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# traffic_chapter - Chapter 7 Modeling Network Trac using...

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Chapter 7 Modeling Network Tra ffi c using Game Theory 7.1 Tra ffi c at Equilibrium We began Chapter 5 by noting that the problem of traveling through a transportation network, or sending packets through the Internet, falls squarely in the domain of game theory: rather than simply choosing the shortest route, travelers need to evaluate routes in the presence of the congestion resulting from the decisions made by themselves and everyone else. In this chapter, we develop models for network tra ffi c using the game-theoretic ideas we’ve developed thus far. In the process of doing this, we will discover a rather unexpected result — known as Braess’s Paradox [38] — which shows that adding capacity to a network can sometimes actually slow down the tra ffi c. Let’s start by modeling the e ff ects of congestion, and then introduce the game-theoretic aspect after this. First, we model our transportation network by a directed graph: we imagine the edges to be highways, and the nodes to be exits where you can get on or o ff a particular highway. There are two particular nodes, which we’ll call A and B , and we’ll assume everyone wants to travel from A to B . (We can imagine that A is an exit in the suburbs, B is an exit downtown, and we’re looking at a large collection of morning commuters.) Finally, each edge has a designated travel time that depends on the amount of tra ffi c it contains. To make this concrete, consider the graph in Figure 7.1. The label on each edge gives the travel time (in minutes) when there are x cars using the edge. Now, suppose that 4000 cars want to get from A to B as part of the morning commute. There are two possible routes that each car can choose; for example, if each car takes the upper route (through C ), then the total travel time for everyone is 85 minutes, since 4000 / 100 + 45 = 85. The same is true if everyone takes the lower route. On the other hand, if the cars divide up evenly between the two routes, so that each carries 2000 cars, then the total travel time for people on both routes is 65. 123

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124 CHAPTER 7. MODELING NETWORK TRAFFIC USING GAME THEORY B A C D x/100 45 45 x/100 Figure 7.1: A highway network, with each edge labeled by its travel time (in minutes) when there are x cars using it. When 4000 cars need to get from A to B , they divide evenly over the two routes at equilibrium, and the travel time is 65 minutes. Equilibrium tra ffi c. So what do we expect will happen? The tra ffi c model we’ve described is really a game in which the players correspond to the drivers, and each player’s possible strategies consist of the possible routes from A to B . (In our example, this means that each player only has two strategies; but in larger networks, there could be many strategies for each player.) The payo ff for a player is the negative of his or her travel time (we use the negative since large travel times are bad).
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