[For the first part of this two-part post, see here]
Alas, we are all offenders. When
we get stuck in a highway queue, with the clock ticking and our meeting time
dangerously approaching, we often act irrationally, driven by instinct rather
than rational thinking. But this needs not be so. Back home, as we sit in our
old armchair with a long drink in our hand and coolly ponder at the problem, evaluating
the odds of the various outcomes following different possible decisions, we may
gather enough insight into the matter
to recognize what was the correct way to handle the situation. This may be all
we need to be able to act wiser the next time we hit the road.
Changing lanes
while queueing up on a highway is bad: please don’t do it! The reason does not
only have to do with the danger of the maneuver. Even assuming that a change of
lane is performed in a perfectly safe way, by moving your car from one lane to
the next you slow everybody else down! That is because you force several
trailing vehicles to brake as a consequence of your maneuver. The dynamics of
car queues have been studied in detail by mathematicians, and the gathered
wisdom is that “waves” of braking vehicles propagate backwards, creating queues
in what would otherwise be a smooth flow, with a large decrease of the overall average
speed.
Ah, but I can
hear your objection: “I do not give a dime about the cars behind me, if all I
am optimizing is my own travel time” (or maybe you have other ideas on what you
would not give). Although morally reprehensible, your reasoning is correct: it
is perfectly reasonable for your loss to only account for your personal gain. Indeed,
it is your loss! Decision theory is based on precise maths, and as such
it does not include ethical considerations.
Yet, it is important to realize
that we live in a complicated and interconnected society, where individual
behavior sometimes modifies global trends. Hence we have to always keep in mind
that our own profit does eventually depend on that of all others, and usually
they correlate positively – meaning that in most situations you should care for
the profit of everybody else, to maximize your own gain. Besides, even if we
insist to not account for other people’s gain, we are likely to be influenced
by it anyway, to some extent. It is an uphill struggle for altruism to emerge
in such a complex dynamical system as today’s society; we are not ants, but we
still share resources and common goods. Eventually, our individual behavior
evolves from the feedback we get from our actions.
But then there
is another issue: more often than not, by changing lanes in the highway you do
not actually gain any travel time! The situation has been studied with computer
simulations [Tibshirani, Redelmeyer, Nature 1999 doi:10.1038/43360], and the
results –also tested in real environment— clearly show that there is an
illusion at work: even if two lanes are moving at the same average speed,
drivers will “see” more cars overtaking them in the adjoining lane than they
themselves overtake. This is due to the fact that a vehicle travels faster when
it overtakes vehicles in the other lane, than when it is overtaken. The time
difference results in a false perception.
The “faster lane
illusion” certainly makes the decision of changing lane harder than it could
be. To be fair, the advice I gave above - stay in your lane regardless of the
specific conditions you are in - is driven by safety considerations, as well as
by having in mind a loss function that appraises the common good: if everybody
followed it, we would all gain a lot, as driving would be smoother. But again,
if you only care for your own loss, let us consider that alone. Indeed, in
practical situations it does happen that a lane is faster than another. This
may occur because of a number of causes: mergings, splittings, lane closures,
accidents. How to assess the situation rationally, then?
From inside your
vehicle you are subjected to the “faster lane illusion”, so you should not base
your decision on the frequency at which you get overtaken by vehicles in the
other lane. What you should do is to stick to a recognizable vehicle in the
other lane that is in your field of view, maybe fifty yards ahead of you. A
truck will be an excellent target for this. If after some time (but don’t be
too hasty… Collect enough data first – let at least one or two minutes go by!) you
lose sight of it, and you are sure you did not pass it yourself, chances are
that indeed you are on the slower lane! This “life hack” has little to do with
probability theory, admittedly, but it is close to the best one can do in that
particular situation.
Probability
theory is still useful when you are driving your car in smooth flow situations.
A case to discuss is the choice of a route to your destination. Except when
your driving distance is very short, there are typically several possible
options. Today we are commonly helped here by automatic systems that not only
know the road map exactly, but have also access to real-time information on the
road conditions and delays, as well as on the average speed of cars in every segment
of the road.
Usually you cannot beat that: the instructions of your navigator,
or Google maps itself, will get you there sooner than any other choice. Indeed,
what these systems do is to use probability theory to make precise estimates of
the expected travel time for all possible routes, and they suggest the best one
– optionally conditional to the constraints you pre-set on the device (such as,
e.g., avoidance of tollways, or the need to pass by a certain stopover
point). They are powerful predictors, and usually they are quite reliable. Yet
they use models, and as we pointed out earlier, models may be wrong. A
statistician named George Box in fact once famously said that “all models are
wrong, but some are useful”! This is as true a statement as they come.
