Recently, I have been reading up on game theory and how it can be applied to data science situations where multiple parties are making decisions that affect the system as a whole and each other. During this, I came across the argument called Tragedy of the Commons (TOTC), that illustrates the flaw in allowing individuals make decisions based on self fulfilling motives. TOTC refers to the conflict over scarce resources that results from the tension between individual selfish interests and the common good. This concept was popularized by Hardin (1968). The original example of TOTC is as such:
Consider a pasture shared by local herders. Each herder wants to maximize his yield, increasing his herd size whenever possible. However, each additional animal degrades the quality of the pasture itself. In a selfish situation, all the herders will want to maximize their herd size regardless of the damage it does to not only other herders but also the pasture itself. This is the TOTC. For those who have studied economics, this situation will be very familiar. In fact, TOTC is essentially the “free-rider” problem.
Now the natural question is to ask if there is a possible solution to the situation, perhaps some form of an equilibrium, that maximizes the yield for the herders, yet minimizes the damage done to the pasture. Let us consider a more analytically tractable problem so that we can understand the concepts better.
Imagine that there are n firms in the world, each able to choose how much air they want to consume (to manufacture some product perhaps?), and that the total air available is a constant K. Let the amount of air they each consume be k_i, for i from 1 to n. Also, each of the firms benefit from the remaining air left after consumption. For simplicity, we let each of the utility functions be logarithms. So the value function for each of the firm, given some belief on what the other firms are gonna choose is
The notation k_-i denotes what the firm believes the other firms are going to choose. This belief can be founded on an informational advantage or some common knowledge of behavior. In most introductory game theory examples, we assume that all parties are rational and that information is also common knowledge ad infinitum.
Before we can proceed to find a solution, we need to introduce the concept of Nash equilibrium, discovered by mathematician John Nash. Nash equilibrium is informally defined as a solution where the individuals make a choice that maximizes their personal good for all strategies of the other individuals. Formally, it is the solution that maximizes the value function for all possible beliefs of all other parties. A simpler iterative way to understand is this:
- For A, find the choices that maximizes his value function for every choice of B.
- For B, find the choices that maximizes his value function for every choice of A.
- If there is a solution that is both the optimal for A and B (from steps 1 and 2), then that solution is the Nash equilibrium.
This iterative procedure gives us some clue as to how to approach the problem at hand.
From our iterative procedure, we evaluated the best choices for a party for every other choice of other parties. This can also be done in the continuous analog by the following:
- Find the maximum point in the value function of A. (Note: it will be a function of the choices of the other parties)
- Repeat for all parties
- Now we will have n equations with n unknowns, which is most cases will give a intercept point (or plane, depending on the geometry of the value functions). This point is the Nash equilibrium.
Now we can attempt it for our problem. The stationary point of the value function is
Now for n parties, this linear system becomes
The solution to this linear equation is simple to derive and it gives
So the optimal amount of air to consume for the firms is the total air available divided by the number of firms plus one. This does make intuitive sense, the plus one comes from the fact that your consumption also impacts to indirectly.
As we saw from our analysis, it is important to take into account how your actions affect the resource you are using, especially because the resource quality/quantity affects your output as well. I hope you have enjoyed this exposition into introductory game theory.