Lesson 4: Probabilistic Models of Uncertainty

Politics Under Uncertainty

On August 29, 2005, Hurricane Katrina made landfall in southern Louisiana and rapidly moved northward, passing over the area around the New Orleans. The tidal surge caused by Katrina overwhelmed the levees and floodwalls meant to protect the city and a catastrophic flood ensued. The exact number that died will never be known. About seven hundred persons were drowned during the night of August 29 and hundreds more died from lack of water, shelter and medical care in the days following the storm. Deaths and permanent displacements caused by the storm were concentrated among the area’s poor, and seemed to reveal an inability or unwillingness on the part of the city, state and federal governments to provide basic physical security for their citizens.

Either of two paths of thought might occur to you as you consider the Katrina disaster. One is retrospective: What caused all those people to die and be displaced in August of 2005? Who or what is to blame? What decisions were made prior to or during the storm that, if made differently, might have averted so much loss?

Another is prospective: Having learned from Katrina that a hurricane passing near New Orleans can cause catastrophe, what can be done that might protect the city and its residents from future storms? What would such measures cost and who should pay? Can those things be done, or must we just wait helplessly and watch it happen again?

In practice, these two directions of thought are not quite as distinct as they might at first seem. To see why, check out this image:

Hurricane paths in the New Orleans region, 1945-1960

This shows the paths of all the hurricanes that passed through the region around New Orleans during the fifteen year period from 1945 through 1960. There’s nothing special about that particular range of dates. In just happens that the U.S. Government made a visualization of the paths of hurricanes over that fifteen year period, and that visualization is an extraordinarily vivid depiction of an essential fact about New Orleans: Hurricanes regularly and frequently threaten the city. If you created a similar graphic spanning any fifteen year period in the city’s history you would see a similar thick tangle of storms, with between one and three storms passing near the city every year, and one or two storms passing directly over the city every fifteen years.

So Katrina, terrible as it was, was not seen by longtime residents of New Orleans either as unprecedented or even unexpected. In fact, floods caused both by hurricanes and inland storms occur in the city almost every year. Sometimes the water causes no more than minor damage to a handful of properties. Other times waters flood many blocks with several inches of water, causing millions of dollars in damage, and at most a few injuries or deaths. Rarely, but predictably (between once and twice every hundred years), something like Katrina occurs.

So, if we think retrospectively and ask “who is responsible for the death and destruction in New Orleans in 2005?”, any answer must hinge on what we know or suppose about the prospective thinking of residents of New Orleans and their political representatives in the years before 2005. They knew the city and region well. Just by living there – by experiencing floods throughout their lives and hearing stories of past floods handed down by their elders – they knew that hurricanes and other flood hazards were frequent, and in particular that monstrous storms were rare but always possible. Indeed, a hurricane nearly as destructive as Katrina inundated the city and killed hundreds of residents as recently as 1965. So the prospective question “what should we do to protect ourselves from the next storm”, has always been on the agenda in New Orleans. Whatever it was that turned Katrina into a catastrophe, then, stems from the politics resulting from that prospective question, asked every day of every year in the decades leading up to Katrina. Politics in each of those years at the local, state and federal levels culminated in choices by politicians to build sufficiently robust levees and other flood control infrastructure, or not; To permit and subsidize the construction of homes in flood-prone areas, or not; To destroy wetlands that if left in tact would dampen storm surges, or not; To prepare and fund adequate evacuation plans, or not.

Evidently, those politics did not produce policies and investments adequate to protect New Orleans residents and their homes from a storm the size of Katrina. In a series of decisions made throughout the 20th century, politicians at the local, state and federal level permitted the construction of thousands of homes on low-lying and sinking ground, drained and dredged swamps, marshes and wetlands that would have absorbed much of Katrina’s storm surge, built a system of levees and floodwalls too fragile to withstand a storm of Katrina’s size, and failed to plan and fund an evacuation program adequate to move the tens of thousands of New Orleans residents too poor to own personal vehicles out of harm’s way. Katrina, it turns out, is more aptly called a political disaster than a natural one. It amounted to a failure of political institutions to produce the investments required to protect persons from a natural hazard that, although irregularly occurring, was certain to occur at some point.

Can we hope for better? As climate change brings a surge of increasingly frequent and severe, and irregular but certain-to-occur natural hazards – floods, droughts, wildfires, extreme heat etc. – will our politics produce the public investments required to protect our lives and homes?

