## Scenario

You are standing by the side of a road with no traffic lights. You want to cross the road as soon as possible, but there’s a car from afar that’s about to drive through the crossing. You look down at your watch and realize that you are about to be late for school, so you decide to cross the street without waiting for the car to pass. But little did you know that the driver, too, is in a hurry. As a result, with the sharp shrieks of the tires gripping the ground, the car abruptly stops only a few inches away from you, while you and the driver engage in a heated stare-down. Or worse still, you might get hit by the car while crossing the road!

## Introduction

The seemingly artless task of crossing the street that most of us engage in every day is not as easy as it looks. It requires multiple parties to reach a mutual agreement so that the crossing can proceed in an efficient manner. The scenario described above aptly demonstrates how the failure of crossing the street can be socially awkward and at times even quite dangerous: in Hong Kong, the number of cases of pedestrians getting hit by a car while crossing the street has been consistently over 2000 from 2016 to 2020.

While in many countries such as the US, there are laws that require vehicles to stop for pedestrians at the crossing, people–the pedestrians as well as the drivers–mostly follow some sort of inexplicit social protocol that has more to do with their moral values rather than legal obligations when they are at crosswalks. As a result, pedestrians often rely on an equivocal agreement with the drivers on what appropriate action to take for both parties. The difficulty of crossing the street, therefore, lies in the reconciliation of individual interests and the communication of intentions. While people tend to do what’s best for themselves, there might always be ideological, and in this case, physical, collisions among individual needs. In addition, there are many communication barriers that deter pedestrians and drivers from effectively communicating with each other. First of all, since verbal communication is ineffective, people tend to resort to hand gestures, which are often proved inefficient. Moreover, the time window for communication is often very limited, as the car might be approaching at a relatively significant speed. Considering all the difficulties mentioned above, it’s definitely important for pedestrians to find some premeditated strategies for their safety in case they cannot effectively communicate with the drivers: and that’s where game theory comes into play.

### Model 1.0

If we were to depict the scenario above into a game theory model, there would be two players in the game: the pedestrian and the driver. The pedestrian can either wait for the car or cross the street without waiting for the car. Conversely, the driver can either wait for the pedestrian or drive past the pedestrian before the latter crosses the street. Thus there can be four possible outcomes to this game.

Driver/Pedestria |
A’. Wait for driver |
B’. Walk on |

A. Wait for pedestrian |
(a,b) |
(c,d) |

B. Drive on |
(e, f) |
(g,h) |

Outcome AA’: The driver and the pedestrian both choose to wait for each other. Nobody makes any progress on their trip, time wasted.

Outcome AB’: The driver waits for the pedestrian to walk by first.

Outcome BA’: The pedestrian waits for the driver to go first.

Outcome BB’: Nobody waits for anybody. The car collides with the pedestrian.

**Cardinal Ranking (The quantified payoff for each)**

AA’: This scenario is equally bad for both parties since it’s the least efficient in waiting time. Therefore, the payoff for this outcome is -10 for both players.

AB’: This scenario benefits the pedestrian yet costs some waiting time for the driver. Therefore the payoff for this outcome is +5 for the pedestrian and -5 for the driver.

BA’: This scenario benefits the driver because it saves the driver’s time. However, since waiting for pedestrians is considered a social etiquette in many places, this outcome would do some damage to the driver’s social reputation. Let’s say the damage in reputation is -3, the payoff for the driver is consequently +2 and -5 for the pedestrian who has to wait for the car to pass.

BB’: This scenario creates mostly financial and reputational damage to the driver and physical damage to the pedestrian. Therefore, I would argue that the pedestrian has it slightly worse than the driver. The payoff in this one is consequently -20 for the driver and -30 for the pedestrian.

Driver/Pedestrian |
A’. Wait for driver |
B’. Walk on |

A. Wait for pedestrian |
(-10,-10) |
(-5,+5) |

B. Drive on |
(+2,-5) |
(-20,-30) |

Solving the model above using a movement diagram in game theory, one can see that the Nash equilibria are at BA’ and AB’.

This means that the two optimal outcomes for both players are either

BA’: in which the pedestrian waits for the driver to go first.

or

AB’: where the driver lets the pedestrian go first.

However, the solution above is far from satisfactory. First of all, since there are two optical outcomes in the solution, there isn’t a good approach that the pedestrian can take every single time. More importantly, this game is far more complicated than what’s depicted in the model above. The most obvious reason is that the game is not exactly a simultaneous game. In other words, the driver and the pedestrian doesn’t make their decisions at the same time. More often than not, they act accordingly in response to the other player’s move.

## Model 2.0

To account for this interactive nature of players’ decision-making process, a model of a sequential game was built. Since most collision incidents between vehicles and pedestrians happen within 20 seconds, the game has two rounds, assuming players are able to make one move in every ten seconds.

In this case, the general rule of assigning utilities to each outcome is that when a player chooses to wait, that player expects the other player to proceed and vice versa.

