Pedestrian crowds as large games

This article has been previously published in our blog the 28th of November, 2012.

In 2006, Lasry and Lions started to work on Mean Field Games. One of the main ideas they had in mind was the modeling of individuals’ behavior in crowded places, e.g. Times Square in New York City.

How people individually behave to avoid each other and to eventually reach their exit?

Mean Field Games allow to model a large number of pedestrians which have strategical interactions and are able to anticipate what other people do.
In a crowd, pedestrians can be seen as competing players who try to find the best strategy to achieve their goal (e.g. reach an exit), given what the others do. The classical game theory is the theoretical framework to be used for such a situation. Nevertheless, the usual equations describing the strategy for the pedestrians (the equilibrium) result to be very hard to solve as the pedestrians become numerous (which by definition always happens in a crowd).

In Mean Field Games, things are made simpler because every pedestrian takes into account only statistical data about others (namely the statistical distribution of the crowd) rather than all the positions of each other pedestrian.

In other words, if I am a pedestrian in the crowd, I’ll try to optimize my path as follows: « if I anticipate that many people in the crowd will have to pass by the center of the place, then I’d probably better use another path ». This is clearly simpler than considering the following: « if this guy with a red hat and this other guy with pink shoes and this dark hair girl and so on … will pass by the center, then I’d better use another path ». There is evidence that considering the global dynamic of the crowd (or its approximation) is more truthful and simplifies the decision process.

An equilibrium in the mean field game takes place when the anticipated crowd distribution (that pedestrians use to find their optimal path) coincides with the distribution resulting from the optimal paths chosen by the pedestrians when they take this distribution as a forecast. This loop in the decision process is the statistical equivalent of the similar loop in the Nash equilibrium for game with n players.

The movie above shows an example of pedestrian crowds modeled as a mean field game. We see two groups of many people that are initially on the left hand side corners of a square. The density of pedestrians is colored from blue to yellow and red as it increases. Each group would like to move to the opposite corner, thus having to cross the other group. We can observe that, after spreading all over, one group passes first thru the center of the square and reaches its destination quickly. The other group waits until the road is free again. Here two symmetric equilibria can occur depending on the shared belief of all pedestrians about which group goes thru first. As in classic game theory when there are several Nash equilibria, to answer the question about which equilibrium will appear in real world, one needs more data. Also, if there are too many equilibria it is expected that unless some exogenous coordination is implemented, the crowd is likely to behave chaotically.

One can for instance try to figure out how all this can be generalized to social network interactions.

To be continued… that’s our job.