In this and the next two lessons, you’ll learn a set of tools political scientists use for modeling the politics of decisions like these – i.e. decisions made under uncertainty. At any given time, no one can know for sure when or where the next major natural hazard will occur or how severe it will be. Thus when governments make investments meant to protect persons from natural hazards like hurricanes, earthquakes and forest fires, the benefits of those investments are always uncertain. When politicians, say, raise taxes to fund seawalls that would divert storm surges away from populated areas, or burden landlords with building standards that require structures to be hardened to earthquakes, or fund controlled burns that inconvenience residents in the short run but lessen the chances of uncontrollable wildfires in the long run, no one can know for sure who will benefit, when, or how much.

These lessons teach two tools for depicting decisions under uncertainty: Probabilistic models of uncertainty and expected utility. These tools have applications far beyond the politics of natural hazards. Uncertainty is, after all, present in every aspect of life. But throughout we’ll use examples of natural hazards and the political decisions that can mitigate or exacerbate their effects to illustrate the concepts and techniques.

Probabilistic Models of Uncertainty: An Example

Imagine a city that sits on the banks of a river, just inland from where the river empties into the sea. Any such city can flood when a hurricane or cyclone comes ashore nearby, as the storm’s winds and low atmospheric pressure pull water from the sea up the river, and dump torrential rains into the river’s watershed. Imagine this city has experienced many coastal storms over the years, and scores of resulting floods. Most of these floods were small and caused no more than inconvenience. Some have been large but limited, damaging much property but harming no one. Two or three of the floods in the city’s recorded history have been catastrophic – killing hundreds and levelling whole neighborhoods.

Imagine this city has a new mayor. This mayor, rather than just stand by and hope for the best, would like to do something to protect the city from the next hurricane.

The mayor’s idea is to build a seawall between the city and the river. The wall will be impervious to water and will stand some meters higher than the river’s normal level. Its foundations will be anchored in the bedrock that lies several feet below the city’s sandy soil. So when the river rises, the water will be held behind the wall, keeping the city’s people and buildings dry. When finished, it will look something like this:

To build her seawall, the mayor must first resolve a very basic issue: Just how tall should the seawall be? As an elected politician, dependant for her job on campaign contributions from wealthy city residents and on votes from everyone else, this question is critical. The cost to taxpayers of the seawall will be exorbitant, and these costs will rise with the height of the wall. The mayor, then, will want to build the wall just high enough to keep the city dry most of the time, and no higher. So how high is that? This is a difficult question to answer, because no one can know how strong the hurricanes that pass near the city will be in any given year. A seawall too tall, on the one hand, will be more than is needed against the relatively small storms of most years. A seawall too short, on the other hand, will fail in the face of a large storm.

To depict the mayor’s choice and its stakes, we’ll use a probabilistic model of uncertainty. Suppose that the city’s historical records show that in most years, no storm pushes the river’s waters above their normal level. Storms that push the river about one-half a meter above its normal level, on the other hand, occur occasionally. Storms flooding the river to a level about one meter occur somewhat less frequently than that. Storms pushing the river to one-and-one-half meters above normal occur less frequently than that, and so on up to the rarest event, which has occurred only once in recorded history and in which the rivers surges to three meters above its normal level. Thus, supposing that surges in the river’s level continue to occur at roughly the frequency of those surges seen in the past, the relatively likelihood in any given year with which the river will rise to each possible height above its normal level looks something like this:

Relatively likelihoods of river surges of various heights

The key idea this graph depicts – and the key idea that probabilistic models of uncertainty depict in general – is that it is possible to be uncertain which of a set of events will occur, yet to have definite beliefs about the relative likelihoods of each of those events. Our imaginary mayor, for instance, does not know how high the river will rise in any given year after she constructs her seawall. But she knows how likely a surge in the river’s level of any given height is to occur relative to a surge of any other height. Examining the figure above, for instance, you can see that she knows that the river surging to one meter above its usual level is about twice as likely as the river surging two meters above its usual level.