U: This scenario resembles the outcome AA’ in the first model, where the pedestrian and the driver both choose to wait for each other. However, the caveat here is that none of them gets the expected outcome from their previous action. Therefore, this result is equally bad for the two players both in terms of the unnecessary cost of waiting time and the frustration from not receiving the desired outcome for their action. As a result, U is (-15,-15) instead of the (-10,-10) payoff in AA’, with the players’ frustration quantified as -5 in each of their payoffs.

V: This scenario resembles the outcome AB’ in the first model, where the driver waits for the pedestrian to pass. The caveat here is that the pedestrian does not receive the desired outcome for their previous move. Following the same logic as the calculation of U, V is (-5,0).

W: This scenario is exactly the outcome BA’ in the first model. Thus the payoff is (+2,-5)

X: This scenario resembles the outcome BB’ in the first model, where the pedestrian and the driver both choose to proceed. The caveat here is that the driver does not receive the desired outcome for their previous move. Following the same logic as the calculation of U, X is (-25,-30)

Y: This scenario resembles the outcome AA’ in the first model, where the pedestrian and the driver both choose to wait for each other. However, the caveat here is that none of them gets the expected outcome from their previous action. Following the same logic as the calculation of U, Y is (-15,-15)

Z: This scenario is exactly the outcome AB’ in the first model, where the pedestrian safely passes as the driver waits. Thus the payoff is (-5,+5)

S: This scenario resembles the outcome BA’ in the first model, where the pedestrian waits for the driver to pass. The caveat here is that the pedestrian does not receive the desired outcome for their previous move. Following the same logic as the calculation of U, S is (+2,-10).

T: This scenario resembles the outcome BB’ in the first model, where the pedestrian and the driver both choose to proceed. The caveat here is that none of them receives the desired outcome for their previous move. Following the same logic as the calculation of U, X is (-25,-35)

To sum up, the payoff for each outcome is listed in the graph below.

Using the backward induction technique in game theory, assuming each player will want to select the outcome with the largest payoff, the pedestrian might make the following choices:

Knowing the pedestrian’s potential choices, the driver, in turn, has to choose from the following options.

Assuming the driver wants to optimize their payoff, he will make the following decisions:

Understanding that the driver will likely choose to proceed, what’s left for the pedestrian is quite simple:

From the two possible outcomes, it’s only reasonable for the pedestrian to choose “w” over “s,” as the former gives a -5 payoff, whereas the latter gives a -10 payoff. Therefore, through using the backward induction method, one can conclude that the pedestrians, before receiving any signals from the drivers, should always wait first. While doing so will likely encourage the driver to go first and consequently cost a longer waiting time for the pedestrians, it’s the safest way to cross a street among all other possibilities.

## Conclusion

Combining the findings of the two models above, one can conclude that in order to secure their personal safety when crossing a street, the pedestrians should always wait for the vehicles. While this conclusion might contradict the social norm in some places where vehicles should always stop for pedestrians, it is a universal approach to ensure pedestrian safety on the crosswalks. Along with this theoretical approach to improving pedestrian safety, attached below is a video with some more practical tips on how to safely cross the streets.

## Works Cited

Hong Kong Transport Department, Hong Kong Monthly Traffic and Transport Digest, January 2021

https://www.td.gov.hk/filemanager/en/content_5041/2101.pdfWHY DID

Gary L. Wicker, THE PLAINTIFF CROSS THE ROAD? Understanding Pedestrian and Crosswalk Laws in All 50 States, October 29, 2018 https://www.mwl-law.com/plaintiff-cross-road-understanding-pedestrian-crosswalk-laws-50-states/

*Impact Speed and a Pedestrian’s Risk of Severe Injury or Death*(Technical Report). Washington, D.C.: AAA Foundation for Traffic Safety, 2011

Hi Zhiyuan, I really liked how you took what appears to be a simple problem and expanded it in a way that is meaningful by connecting it to game theory. I could see the amount of thought you put into it with both the matrix and the sequential game. While I was initially surprised by the result that you came up with, I think that it does end up making sense, especially in situations with jaywalkers. After all, we are told from a young age to “look both ways while crossing the street” even on walkways. However, I do wonder how the payoff matrix would change if we considered the legal consequences as well to the driver for not stopping for the pedestrian, especially in areas where the pedestrian has the right of way?

Hi Zhiyuan, I’m sorry to hear about your friend, I am glad he is okay! I was originally compelled to click on your project because I was a little confused at how simple your question seemed. After going through your whole project, I can I was quite wrong. I do not really any understand game theory but I think I was able to get most of you presented. I actually think your conclusion makes sense, as in personal experience it is harder to stop a car than a person. I think it is pretty crazy that you managed to somewhat prove that the way most of our pedestrian systems are designed is faulty. I wonder if traffic engineers would agree with your conclusion. Also, all the charts you made were super helpful!