## Why do we break traffic lights? – A Quantitative Analysis

January 4, 2014 at 12:05 am | Posted in india | Leave a commentTags: bayesian, break, game theory, himanshu joshi, india, probability, traffic signal

India is my country and all Indians are my brothers and sisters. So this analysis believes in right to equality and assumes each of us to be equally prone to break a traffic signal. I am sure most of us would have broken a traffic signal atleast once in your lifetime or have seen someone breaking it.

P_{b} = Probability of breaking a traffic signal by you

P_{c} = Probability of getting caught when you break a signal

P_{p} = Probability that a policeman is standing on the traffic light to catch you

This analysis uses some basic approaches of game theory and Bayesian probability models. Please drop a comment if you have doubts in the analysis.

**The Decision Tree**

To start any analysis that involves probability, we should start with creation of the decision tree for various outcomes that can happen. If we draw a decision tree for a normal scenario:

Here we assume some more things:

- The payoffs for breaking a traffic signal are definitely more than those for not breaking a signal.
- When a rider does not break a signal, he loses some time and we can define the losses in financial terms.
- When a rider breaks a signal, he saves time thus he does not lose any money (arbitrary)
- When he gets caught, he has to pay some fine and some more extra money to the policeman to get away.

**Assigning Values to Variables:**

We do not know how much the loss is, in terms of money, for the rider when he does not break a signal. We assume it to be Rs. 100 to start and then proceed with our calculations. Similarly the gains that he gets by breaking the signal is equal to Rs. 100/-

Number of vehicles at any signal: 10

Number of vehicles breaking signal: 2

Number of vehicles getting caught: 1

Probability of breaking a signal, P_{b} : 2/10 = 0.2

Probability of getting caught, P_{c}: 1/2 = 0.5

Probability that a policeman stands (we can safely assume that in India, we see a policeman 1 in 10 times), P_{p} : 1/10 = 0.1

Fine when the rider is caught: Rs. 500

**Updating the Decision Tree:**

Next we add the payoffs / losses at various options and try to calculate the average payoffs at the various option nodes.

All we are left to is to calculate the payoffs/losses at the nodes C, B & A and try to see if breaking a signal sounds more logical.

At node C:

Nett payoff = 0.5 * 100 + 0.5 * -500 = 50 – 250 = Rs. – 200/-

At node B:

Nett payoff = 0.1 * (-200) + 0.9*100 = -20 + 90 = Rs. 70/-

At node A:

Nett payoff = 0.2 * 70 + 0.8*(-100) = 14 – 80 = Rs. -66/-

This essentially means that if there is a policeman, it is more logical for the rider not to break the rule because average expected payoff (Rs. 100 * 0.5 = Rs. 50) is less than the average expected loss (Rs. 500 * 0.5 = Rs. 250). However, if there is no policeman, the payoffs for the user to break the red light are much higher (0.9 * 100 = 90). So when there is no policeman, it sounds better to him to break the rule because he is atleast getting Rs. 90/- out of his payoff of Rs. 100/- on an average.

Overall at node A, there is only one option through which he gets a positive payoff and that is by breaking the traffic signal. So it sounds more logical to the rider to break the rule.

Next Steps:

If someone is interested in further analysing the behaviour using Game Theory of economics, he can approach the problem as under:

1. Change the payoffs and probabilities with real ones.

2. Calculate the threshold value of fine below which the rider will be tempted to break the law.

3. Calculate the optimum probability of a policeman standing on a traffic signal.

Please contact me in case you need any suggestions on game theory or Bayesian probablistic models.

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