Superimposing our graph of the relatively likelihoods of river levels of different heights on our cartoon sketch of the mayor and her city helps to clarify the implications for decision making of the information about relatively likelihoods that a probabilistic model depicts. Recalling that our mayor needs to decide how high to build her seawall, consider the choice between a seawall just tall enough to protect the city from a one-meter surge in the river versus a seawall just tall enough to protect the city from a two-meter surge, like this:

One-meter seawall versus two-meter seawall

Notice that a difference in performance between these two seawalls only occurs in very particular circumstances – i.e. when a storm raises the level of the river either 1.5 or 2 meters above its typical level. In contrast, surges smaller than one meter and surges larger than two meters result in the same outcome regardless of which of these two possible walls are built. So, in choosing between a one-meter a two-meter wall, what really matters to the mayor is how likely a surge of 1.5 or 2 meters is. If surges in that narrow range are relatively likely, the additional cost of the taller seawall may be well worth it.

For instance, suppose that, in contrast to what we assumed above, only very big (higher than 2.5 meters) and very small (0.5 meters or less) surges in the river’s level are likely, while middle-range surges (between 1 and 2 meters) are very unlikely. Something like this:

A probabilistic model in which middle-range floods are relatively unlikely

Superimposing this graph on our sketch of the mayor and her city shows that the choice between a 1-meter seawall and 2-meter seawall looks quite different:

Choosing between two seawalls when middle-range floods are relatively unlikely

In this case, the mayor gains relatively little by increasing her seawall from just high enough to stop a 1-meter surge to just high enough to stop a 2-meter surge.

Defining Probabilistic Models of Uncertainty

A probabilistic model of uncertainty has two elements: A description of a set of exhaustive and mutually exclusive possible events, and a probability distribution that depicts the relative likelihood with which each possible event will occur.

For example, we can model the mayor’s uncertainty about the height to which the river will surge above its normal level in any given year using the following probabilistic model of uncertainty:

A Probabilistic Model of Flooding

Possible Events

In any given year, the river will surge to exactly one of the following heights above its normal level: 0 meters, 0.5 meters, 1 meter, 1.5 meters, 2 meters, 2.5 meters, 3 meters.

Probability Distribution

Event	Probability
0 meters	0.25
0.5 meters	0.23
1.0 meters	0.20
1.5 meters	0.15
2.0 meters	0.10
2.5 meters	0.06
3.0 meters	0.01

There are two things to notice about this model. First, the list of possible events is highly simplistic, in that there are only seven possible heights above its usual level to which the river will surge. These heights, moreover, are laid out in precise, half-meter intervals. While a more realistic model is possible, the simplistic model here is useful for illustrating the most important feature of any probabilistic model’s description of possible events: Any such description must be exhaustive (i.e. it must specify all possibilities), and it must be precise about which events are mutually exclusive of each other. By making the simplistic assumption that the river will rise to exactly one of only six particular heights, and by specifying each of these six particular heights, this model of flooding meets both of these requirements.¹

Second, the probability distribution this model uses to depict the events’ relative likelihoods amounts to a list of numbers, with one number assigned to each possible event. More generally:

Probability Distribution

Given a description of an exhaustive and mutually exclusive set of events, a probability distribution depicts the relative likelihoods with which the events will occur by assigning a number to each event, with relatively higher numbers depicting relatively higher likelihoods.

The numbers assigned to the events are called probabilities and must satisfy the following rules:

Each number must lie between 0 and 1, inclusive.
The sum of the numbers assigned to the full set of mutually exclusive events must equal 1.

More Examples of Probabilistic Models

This section illustrates key aspects of probabilistic models used in PPT through a series of examples. We’ll start with what might be the simplest probabilistic model possible – a model of the uncertainty you face when you flip a coin. When you flip a coin, the outcome – i.e. whether the coin lands heads-up or tails-up – is uncertain. Moreover, each of these two outcomes is just as likely to occur as the other. Here is a model that captures both of these aspects of the uncertainty about the outcome of a coin flip.

Flipping a Coin

Possible Events

The coin will land either heads-up or tails-up.

Probability Distribution

Event	Probability
heads-up	0.5
tails-up	0.5

Why does this model assign the number 0.5 as the probability for both “heads-up” and “tails-up”? Recall the two rules of probabilities: Each probability must be a number between 0 and 1, and the sum of all the probabilities assigned to the full list of mutually exclusive events must be 1. Since we want to depict each face of the coin as equally likely to occur, the numbers assigned as probabilities to those events must be equal to one another. And there is only one number between 0 and 1 that when added to itself is 1 – i.e. 0.5!

A similar line of thinking is at work in the only slightly more complex uncertainty faced when one rolls a six-sided die. The die has six faces, and thus there are six outcomes instead of the two outcomes that can occur when one flips a coin. Each face is equally likely to occur. Thus, since each probability must be a number between 0 and 1 and since the six probabilities must sum to 1, the probability assigned to each face of the die must be \frac{1}{6}.

