Friday, May 21, 2010

The Efficient Allocation of the Right of Way

This may not be representative of society in general, but lately when passing by crosswalks, I have noticed two things:
1. Cars being less willing to stop for waiting pedestrians.
2. Pedestrians being more willing to wait for a break in the traffic before starting to cross, instead of demanding that cars stop for them.
This got me thinking about the traffic rules for crosswalks (giving pedestrians the right to cross whenever they want to), and has led me to some interesting insights into why the rules are the way they are. I do not know, however, why I see so many drivers and pedestrians acting contrarily to the rules. What follows is a meditation on proper crosswalk behavior, and how economic efficiency dictates what customs we follow (or rather should follow).
To begin with, let's make some assumptions that will allow this thought experiment to take place.
Let's assume that drivers and pedestrians can all be categorized into four groups, and that people in these groups behave the same each time they approach a crosswalk. The four groups are as follows:
1. Pedestrians who wait for the cars to pass before they walk, denoted by Pw (pedestrians who wait)
2. Pedestrians who do not wait for the cars to pass before they start walking, denoted by Pnw (pedestrians not waiting)
3. Drivers who wait for pedestrians to cross, following the law as it currently stands, denoted by Dw (drivers who wait)
4. Drivers who do not wait for pedestrians to cross, just driving through and forcing the pedestrian to wait or get run-over, denoted by Dnw (Drivers not waiting)

Thus there are the following four scenarios that could happen at the crosswalk:
Pnw meets Dw (pedestrian crosses easily)
Pnw meets Dnw (pedestrian and driver both try to go, resulting in a dangerous face-off)
Pw meets Dw (pedestrian and driver both wait around like idiots, resulting in a delay until they sort out who should go)
Pw meets Dnw (Driver goes and pedestrian waits for break in traffic)

Now lets consider the costs involved in each scenario.
(denoting "meets at the crosswalk" with a /)
Pnw/Dw: the cost of the driver having to waste time and gas to stop.
Pnw/Dnw: the cost to both pedestrian and driver of a possible harmful or fatal accident.
Pw/Dw: the cost of the driver's time and gas, as well as the pedestrian's time.
Pw/Dnw: the cost of the pedestrian's time.

Clearly Pnw/Dnw entails the highest cost.
We'll return to this discussion of cost shortly, for now we have more assumptions to consider.
Let's assume that 80% of pedestrians don't wait for a break in traffic, and the remaining 20% wait for the cars to pass. This gives us probabilities that a random pedestrian will be of each group. Here are the probabilities:
("probability of an event" is here denoted by P(event))
Secondly, lets assume that 80% of drivers are those who stop for people at crosswalks, and the remaining 20% are the jerks who just plow through. This gives us probabilities that a random driver will be of each group. Here are the probabilities:
From these probabilities we can create a probability distribution for each of the possible scenarios at a crosswalk.
Because a certain type of pedestrian coming to a crosswalk, and a certain type of driver coming to a crosswalk are totally unrelated, independent events, we can find the probabilities for each situation by multiplying the driver and pedestrians' probabilities together.
So this gives us the following probability distribution:
P(Pnw/Dw) = 0.8*0.8 = 0.64
P(Pnw/Dnw) = 0.8*0.2 = 0.16
P(Pw/Dw): = 0.2*0.8 = 0.16
P(Pw/Dnw): = 0.2*0.2 = 0.04
So in this imaginary world I have created, 16% of all crosswalk encounters create a possibly dangerous showdown of pedestrian versus car, the most costly of the scenarios. 4% of the time there will be the boring situation of both driver and pedestrian wasting their time and/or gas. And the remaining 32% are efficient situations where either the driver or the pedestrian waste time/gas, but not both.
The first conclusion to be drawn from this hypothetical situation is that fewer costs will be incurred if all drivers and pedestrians knew what the rule was and followed it consistently. Let's imagine another world with different laws, where 100% of pedestrians were Pws and 100% of drivers were Dnws. Pedestrians would spend more time waiting than they do under the current rules, but the dangerous Pnw/Dnw scenario, and the extra time-and-gas-wasting Pw/Dw scenario would both be eliminated. This shows that there are clearly efficiencies to be gained from people behaving consistently as the result of clear property rights, regardless of who is given the rights in the first place. I believe this is the essence of the famous "Coase Theorem" in economics. In this case it would be cars that "own" the right of way. If pedestrians uniformly respected this property right, the result would be better than if pedestrians and cars didn't know or care about who has the right of way, resulting in accidents and delays.
With that being said, cars clearly should not have the right of way. The "transaction costs" (also a key element of the Coase Theorem) pedestrians face in crossing the street (e.g. the chance of getting run-over) are obviously much higher than those that cars face (e.g. wasting some gas). Thus to create a more efficient society, the law allocates the property right in a way that minimizes costs.
If drivers and pedestrians would just act like they understand who owns the crosswalk, it might be safer out there.

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