Okay, in the last lecture, I explained traffic allocation on different routes, can be determined by Nash equilibrium in a very much simplified version of traffic game. So, now I'd like to explain how it works in reality, okay? So I'm going to present an empirical or Theo- theoretical study. About, the traffic around Hamamatsu city in Japan. Okay, so this is a map of the area around Hamamatsu city, which is in the middle of Japan. Okay. So in this empirical study conducted by Japanese researchers, they are going to compare, the real traffic allocation and Nash equilibrium prediction. So to perform the comparison between the real traffic allocation and the Nash prediction, first you need must collect some data. Okay? First set of data you must collect is the eh, is origin and destinations and number of cars. Okay. So for example, there might be lots of cars, commuting from City X here to City Y. Okay? And then you have to get some data about how many cars are commuting from City X to City Y. That's the first step. Okay? So you must collect the data and, and you can see for example, maybe 60,000 cars are daily commuting from X, City X to City Y. Okay. And you, you have to do this Destination-origin statistics. You have to collect data, from all combination of origins and destinations. It's a lots of tasks. And secondly, you have to examine, you know, the traffic allocation of each segment of road. So maybe from city x to y. One- 500 cars are commuting on this route. This brown route has 5000 cars, commuting from x to y. And maybe 3000 cars are commuting on this purple road. Okay, so for each road segment. You have to get a statistic, showing that how many cars are commuting on each segment. Okay? That's the empirical part. That's a lots of work. Okay, second, you have to compute theoretical prediction, Nash equilibrium. Okay? Nash equilibrium traffic allocation can computed. Based on the following two in- two pieces of information. Okay? First, you have to know that, you have to know, how many cars are commuting from each origin to each destination. Okay? And that's coming from the data you collected. And secondly, you have to know, the relationship between travelling time and the traffic on a given road segment. Well, you can estimate the relationship by using the data. Okay? And let me just show you the final estimated result. So, in the very simple traffic game, in the last lecture, I assumed that travelling time is a summation, between the lengths of the road and the amount of traffic. So, the relationship was linear in my simple example. But according to the estimation of those scholars in Japan, the relationship between travelling time and traffic is not linear and it looks like this. As you increase the number of crowds, traffic time or, or travelling time really doesn't change at the beginning. But if you get more and more traffic. Traveling time sharply increase. Okay? So this is the relationship between traveling time and traffic, estimated from real-life data. Okay. So by using those two pieces of information, how many cars are commuting from, say, city x to city y? And how many cars are commuting from each origin and each destination and the relationship between travelling time and traffic. You can compute a Nash equilibrium traffic allocation. Okay. Nash equilibrium traffic allocation has a property, that no single driver can save their traveling time by deviating to another route Well, it's a lots of calculation you have to perform to find out the Nash equilibrium allocation. But fortunately, there is a computer program to find out or to compute, Nash equilibrium traffic allocation. Okay? And actually, this is the end result. So in this map, different route, have different thickness. And basically thickness of routes- thickness of route indicates the amount of traffic on the route. So, this thick route here, have, has lots of traffic. And this thin line here has very little traffic. So this is the prediction performed by computer computation, to find out the Nash equilibrium around Hamamatsu city. Okay? So let's compare, the prediction of game theory, with the real traffic allocation. Okay, so let's confine our attention to this very, let's confine our attention to this very small segment. Okay? So for this segment, you can compare the real traffic and Nash equilibrium prediction. Okay? If Nash equilibrium prediction is perfect, it should be on the 45 degree line. Okay? And you can plot the comparison between actual traffic and Nash equilibrium like this point, for all road segments. There are lots of small segments. Okay, and here is a picture. Okay? The horizontal axis measures actual traffic on each segment, and the vertical axis measures the Nash equilibrium prediction of the traffic on each segment. Okay? If the prediction is perfect, those data points should be on the 45 degree line. And you can see that it's not perfect, but it does reasonably good job of predicting traffic in reality. Okay. So this number here 0.85 roughly speaking means, that 85% of variance of trans- traffic is explained by Nash equilibrium prediction. So Nash equilibrium is a very simple concept, but it does a fairly good job of predicting traffic allocation in reality. I must emphasize, that Nash equilibrium is a general purpose solution concept. Okay? So it can be applied to any social situation, but if you apply it to this particular traffic problem, it did a wonderful job of predicting real- real traffic.