Rolling a Die

Possible Events

The die will land with one of its six faces up.

Probability Distribution

Event	Probability
	\frac{1}{6} \frac{1}{6} \frac{1}{6} \frac{1}{6} \frac{1}{6} \frac{1}{6}

\frac{1}{6}

Pause and complete check of understanding 1 now!

The two examples above are simple and clear but do not obviously resemble any important aspects of politics. Can the uncertainties we face in politics – which are vastly more complex and ambiguous than the uncertainty involved in flipping a coin or rolling a die – be depicted using a probabilistic model?

At the time of this writing (August of 2023), there is apprehension across the world that the Peoples Republic of China might launch a military invasion of the island of Taiwan at some point in the next several years. No one, excepting perhaps the paramount leaders of the Chinese Communist Party, can know for sure whether such an invasion will occur, or whether an invasion, if attempted, will end in a successful military occupation of Taiwan by the PRC.

The complexity of a potential military invasion of Taiwan by the PRC and the possible reactions to an invasion by the world’s other militaries is overwhelming. Hundreds of thousands of soldiers, pilots, and sailors from dozens of nations would likely be involved in fighting triggered by an invasion, and the fighting would set off cascades of reactions in multiple non-military domains – including in trade flows and trade routes, diplomatic relations and alliances between nations, and the balance of political power between competing factions within multiple nations. Such an invasion could even trigger an exchange of nuclear weapons, opening up the possibility for levels of destruction that could lead to the collapse of nation-states.

Can such a bewildering tangle of momentous possibilities be reasonably depicted using a probabilistic model? Yes, partially. Consider two simple questions about a possible invasion:

Will the PRC launch an invasion of Taiwan at some point between now and August 1 of 2028?
If the PRC launches an invasion of Taiwan at some point between now and August 1 of 2028, will that invasion culminate in a successful military occupation of Taiwan by the PRC at some point between now and August 1 of 2030?

These two questions set aside all of the uncertainties of a potential invasion except for two definite and conceptually simple questions – within a certain period of time, will there be an invasion, and if so will that invasion succeed? These questions are deliberately phrased so that they each have exactly two possible answers – i.e. “yes” or “no”. Thus we can use a probabilistic model just as we use all other PPT modeling tools: To depict one aspect of an otherwise un-manageably complex reality.

War and Its Outcome

Possible Events

The PRC does not launch an invasion between now and August 1 of 2028.
The PRC launches an invasion at some point between now and August 1 of 2028 but does not achieve a successful military occupation of Taiwan at any point between now and August 1 of 2030.
The PRC launches an invasion at some point between now and August 1 of 2028 and achieves a successful military occupation of Taiwan at some point between now and August 1 of 2030.

Probability Distribution

Event	Probability
no invasion by 2028	0.75
invasion by 2028 but no successful occupation by 2030	0.125
invasion by 2028 and successful occupation by 2030	0.125

Pause and complete check of understanding 2 now!

One thing about the model above is absolutely essential to recognize: In the absence of a large number of previous cases of potential invasions like the PRC’s potential invasion of Taiwan, we have no evidentiary basis for deciding whether any precise numerical probabilities assigned by the model are reasonable or valid. Consider, in contrast, the heights to which a river might flood in any given year. Using a historical record of the river’s flood levels over several decades, we can calculate the frequency with which each flood level has occurred and estimate probabilities for each possible flood level in a future year accordingly. No such calculation is possible for an event as historically idiosyncratic as a potential invasion of Taiwan by the PRC.

Thus it does not make sense to interpret the probabilities in the above model as representations of objectively knowable relative likelihoods. Instead, models of uncertainties like the one we currently face about the future of Taiwan – i.e. uncertainties in which we have little or no empirical basis from which to determine relative likelihoods – are best interpreted as depictions of subjective beliefs that different persons might hold about uncertain events. For instance, military and political leaders throughout the world differ widely in their assessments of how likely the PRC is to launch an invasion of Taiwan sometime in the next several years and the PRC’s prospects of success if it invades. A model like the one above would be useful for depicting variations across political leaders in these assessments. Specifically, it could be used to depict some leaders who believe an invasion is very likely to occur but very unlikely to succeed if it does occur, and thus have beliefs like this…

