[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.