Lesson 3 from Game Theory - Open Yale Course

Tipping point of Thomas Schelling

In this model, teacher told us point B (1, 0.5) is a strict equilibrium, and point C (0.5, 1) is weak equilibrium. Equilibrium B is a stable one, and it is difficult for people to move from equilibrium B to C. Equilibrium C is not stable one, and we are going to go back at equilibrium B, if we move away from C a little bit. There is a tipping point here. If you go beyond it, you will go to the other equilibrium. This point was called tipping point, proposed by Thomas Schelling.

I agree with Schelling very much, but teacher didn’t mention which the tipping point in this model. I have some personal thoughts about tipping point here and please let me explain for you.

If we want to move from equilibrium B to C, and it means there must have some people who can abandon their immediate benefits 0.5 to 0 right now, I think it is really hard for people, because people always only focus their own immediate benefits. Conversely, we have been equilibrium C (0.5, 1), a little bit people maybe 1 or two person move away, the payoffs of minority go down to the point K^/, and the majority’s payoffs go down to the point P^/. Although the payoffs of both are not equal, I don’t think the move can cause collapse of the system, because whether a system collapse or not depends on the choice the minority make. When they go down to the point K^/, they suffer a loss, but they still get payoffs more than 0.5 (equilibrium B). At this moment, the minority knows they can’t get more payoffs through their own efforts, and they will lose more and make everyone to 0.5, if all minority people move away, they have no choice, only depend on majority’s intelligence and rationality. Now the minority still has faith to this system, until the minority go down to the point K and get payoffs equal to 0.5 (equilibrium B). I think point K is the minority’s deadline, because if they go down to the point K^//, they suffer a loss than 0.5 (equilibrium B). At this moment, they don’t have to depend on others to increase their payoffs, but only on their own. They will betray immediately when a little bit down to the point K, and then system collapse just happens suddenly, because they don’t still have faith to this system. So I think the point K is tipping point in Thomas Schelling’s theory.

In the investigation of reality, a lot of tipping points are less than 50%. I think it because there is not linear function in reality. We should change our model for a little bit.

Let me change the linear to convex, it is more difficult to make equilibrium B to C, because of the low slope during AK, if someone wants to move equilibrium B to C, he will face more difficulty to get the point K. Conversely, it is easier to make equilibrium C to B, and only 20% of the minority betray to the other side, the system will collapse down. The system existence is only between X (0.4, 0.6), and the amplitude is 0.1. Well, what circumstance belongs to this function? Some system is with serious polarization between the rich and the poor, and under and middle classes are difficult to get rich through their efforts. The vast majority of people in the middle-lower class and a few people control the vast majority of the wealth. The rich get richer and the poor get poorer. The more polarization, the less X, and the system more easily collapse. This social system is not stable.

What if I change the linear to concave function? It is just the opposite to the convex function. The vast majority of people are in the middle-upper class, and the bottom of the people can easily reach the middle through their own efforts. The vast majority of people control the vast majority of the wealth. At least about 80% of the people betray to the other side, the system will collapse down. The system existence is between X (0.1, 0.9), and the amplitude is 0.4. This social system is the most stable and difficult to be subverted.

There is another thing we should consider, except the slope of the curve. What if we change the payoffs of betrayers? If we increase the payoffs from point B to B^/, the tipping point K immediately move from point K to K^/. System existence X is between (0.375, 0.625), and the amplitude will reduce to 0.125. If we decrease the payoffs from point B to B^//, the tipping point K immediately move from point K to K^//. System existence X is between (0.125, 0.875), and the amplitude will amplify to 0.375. So apparently the payoffs the betrayers get directly affect the system’s stability. 

I think research tipping point is very useful, particularly in the study of system stability. I also think everyone has their own tipping point in everything, there is no absolute loyalty, and everyone could betray you, when you cross his tipping point. Conversely, if you really want to overturn a system, you should pay attention to two things:

1, make the system more polarization.

2, pay as much to betrayers as you can.

Tipping point applies not only to a system, but also to the maintenance of the relationship between two people or two countries. “A nation has no permanent enemies and no permanent friends, only permanent interests." Winston Churchill said. Whether a man chooses betray or not totally depends on the position of tipping point. The theory of tipping point can apply to every area in our daily life.

By the way, because of the two points above, if I were Mr. President, I will do the opposite to the political asylum of Wang.