Event	Probability
no invasion by 2028	0.25
invasion by 2028 but no successful occupation by 2030	0.675
invasion by 2028 and successful occupation by 2030	0.075

…and other leaders who believe an invasion is very unlikely to occur but very likely to succeed it does occur, and thus have beliefs like this:

Event	Probability
no invasion by 2028	0.75
invasion by 2028 but no successful occupation by 2030	0.025
invasion by 2028 and successful occupation by 2030	0.225

More generally, differences between persons in their beliefs, hunches and suppositions in response to uncertainty are a ubiquitous source of conflict in politics. Consider, for instance, Americans’ beliefs about the rightful and legitimate winner of the 2020 presidential election. Most (perhaps all?) Americans do not have the ability to directly check the validity of the numerous processes of voting and ballot counting that culminated in the 2020 election outcome. Instead, their beliefs about the legitimacy of the 2020 election depend on what they have heard about the election from voices in the media, and which of those voices they trust. Since November of 2020, a small but powerful faction of Republican politicians and right-wing media outlets have insisted that the election was fraudulent and Donald Trump was the rightful winner. A somewhat larger group of moderate Republicans, Democrats and mainstream and left-wing media outlets have maintained that the election was valid and Joe Biden was the rightful winner. The result is a variety of beliefs in the general population about who was the actual and rightful election winner.

An organization called Bright Line Watch, for instance, has run a twice-yearly poll of Americans since 2017 meant to monitor the state of American democracy. In an October 2022 poll, they asked a sample of about 2700 respondents “Do you consider Joe Biden to be the rightful winner of the 2020 presidential election or not the rightful winner?” Respondents were given the option to select one of “Definitely the rightful winner”, “Probably the rightful winner”, “Probably not the rightful winner”, or “Definitely not the rightful winner”. Weighting the sample to adjust for under-response to surveys by some groups, the results suggest the following:²

Belief about the 2020 election	Percentage of Americans holding that belief
Biden was definitely the rightful winner	47.0\%
Biden was probably the rightful winner	19.2\%
Biden was probably not the rightful winner	15.4\%
Biden was definitely not the rightful winner	18.4\%

We can depict any one of these four different beliefs about the rightful winner of the 2020 presidential election using a probabilistic model as follows:

Various Beliefs about the Rightful Winner of the 2020 Election

Possible Events

Joe Biden was the rightful winner of the 2020 election.
Joe Biden was not the rightful winner of the 2020 election

Probability Distribution…

…depicting a person who believes Biden was “definitely” the rightful winner:

Event	Probability
Biden rightful winner	1
Biden not rightful winner	0

…depicting a person who believes Biden was “probably” the rightful winner:

Event	Probability
Biden rightful winner	0.75
Biden not rightful winner	0.25

…depicting a person who believes Biden was “probably not” the rightful winner:

Event	Probability
Biden rightful winner	0.25
Biden not rightful winner	0.75

…depicting a person who believes Biden was “definitely not” the rightful winner:

Event	Probability
Biden rightful winner	0
Biden not rightful winner	1

Pause and complete check of understanding 3 now!

Parameterized Probability Distributions

Probability models are most often used in PPT to depict differences in the relative likelihoods of different events from one situation to another. In practice, these differences are most often captured using parameterized probability distributions, in which probabilities are expressed in terms of mathematical expressions of variables or parameters. This section will walk you through the process of understanding, interpreting and modeling with parameterized probability distributions.

Consider once again our imaginary mayor who is trying to decide how high to build the seawall that will protect her city from flooding. Suppose that the mayor’s beliefs about the height to which the river will rise in any given year are given by the following model:

A Parameterized Model of Flooding

Possible Events

In any given year, the river will rise to one of five heights above its normal level: 0 meters, 1 meter, 2 meters, 3 meters, 4 meters.

Probability Distribution

Let \delta be a number between 0 and 1. (\delta is the Greek letter pronounced “del-tah”).

Level above normal	Probability
0	\frac{1}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
1	\frac{\delta}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
2	\frac{\delta^2}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
3	\frac{\delta^3}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
4	\frac{\delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4}

This is a parameterized probabilistic model because it specifies the probabilities of the possible events in terms of a parameter – in this case the parameter \delta. The expressions for the probabilities in this model may look obscure. But in fact they depict two simple ideas. First, each level above normal is less likely than a slightly lower level. Second, as \delta increases, higher levels become relatively more likely.

