Car Game Theory

Large, heavy cars will forever roam our roads, adding to overall pollution and drinking up our gas.

There are all sorts of factors that people consider when buying a car. One of them is “I need a machine which will get me from point A to point B the fastest way possible”. While this may seem to be the essence of a car, it is probably one of the least major factors considered by buyers when purchasing a model. There are many other aspects: How much does it cost? How much maintenance will the car require? How safe is it? How much millage does it get? Does it come with any built-in extras such as navigation systems? How roomy is it, and how many people can it accommodate? Can I find parking for it? Does it look fancy? Is it a chick magnet?

The Happy Case
Let us suppose for now, that every person can design and drive his or her own car. Instead of buying pre-built models, customers can choose their features as they see fit (within engineering limits, of course). Each one would customize the car to his liking. Out of the many aspect which the consumers control, such as shape, color, engine type, etc. let us consider just one parameter: weight.
How does weight affect the car? There are some important upsides to minimizing weight. The lighter the car is, the less fuel it consumes, since less power is needed to move it. This is both an economic benefit, as well as an environmental one. The break distance is also smaller, since less kinetic energy needs to be dispersed. On the downside, light cars are less stable (and thus tend to flip or drift in strong winds), and react more poorly in accidents. **
Let us assume, however, that users can calculate the risks of crosswinds and traffic accidents, and compare them against the benefits of fuel and breaking efficiency. It might seem to you that real users don’t do this, and always opt for the safest possibility available. First of all, if it were so, then people would hardly ever use their cars at all, since staying at home is almost always safer than venturing on the dangerous roads. Second, even once we establish that people do go outside and live their lives, they obviously make some sort of trade-off between cost and safety. If they really wanted the most crash-resistant car possible no matter the cost, then they would not buy one from normal vendors, but rather have one custom made, with a body not of aluminium but of some stronger alloy, such as GLARE (we are talking about real car owners here; our fictional customizing users can choose any material they want for their frame).
Having certified that customers can partake in some sort of calculation regarding the mass of their vehicle, let’s say that they reach some optimal, relatively lightweight machine. Regardless of other features, users would opt for this weight. We might think that the entire car community will be comprised of cars all weighing pretty much the same (certainly this parameter has less variance than, say, color, which is much more suspect to user opinion). I say “relatively lightweight”, because indeed, strong winds are not much of a problem in most areas, and assuming everyone has the same light car, damage in traffic accidents is not that significant. However, the following argument is sound even if the eventual calculated weight is not what most people would call “light”.
While the idea of lightweight, efficient transportation certainly sounds good, especially to all of us environmentalists out there, it is unfortunately not a steady solution.

Collisions Are Not Nice
First, let’s talk a bit about collisions. The amount of damage caused by crashes is determined by the forces (and therefore, accelerations) involved when you collide. In general, the faster and stronger your speed changes, the more catastrophic the collision. We can divide accidents into two types: collisions with sturdy inanimate objects, such as brick walls, and collisions with smaller, more deformable and movable objects, such as other cars. Collisions of the first type tend to be quite disastrous – there is no use arguing against a slab of concrete. When you hit such a barrier, you almost immediately stop, and going from 70 kmph to 0 in a fraction of a second means a very high acceleration, which usually means total loss. Impacts of the second kind, however, are more forgiving. First, the collisions tend to last longer, as both objects are deformable. Second, there is a dependence on mass: the larger your mass is compared to theirs, the better off you are.
Consider what would happen, if a person, or a group of people, decided that they don’t like the outcome of car-car accidents. They thus decide to buy a heavier car. Sure, it would be a bit more inefficient in terms of gas, and it would pollute slightly more, but it will tend to survive accidents with other cars. Seeing that all the other cars are lightweight, there is much benefit in switching to a heavier car in this instance (for an exaggerated scenario, think of a collision between a semitrailer and a Volkswagen Golf. Who wins?) However, once these semi-selfish users have switched, the other users are at a loss: suddenly, there are massive cars out there. In fact, every one of the lightweight cars is now below the average mass. Although they haven’t done anything of themselves, in the current situation their cars tend to come out more heavily damaged in accidents. So, while they may not have wanted to in the first place, they will have to purchase similar cars, to avoid the increasingly larger disadvantages of small weight.

The Technical Details
Does this mass factor really have such a large impact on crashes? Suppose that you are unlucky enough to collide head on with an incoming car. Both your car and the oncoming one have the same mass, and are travelling at 70 kmph. Assuming an inelastic collision (none of the cars bounce back after hitting each other), both of your speeds will be reduced to 0, and the result will be a destroyed wreck in the middle of the road. If the collision takes just one twentieth of a second (about the time it takes to tear through the hull of a car at 70 kmph), that means we went from 70 kmph to 0 in 0.05 seconds, which is 389 meters per second per second – about 40g’s, or forty times the strength of gravity. Not nice at all.
Suppose now, that you yet again collide head on with an incoming car, but this time you are driving a Hummer H2, which weighs about 3200 kilograms (6400 pounds). The car in front of you, however, is a Golf Mk5, which weighs about 1600 kilograms. The collision will turn out quite differently now. Instead of a flaming wreckage in the middle of the road, the large Hummer will continue forward, pushing the smaller Golf back the way it came. Again assuming an inelastic collision, the Hummer’s speed will be reduced from 70 kmph to 23.3 kmph. This means the overall acceleration was 47 kmph in 0.05 seconds, which is 261 meters per second per second – about 26.5g’s – considerably less than when the cars were of equal mass. The poor Golf, however, got the worst of it: it went from 70 kmph to 23.3 kmph in the other direction. With an acceleration of 93.3 kmph in 0.05 seconds, it experienced well over 50g’s. So to conclude this technical portion – masses do indeed matter in car-car collisions.

The Sad Truth
The more people that buy heavier cars, the more urgent it is for the others to switch over as well. Consider the case where there is only one person with a light car; all the rest are heavy. Every accident he gets into is to his disfavor. It would be wise, in this case, to increase the weight of his car. Of course, there are intermediate states as well. It may be that a certain “critical mass” of heavy cars is required, after which the entire population makes the switch. And even if not, we can still expect an additional certain percentage of the people to switch over, their threshold for safety/efficiency having being crossed over because of the increase in heavy cars.
In the “critical mass” scenario, we are quite worse off. We started with a state in which all the cars weighed the same, and were relatively light. After everyone has switched, we are still in the same state: all the cars still weigh the same, only this time the value is higher. The current state is again susceptible to the same argument. The average vehicle mass will keep growing and growing, until it no longer pays off to buy a heavier model in the first place. It might have been better if everyone had cooperated and kept their lighter cars, but once we introduce heavier models, everybody loses.
Thus, it is much unfortunate, but a population consisting entirely of lightweight cars is not stable. The introduction of heavy cars forces some members to abandon their lightweight transportation, and perhaps even the entire population. Even worse, we cannot say that the initial group of heavy cars will never form, because, as a matter of fact, that are definite cases when large vehicles are needed (buses, media vans, trucks). It would indeed be nice, if all manufacturers started producing only efficient, lightweight cars. But if no regulation is provided (say, by the government), there is no hope that these will catch on in the majority of the population.

** As a side note, the weight and size of a car are often correlated. If we make a car larger, for example, we obviously add weight, as there is now more material in the frame. On the other hand, if we make the car heavier, we require a larger engine to run properly, so that car will need to have more space. However, we will focus only on the property of mass.

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One comment

  1. A technical note:

    It is not wise, from a safety perspective, to build your car body from a stronger material. What you want are “weaker” (i.e. more plastic) materials that would absorb more of the shock. E.g. Aluminum.

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