If you
take the same route every day, you will at some point learn nuances about the
road conditions and patterns of traffic connected, e.g., to special
dates or weather conditions. Your information bank may eventually outweigh the
precision of the navigation system! This is not as uncommon as it sounds. If, e.g.,
your route goes by a stadium, and you know that a special game of baseball (not
one that takes place every Sunday) is close to the end time there, you may
decide to take a detour to avoid the area; a state-of-the-art navigation system
would not be able to catch that nuance, as it may only detect slow traffic once
it starts, not before, and it otherwise relies on a database of slow traffic
situations that cannot account for non-periodic events. In a not-too-distant
future a navigation system may learn to scout the web for events in the area
and account for them in designing the fastest route, but so far I have not
heard of similar advances.
As in many other
situations, what you are doing when you decide for an alternative route is to
leverage your assessment –often unconscious— of the loss function. Route A,
which goes by the stadium, is quicker at this time of the day by 10 minutes
than route B, which avoids the area; but you may assess, based on your
experience and maybe even by listening to the game on the radio, a 50% chance
that by the time you get to drive by the stadium, the game will have ended and
heavy traffic will have set on in the area. In that case, you estimate a delay
of 30 minutes, again based on -alas- your road experience.
The calculation is
then easy to carry out: if you are interested in minimizing the expected travel
time, you should take route B, accepting the 10-minute delay with respect to
optimal conditions of route A. Your reasoning is that by taking route A you
would lose 30 minutes with a 50% chance, so the expectation (which is an
average over many hypothetical repetitions of the same situation) is that route
A delivers a delay of 15 minutes. In other words, given the special conditions
of the road this time, route B is faster by 5 minutes than route A today. As
you can see from this example, the loss we have used is measured in minutes
here, as we have only been concerned with our travel time. A more nuanced
discussion might have factored in other effects of our decision, such as the
payment of tolls, the safety of the route, etcetera, but at this point you
should be able to include them in the equation by yourself: all you need is to appraise
each factor and multiply it by the probability of its occurrence.
The
last situation connected to driving decisions I wish to touch on is about
speeding. Many of us consider the speed limits on most roadways out of line
with our own perception of safety at the wheel. Hence we tend to disregard
them, with a varying menu of possible outcomes. The fact that a speeding ticket
can be metabolized differently depending on your personal wealth (unless you
are driving in Finland, where speeding tickets amount to a percentage of your
income) is already an indication that in this problem the loss function is
indeed your loss, which will differ from that of anybody else,
regardless of your personal assessment of the odds of getting a speeding
ticket; similarly, the “benefits” of speeding above the limit are a quite
subjective input in the equation. This makes the example at hand an interesting
one to work out – you will have to find your own optimal working point, which
will be different from that of everybody else.
In
order to decide whether you can drive above a given speed limit, for a certain
segment of road, you need to assess the loss you expect from the different
actions you may choose to take. For each of them, you will certainly want to
size up the odds of two main elements: the probability of a speed ticket and
the probability of an accident. In addition, you need to appraise the “benefit”
of speeding (to your ego or to your endorphins) in a metric system that allows
to measure also the ticket cost and the loss caused by an accident – which can
of course vary greatly. To make matters simple we may choose to convert
everything to US dollars. The fine then comes already in that unit; e.g. it
could be a loss of US $100.
Conversely, the benefit of speeding (with respect
to obeying speed limits) does depend on your personal value of that behavior;
we assume here you personally give it a -20$ value, meaning that in general you
would be okay with giving away a Jackson portrait in order to be allowed to hit
the gas pedal at your leisure. The problem comes with the assessment of the
loss of the car accident. In an earlier example in Chapter 2 we discussed the
overtaking of a truck, and we only concerned ourselves with assessing a loss
corresponding to damaging your car when you had to swerve it off road. But here
we will be more careful, and try to consider several different possibilities.
Car
accidents come in a wide variety of severity and consequences, all the way from
the annoyance of having to be towed and withstand a minor fix at the local car
clinic, to losing your life. It may look silly to believe we can assign
credible odds to every different situation, leave alone giving a monetary appraisal
to them. And yet, decision theory is about that. If we make the exercise we
will gain some insight in the whole mechanism, and draw a few interesting
conclusions.
