By Molly Kozminsky

Although we have to wait until December 18 for the next Star Wars movie, some characters have already made an appearance in our lives as toys. Eager fans lined up for the release of the new toys at midnight on Force Friday, September 4, giving retailers a taste of the Black Friday crowds to come. So how should stores prepare for the upcoming sea of shoppers craving Star Wars swag?

Holiday shoppers on the warpath
Image credit

Scientists have looked for analogies to study and model crowds and in the process have turned up some surprising results. One way to investigate how large groups of people move is to consider them as a “thinking fluid,” or a literal sea of people. To describe how people move, we need to make a few assumptions. First, we assume that the speed of each person is dependent on how many people are in a given area (the density), how the surrounding people are moving, and the physical characteristics of the ground on which the person is moving. The second assumption is that we are describing a crowd that moves with purpose – that is, there is a reason (desire for Star Wars swag) the crowd is moving in a common direction. Our last assumption is that people want to move to their destination in the shortest amount of time while at the same time avoiding areas that are too crowded and therefore uncomfortable, such as a horde of people rushing to grab the hottest new toy.

As with many assumptions used to help study complex situations, they seem reasonable and work for many situations, but are not universal. For example, some of these assumptions start to break down when the people don’t have enough information to make decisions about where their target is or what path minimizes the time to reach that target. However, these assumptions can be used to build a set of equations that have allowed us to understand some apparent contradictions in crowd flow.

Braess’ paradox: Why adding roads doesn’t always reduce traffic

One such contradiction came up in the study of vehicular traffic flow, known as Braess’ paradox. This paradox describes the situation where adding more roads to try to clear up traffic sometimes makes travel time worse. This is because the traffic flow can think, and there is a difference between optimal routes for individuals and the group. Imagine you have the choice between two routes to work. One route might take you a bit longer on average, but shortens the overall commuting time for all the drivers. This is optimal flow. Alternatively, if you take the other route, it might shorten your commute but negatively affect other drivers. This is critical flow. The paradox of the contrast between optimal and critical flow does not occur in every example of an expanded road network, but when it does, it can be corrected for in some instances by using tolls, which nudge crowd patterns closer to optimal flow. Additionally, this paradox can be applied to come up with a unique solution to avoid crowd formations: placing obstacles in the way.

So what should retailers do this Black Friday?

This reasoning is why if I were a retailer peddling Star Wars merchandise, I would hide my toys behind things rather than create a massive display at the entrance. Braess’ paradox for pedestrians proposes the placement of an obstacle in front of a target sought by pedestrians. This solution can be controversial; putting barriers in front of emergency exits to prevent stampedes seems like it could be dangerous. Turns out, it actually allows people to leave more efficiently since it relieves the “pressure” built up by everyone seeking the same path to the exit, an example of critical flow. In our example, a conspicuously placed barrier in the form of the light saber-wielding villain Kylo Ren might be just the thing to ease the rush of consumers craving the world’s cutest droid.

Further reading about crowd flow:

About the author


Molly is a PhD student in Chemical Engineering at the University of Michigan. Her research involves the use of microfluidics (tiny channels) to isolate and study cancer cells traveling in the blood stream. Molly’s undergraduate degree is from MIT, and her master’s degree is from the University of Michigan. Outside of the lab, Molly enjoys reading, running, and optimizing her baking experiments.

Read all posts by Molly here.



Hughes, Roger L. The Flow of Human Crowds’. Annu. Rev. Fluid Mech. 35.1 (2003): 169-182. Web. 19 Oct. 2015.

Braess, Dietrich, Anna Nagurney, and Tina Wakolbinger. ‘On A Paradox Of Traffic Planning’. Transportation Science 39.4 (2005): 446-450. Web. 19 Oct. 2015.

Pas, Eric I., and Shari L. Principio. ‘Braess’ Paradox: Some New Insights’. Transportation Research Part B: Methodological 31.3 (1997): 265-276. Web. 19 Oct. 2015.



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