Nash Equilibrium, Traffic Laws, and Cyclists
The concept of Nash equilibrium can be used to model a variety of everyday situations. Torkel Bjørnskau uses Nash equilibria in a paper titled The Zebra Crossing Game – Using game theory to explain a discrepancy between road user behaviour and traffic rules (2015) to express why the behavior of cyclists and drivers persistently differs from traffic laws at key points of interaction.
In Norway, there is a combined cycling and pedestrian lane that is separated from motorized traffic. Cyclists are expected to share the same areas as the pedestrians including so-called “zebra crossings” where pedestrians cross the street. Pedestrians have the right of way at these crossings and cars must yield. Cyclists, however, are treated as vehicles if they do not dismount and must yield to cars and pedestrians, but are treated as pedestrians if they dismount and walk their bikes through the crossing. Three options are available for cyclists:
- (A) Cyclists remain mounted and yield to cars and pedestrians
- (B) Cyclists cycle over the zebra crossing (potentially risking collision with cars or pedestrians)
- (C) Cyclists dismount and walk over the zebra crossing (and are thus treated as pedestrians so cars must yield)
Two options are available for drivers in these situations. Drivers may either:
- (X) Drive on as the law suggests
- (Y) Give way to the cyclist
Based on traffic laws, A/X and C/Y are acceptable strategies. However, the strategies that follow traffic laws were not commonly observed during surveillance of the crossings undertaken by Bjørnskau. Actual behavior instead seemed to favor the strategy proposed by game theory.
Bjørnskau sets up the interaction and payoffs between drivers and cyclists as a game in “extensive form” because parties know other parties’ previous moves in the game. The ascribed payoffs aim to capture the normal preferences of drivers and cyclists.
We can also model the game using the tools we learned from class and determine the Nash equilibria this way.
As we can see, the two pure Nash equilibria are A/X and B/Y. One solution proposed by traffic laws, C/Y, is not an equilibrium at all. A/X, the other traffic law option is, according to Bjørnskau, a Nash equilibrium but not a “perfect equilibrium” – an extension of the Nash equilibrium applied to dynamic games such as this one – and is therefore unstable. This leaves the option B/Y wherein the cyclist cycles through the crossing and the driver yields which was also what Bjørnskau found in observation of behavior surrounding three zebra crossings.
This is a rare example where real life seems to almost perfectly fit a model, though Bjørnskau’s observations are made in a very limited setting and payoffs are somewhat arbitrarily assigned. Nonetheless, this paper shows how the concepts we are learning in class can model the real world and explain potentially dangerous discrepancies between laws and behavior.