*Inspiration for Zardoz?This is the cover illustration of the first edition of H. G. Wells', The First Men in the Moon.The 1964 film adaptation of this novel is notable for the stop-motion animation effects of Ray Harryhausen (1920-2013).(Wikimedia Commons image, colorized for artistic effect.)*

While lovers would enjoy colliding in their traverse through room space, most of the time people try to avoid other people while they are walking towards each other, as on a sidewalk. Fortunately, there's an established convention that approaching people will veer to the right to avoid collision. This convention is also followed in aircraft visual flight rules (VFR), which specify that two aircraft flying head-on will each move to their right. You can imagine pedestrian collision avoidance to take many forms depending on how closely a person would want to be to another. Such a simple problem inspired my attempt at a simple computer simulation. I decided to use the same inverse-square law that governs the force between repulsive charges, with a change in the force constant determining how closely the approaching people interact. Such physical modeling of pedestrians as interacting particles with repulsive forces has a long history.[1] In my simulation (C language source code here), our two pedestrians start from opposite corners of a room and attempt to reach their far corner. As they approach each other, they veer to their right (actually, towards the opposite wall on their right, since it's easier to program) with an intensity proportional to the inverse square of the distance between them. After passing each other, since knowledge of the direct path that they would have taken is not possible (there are no diagonals painted on the floor), they take the most direct path to their intended corner after the deviation caused by avoidance (see figure).

*Pedestrian collision avoidance simulation. a) The two pedestrians are oblivious to each other and walk directly to their opposite corners. b) As they approach each other, their inverse-square law avoidance kicks in. c) After passing each other, they walk directly towards their intended corner. d) The complete paths taken by the two pedestrians. (Data from my C language program.)*

Real pedestrians are unlikely to follow an inverse-square avoidance. Mathematicians from the Institut für Mathematik of the Universität Würzburg (Würzburg, Germany) and the Université Côte d'Azur (Nice, France) have taken a more advanced approach to this problem by utilizing a statistical mechanics approach afforded by the Fokker–Planck equation.[2-4] This partial differential equation was originally used to describe the time evolution of the probability density of particles in Brownian motion. While Max Planck (1858-1947) is known to most scientists for his discovery of the quantum of energy, Adriaan Fokker (1887-1972) is not as well known. Better known is his cousin, aeronautical engineer, Anthony Fokker, founder of the eponymous aircraft company, Fokker. Adriaan Fokker had a superb physics background, obtaining his Ph.D. in 1913 under Hendrik Lorentz at the University of Leiden. The Fokker–Planck equation was derived in his thesis. Fokker contributed to special and general relativity, but he later became interested in music theory during the Second World War. This was a reaction to the closing of the University of Leiden, where he worked, during the war. While the equal-tempered twelve-tone scale is the foundation of western music, Fokker took an interest in the 31-note equal-tempered scale first researched by Christiaan Huygens.[5] The 31-note equal-tempered scale is notable for having an harmonic seventh. The seventh in the twelve-tone scale differs from 15/8 by about one cent. Fokker composed some 31-note equal-tempered music, and he built the Fokker organ, a 31-tone equal-tempered organ.

*Assembly of physicists at the Kamerlingh-Onnes Laboratory, Leiden, 1926. Adriaan Fokker appears in the first row, fifth from the left, next to Hans Kramers (sixth from left), and Samuel Goudsmit (far right). Other notables are P.A.M. Dirac, second row, second from the left, J. Robert Oppenheimer (third from the left), and Paul Ehrenfest (fifth from left). (Via Wikimedia Commons)*

While my model involved collision avoidance when pedestrians approached head-on, the Würzburg and Côte d'Azur mathematicians examined intersecting paths. In the Fokker–Planck model framework that incorporates aspects of Nash equilibrium game theory, the probability density functions of the two pedestrians are considered. A differential game is formulated, the goal of which is to minimize the cost associated with collision avoidance. The research team demonstrated equilibrium solutions that compare favorably with the outcome of experiments conducted with actual pedestrians.[2]

*Collision theory can apply to more than just pedestrians.Wikimedia Commons image by Shuets Udono (modified).*

Game theory is a new approach to this long standing collision problem, and it seeks to minimize the cost of collision avoidance.[4] The cost is easily described by the idea that if one of the pedestrians chooses not to modify his course, forcing the other to do all the avoidance, the pedestrian who does all the avoidance would think he was treated unfairly.[4] A Nash equilibrium is reached when each pedestrian chooses exactly the one strategy that gives the best possible solution for both of them. As Alfio Borzi, Chair of Mathematics IX (Computational Science) at the University of Würzburg, summarizes,

"When the paths of two pedestrians cross, it basically comes down to the following question: What is the optimal solution of this conflict that is satisfactory for both parties... Each player gets the best possible solution, so they are all happy."[4]The motions of the two pedestrians are assumed to be stochastic drifts along a path with a control function.[2] The Fokker-Planck equation describes the probability of possible motions of a pedestrian from a starting to finishing point.[4] It was found that the calculated paths were strikingly similar to paths taken by actual pedestrians.[4] Such modeling can aid in the design of town squares, and the design of escape routes that are effective even in the event of a mass panic.[4] Says Borzi, "There are signs in current research that more and more fields of biology can be described with this theory," an example being when two animal populations compete for one habitat.[4]

- Dirk Helbing, A mathematical model for the behavior of pedestrians, "Behav. Sci. vol. 36, no. 4 (October, 1991), pp. 298-310, doi:10.1002/bs.3830360405.
- S. Roy, A. Borzì, and A. Habbal, "Pedestrian motion modelled by Fokker–Planck Nash games," Royal Society Open Science, vol. 4, article no. 170648, September 13, 2017, DOI: 10.1098/rsos.170648. This is an open access article with a PDF file available here.
- Supplementary material for ref. 2.
- Gunnar Bartsch, "On a collision course with game theory," Universität Würzburg Press Release, September 22, 2017.
- Anton de Beer, "The Development of 31-Tone Music," Sonorum Speculum, 1965, via the Huygens-Fokker Foundation Web Site.