gets a pecuniary payoff, a politician can get a positive payoff via a new infrastructure project to be built in their district, some favorable laws enacted for their constituents, and perhaps most importantly, these things are to be shown to the electorate in the next election cycle. You have these positive outcomes, and the electorate rewards this politician with a job, to keep doing these things, and to keep playing the Prisoner’s Dilemma with several other players that can also cooperate.

So it’s all an ongoing game that, if played “correctly” by politicians (I’m not sure about prisoners), can lead to tremendous progress in their community. It’s all about mathematics and cooperation.

Guest post by Erasmo Rigoberto Gonzalez

Data Scientist @ Majoritas

Photo Source: theguardian.com