March 29, 2012

Pareto Efficient Crosswalks

Car / Pedestrian and directions of travel
Imagine this scenario, I'm sure it's already happened to you many a times. You're walking down the street, when a car is coming at you from your right. What do you do? The car could stop and wait until you cross the street, then continue on its way. On the other hand, you could also stop and wait for the car to pass by and then continue on your way (Assume there are no stop signs, traffic lights, or police officers).

Either way, somebody must stop. One person must necessarily wait for the other in order to proceed. If we assume that time saved is a normal good and that waiting is an inferior good, then obviously, a rational individual would want to minimize their waiting time and maximize the amount of time saved.

We can show the results in a payoff matrix.

Both Go
Suppose that both the pedestrian and the car decide to go without waiting. Then assume that there will be a car crash and both individuals will die and therefore cannot get to their location (Payoff of 0).

Both Wait
If both the pedestrian and car wait, then neither will ever go resulting in inaction. Prolonged inaction leads to the death by starvation or dehydration of both individuals (Payoff of 0).

Car Goes, Pedestrian Waits
Payoff matrix
If the car decides to go and the pedestrian waits, then the car saves time by not having to stop and wait while the pedestrian has to lose a bit of time before continuing on (Payoff of 1 to Car and 0 to Pedestrian).

Pedestrian Goes, Car Waits
If the pedestrian decides to go while the car waits, then the pedestrian saves time by not having to stop and wait. On the other hand, the car loses that opportunity and gains nothing by waiting (Payoff of 1 to Pedestrian and 0 to Car).

We can see clearly from the payoff matrix that the above scenario has two Nash equilibria--the two cases where one waits while the other goes. Using the Pareto Efficiency criteria, these two results are also Pareto optimal--one must necessarily be worse off for the other to be better off.

Pareto Improvement?
Revised payoff matrix
However, what if there was a way to change the payoffs? What if, you, as a Pedestrian, can alter the payoff matrix so that both you and the Car are strictly better off? That is, what if, under the case that both individuals go, both can save time and not have to wait? (Payoff of 1 to both parties).

We can quickly confirm that if this was the case--that the payoff of both going is 1 for both parties--that the new Nash equilibrium would be this exact scenario. Both the Car and Pedestrian would elect to go and not wait.

How is this possible you ask? I offer two methods that the Pedestrian can take to improve the payoff matrix.

Option 1 - Fake Left
The Pedestrian, as he/she is approaching the crosswalk, pretends to turn left. This can be done in a numerous different ways--turn your head to the left, face your body towards the left, use body language to suggest that you're turning left, step onto the road, etc.

Green arrow indicates the fake direction of travel
This way, the Car will assume that the Pedestrian is turning left and will therefore decide to not wait as there is no need to. This way, the Car receives a payoff of 1, and the Pedestrian can continue forward after the Car has already passed.

A few key assumptions that necessitate this option:
  1. The Car is not turning left, otherwise he/she would probably wait.
  2. The amount of effort / time spent by the Pedestrian in faking left is negligible.
  3. The Car travels faster than the Pedestrian.
    • i.e. The car can pass the crosswalk (if not waiting) faster than the Pedestrian can arrive at the crosswalk.
Note that faking right is not as effective as faking left because the Car cannot discern where the Pedestrian is going and will most likely elect to wait.

Option 2 - Slow Down
The Pedestrian, upon seeing the approaching Car from the right, slows down noticeably. Then the Car will assume that the Pedestrian is not crossing the crosswalk and will elect to go. This effect can be greatly increased if the Pedestrian avoids eye contact with the Car--by looking away, looking at his/her phone, etc.

A few key assumptions that necessitate this option:
  1. The slowing down of the Pedestrian does not affect the payoff matrix.
    • i.e. time lost in slowing down is negligible.
  2. The Car travels faster than the Pedestrian. i.e. The car can pass the crosswalk (if not waiting) faster than the Pedestrian can arrive at the crosswalk.
An obvious benefit or improvement of option 2 to option 1 is that the Car is now allowed to turn left.

However, is all this necessary? Would the Pedestrian want to participate in such behaviors? Is there sufficient incentive to achieve this scenario? We'd have to examine the incentives and utility function faced by the Pedestrian.

One is greater than one-half
Before taking either of the two options listed above, the Pedestrian was faced with essentially two cases--wait or go. Assuming that the probability of waiting is 0.5, then the expected benefit of crossing the crosswalk is also 0.5--that, only half the time will the Pedestrian enjoy the benefit of not having to wait.

However, in our above suggested scenarios, the Pedestrian faces no uncertainty in the payoffs--there will always be a benefit of 1. Thus, if we assume that the Pedestrian's utility is a function of time saved and all other goods, then he/she will elect to utilize either option 1 or 2.

Utility of pedestrian is a function of time saved and all other goods
Possible Economic Analysis
Of course all this can be even further complicated if you added a second good to the model economy--AOG, or "all other goods." Then you could probably do some sort of Edgeworth Box type modeling where you have the Car and Pedestrian bargain with each other before crossing the crosswalk to achieve a Pareto optimal allocation. Of course we'd have to know their utility functions, the various prices of these goods, initial allocations, and the like. Not a problem for economists!

Cars are bigger / faster than you

If all of the above economics jargon does not convince you to let cars go first, then think of this. The amount of momentum and energy lost in the car stopping, waiting, and then moving again is infinitely greater than the amount of energy it takes for a human to stop and wait. However, if you are selfish and think that wherever you are headed to is more important than wherever the car is headed to, then, please, by all means just go.

If you do not decide to use the above options, then you are faced with the choice of going or waiting. If you go, then the Car could possibly get annoyed at you. If you wait, then you could possibly get annoyed with the Car. However, if you do decide to use my suggested methods, then you can feel smug that you improved the Car's utility at no cost to you.

On a side note, come 2013 I my parents will have spent roughly $212,544 (tuition per year times 4) just so that I can use economics to analyze the above scenario.
On a side side note, if you cannot understand the above post, then my apologies. But take solace in the fact that it's just a fancy-ass way of suggesting that pedestrians should stop for cars.


Chris Dembia said...

Yo cousin, learn to make graphics with something else! This is awesome by the way.

1min said...

I purposely draw all my figures in Microsoft Paint. I think it adds character to my blog.

Thanks for the comment!

Anonymous said...

Hey! Great post! Just came across this.... and I went to Brown too! Glad to hear they're still pumping out good material, and gouging their students for tuition in the process.

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