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Chicken Game Theory Applied to Car-Pedestrian Interactions

Walking on a college campus demands thousands of interactions between pedestrians and cars on a daily basis. A given student may cross five or six streets walking from one location to another. Given that there are thousands of students on campus, walking to several locations throughout the day, student interactions with cars are very common. When a student crosses a street while a car is approaching, a small game is being played. Each player, the student or the driver, are prompted to halt or advance.

What should a student decide to do when confronted with a car coming towards them? What would be Nash’s Equilibrium?

To begin we can first analyze a classic game of Chicken defined as the following:

  • Players: Two cars
  • Game: Two cars drive straight towards each other. The first car to swerve to the side is called the Chicken, and thus incurs a negative payoff because they lost the game. The other car wins the game and receives a positive payoff. For simplicity, each player makes their decision beforehand, then reveals their decisions afterwards.
  • Payoff: Referred to Figure 1
  • Strategy: To swerve or not to swerve
  • Information: Each player knows of the payoff matrix

Each player would like to do the opposite of the other player, which means there is no Pure Strategy Nash Equilibrium. By using the Mixed Strategy Nash Equilibrium where we set the expected payoffs of swerving and not swerving equal to each other and define the probability of Car A not swerving to be p and the probability of Car B not swerving to be q, we get that each p and q must equal 1%. This means each player will not swerve 1% of the time. That is to say each player will swerve 99% of the time.

We can modify this game now such that now a person and a car are playing the game of Chicken. All previous rules of the game remain the same, except the negative utility incurred from a crash. The Car will have a higher utility, we will call UC, than the Person. The Person will have a lower utility, we will call UP, compared to the Car. This is to say that UP < UC < 0. The payoff matrix is defined as in Figure 2.

Applying the same Mixed Nash’s Equilibrium as described above where p is the probability the Person will swerve, and q is the probability the Car will swerve we get p = -1/Uc and q = -1/Up. Because we know that UP < UC < 0, that means q < p or that the probability the Car does not swerve is lower than the probability the Person will not swerve. This is to say the Car will swerve more often despite incurring less damage during a collision. Some basic reasoning behind this is that the Person will prefer to serve more often because he is afraid of the large negative utility, UP. However, the Person knows the Car knows this and in order to counter it, they will decide not to swerve. In essence, the player that suffers the most during the incident will have the most motivation to avoid the collision, but if both players know this, then the player who has the most motivation to avoid the collision will attempt to collide to force the other player to swerve. In other words, in order for the payouts for both strategies to be equal, the player who incurs a heavier loss during the collision must have less collisions to balance out for the times they collide.

This is really interesting because the Person, who seemingly has little power in this situation, will force the Car to swerve. And the Person will force the car to swerve because they need to balance out their utilities for when the crash does incur and they get negative payouts. Both players know this and must accept this outcome.

This solution works well for the simple case, however, there are many factors to consider inside of the real world application. First, players are not deciding what they are to do beforehand, and then blindly sticking with their decision. Players are constantly checking the other’s positioning, speed, and distance between each other and point of collision. Because of this, we can develop a Turn-Taking Model. Inside of the paper linked below, they define the model to be a world where players move in discrete squares, moving at speeds 1 or 2 squares per time step. In addition, players would like to minimize their time spent at each interaction so as a result players would have negative payouts in proportion to the time they had to spend to get past the other person multiplied by the amount each person values a unit of time. Using this model, we can build on top of it to have player’s make decisions based on current distance between themselves and point of collision should trajectories continue. This new type of model is defined as a Sequential chicken model, which transforms the game into a sequence of Game Theory Matrix games or “sub-games”. This type of game can be solved using backward induction state probability and the solution is drawn out in Figure 7 of the paper.

In addition this paper describes the basics of the Chicken game and the process to convert the Chicken game to a more applicable model that can be used for autonomous vehicles. The issue that arises when discussing autonomous vehicles or AVs is that it is NOT advantageous for programmers to make it so when the AV experiences an obstacle, it halts. The reasoning behind this is because then pedestrians would then have a dominant strategy to step in front of AVs since the AV will always halt. On the flip side, I would not be good either to have AVs never halt because this would cause an abundance of crashes. So replicating the current decisions making to match the Nash Equilibrium would be advantageous for both players.

 

Topics from CS2850 included: Pure Strategy Nash Equilibrium, Mixed Strategy Nash Equilibrium, Game Theory, Matrix Game Theory, Probabilities.

Source: http://eprints.lincoln.ac.uk/id/eprint/32029/1/Fox2018chicken.pdf

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