For simplicity let us divide the accident in three different
outcomes: a minor problem to your car, fixable with a loss of time and a $1500
check; a severe accident, which destroys your vehicle and causes a two-week
prognosis for a broken rib, with a $20,000 hospital expense; and death. In the
first case you can globally assess the loss at the level of US $2000,
accounting for the hassle of the car repair, and in the second case, you
probably give the loss a value in the US $100,000 range, accounting for the
physical distress, the lost days of work, and the car damage, in addition to
the hospital check. But what should you do with the third? We can leave this
unanswered for the time being, and get back to its appraisal once we have all
the other data, to see how it impacts the final assessment of the two actions -
speeding or not speeding.
If
you stay within speed limits, in the above example you can assume you incur in
no loss, but no gain either. So all the action takes place when discussing the
hypothesis that you do surpass speed limits, by a certain amount – which
determines the odds of the various outcomes in a way that you alone are called
to assess. In a way the problem is then simply to understand whether the “gain”
of an endorphins shot is worth the risk of the various negative consequences.
Let us say that you give the following odds to those consequences: 5% is the
chance of a ticket; 0.01% the chance of a minor accident; 0.001% the chance of
a serious accident; and 0.0001% the chance of losing your life.
The above
numbers are personal estimates: they may look unrealistic or just plain silly,
but they correspond to your own assessment, and since the only one who will
be using them is you, nobody else should complain about their value. Reflecting
your beliefs, they in a sense are perfectly valid inputs to the risk analysis
you make with them. One might argue that there is no such thing as the
probability of dying in a car accident, conditional to speeding in that segment
of the road: it being a practically unmeasurable quantity, one could say that
no number can be associated with it. We will leave this issue aside, after
mentioning that a counterargument to the objection is that by observing that
road segment for enough time we could come up with a frequentist evaluation of
mortal crashes by speeding cars. So let us see where the numbers take us.
In
the given situation we have a negative loss of $20 that we cash in irrespective
of the adverse effects; that is, we give it 100% odds: even if a car crash
awaits us at the next mile, we still got a kick from pushing the gas pedal. But
on the other hand, we have to add to it the following factors: a loss of 100$
times 10 percent, totaling an expected loss of 10 dollars; a loss of 2000$ for
a minor car accident times 0.01%, totaling $0.20 in addition; a loss of
$100,000 for a serious accident times 0.001%, totaling a further expected loss
of $1 dollar; and a loss of life with a one-in-a-million chance (0.0001%).
Alas, we were
doing so well until now: we had on one side a -20$ loss, and on the other a
+11.20$ loss from the possible negative effects of our risky driving. But here
what do we do with the value of staying alive? We can revert the problem, and
observe that if we factor the chance of death out, we have a total loss from
the other factors totaling -8.80$, i.e. a gain. In other words, with the
numbers above, unless we consider the chance of dying on the road, we should
prefer to bust the speed limits.
On the other hand, we have given the odds of
dying one-in-a-million odds. This is not too far-fetched, by the way –
statistics show that the number of fatalities in the US in recent years is of
just above one per 100 million miles traveled. Here we may e.g. be talking of a
10-mile-long road segment, and indeed the chances of dying on a car crash might
be ten times higher than average if you are speeding. In evaluating our loss,
we may thus end up with the following conclusion: if we value our life more
than 8.8 million dollars, we should NOT exceed the speed limits, as the
increased chance of dying, although still quite small (in the one-in-a-million
ballpark), weighs in too much in the total loss, which becomes positive. Not
speeding up will thus be the best option, unless you really give your own life
a ridiculously low monetary value.
The alert reader
might have noticed that we have neglected, in the above calculation, several
negative consequences that may affect the safe-side decision of obeying speed
limits: e.g., even by not speeding we are subjected to a risk (albeit a tenfold
smaller one, in our personal evaluation) of dying in a car crash in that road
segment. That is true, but the effect on the discussion is irrelevant – we can
take that loss factor out by arguing that “one in a million” is the increase in the odds of a lethal car
crash due to speeding. Similar considerations apply for the chances of
accidents and tickets.
In summary, what have we learned by the risk
analysis above? A good deal of wisdom, I would say: having been forced to give
a monetary value to phenomena and possibilities of quite different nature, we
have touched the heart of the matter. Indeed, there are things money cannot
buy. Again, to some extent this also depends on your personal judgement, but
whenever your health and your life are factors in a risk analysis, you are well
advised to err on the side of caution!
More in general, and less dramatically,
there are a number of situations when the possible outcomes of our decisions
cannot be measured on the same scale as others. Does that mean that a risk
analysis is useless in those cases? I would say no. The analysis of the various
outcomes will force us to assign relative chances of the various possibilities,
and already this is a very sound way to approach the decision making process:
you will be thinking forward, and your decision will be much better informed
than one driven by a gut feeling.