To see how the model works, use the following slider to set \delta to various values between 0 and 1…

viewof delta = Inputs.range([0, 1], {value: 0.25, step: 0.01})

…and see to how the value of \delta affects the probabilities of the different river levels:

{
  let prob = (x,delta) => (delta**x)/(1 + delta + delta**2 + delta**3 + delta**4)
  let levels = Array(5).fill(1).map((element, index) => ({
    level: index,
    prob: prob(index,delta)
  }))
  let p = Plot.plot({
    margin: 55,
    width: width,
    height: 300,
    x: {domain: [0,1], label: 'Probability', labelAnchor: 'center', labelArrow: 'none'},
    y: {domain: [4,3,2,1,0], label: 'Level above normal', labelAnchor: 'center'},
    marks: [
      Plot.barX(levels, {x: 'prob', y: 'level', fill: 'lightblue'}),
      Plot.text(levels, {x: 'prob', y: 'level', text: (d) => (d.prob).toFixed(2), dx: 20, fontSize: 14})
    ]
  })
  d3.select(p).selectAll("svg > g[aria-label='x-axis'] > text").style('font-size', '20px')
  d3.select(p).selectAll("svg > g[aria-label='y-axis'] > text").style('font-size', '20px')
  return p
}

With this model in hand, suppose that no matter the height to which the mayor builds her seawall, the city will flood if and only if the river rises to a level above the seawall’s height. Thus, for instance, if the mayor builds a seawall 2 meters high, then the city will flood if and only if the river rises to 3 meters above its usual level or 4 meters above its usual level.

Under this assumption, we can calculate the probabilities with which the city will flood implied by the model above for various possible seawall heights. For instance, if the mayor builds a seawall 2 meters high, the city will flood if and only if the river rises to either 3 meters above usual or 4 meters above usual. Thus the total probability of the city flooding with a 2 meter seawall is the sum of the probability that the river rises to 3 meters and the probability that the river rises above 4 meters: \frac{\delta^3}{1+ \delta + \delta^2 + \delta^3 + \delta^4} + \frac{\delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4} = \frac{\delta^3+\delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4}

More generally, here are the probabilities with which the city floods with different possible seawall heights:

Height of seawall	The city floods when…	Which occurs with probability…
less than 1 meter	The river rises to 1 meter or more above usual	\frac{\delta + \delta^2 + \delta^3 + \delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
1 meter	The river rises to 2 meters or more above usual	\frac{\delta^2 + \delta^3 + \delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
2 meters	The river rises to 3 meters or more above usual	\frac{\delta^3 + \delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
3 meters	The river rises to 4 meters or more above usual	\frac{\delta^4}{1+ \delta + \delta^2 + \delta^3 + \delta^4}
4 meters	The river rises to more than 4 meters above usual	0

Use the following slider to set the parameter \delta to various values between 0 and 1…

viewof delta1 = Inputs.range([0, 1], {value: 0.25, step: 0.01})

…and see the probability that the city floods for each of a range of seawall heights:

{
  let prob = (x,delta) => (delta**x)/(1 + delta + delta**2 + delta**3 + delta**4)
  let heights = [
    {
      height: 'less than 1 meter',
      floodprob: 1-prob(0,delta1)
    },
    {
      height: '1 meter',
      floodprob: 1-prob(0,delta1)-prob(1,delta1)
    },
    {
      height: '2 meters',
      floodprob: prob(3,delta1) + prob(4,delta1)
    },
    {
      height: '3 meters',
      floodprob: prob(4, delta1)
    },
    {
      height: '4 meters',
      floodprob: 0
    }
  ]
  let p = Plot.plot({
    marginLeft: 100,
    marginTop: 10,
    marginRight: 10,
    marginBottom: 60,
    width: width,
    height: 300,
    x: {domain: [0,1], label: 'Probability that the city floods', labelAnchor: 'center', labelArrow: 'none'},
    y: {domain: ["4 meters", "3 meters", "2 meters", "1 meter", "less than 1 meter"], label: 'Height of seawall', labelAnchor: 'center'},
    marks: [
      Plot.barX(heights, {x: 'floodprob', y: 'height', fill: 'lightblue'}),
      Plot.text(heights, {x: 'floodprob', y: 'height', text: (d) => (d.floodprob).toFixed(2), dx: 20, fontSize: 14})
    ]
  })
  d3.select(p).selectAll("svg > g[aria-label='x-axis'] > text").style('font-size', '20px')
  d3.select(p).selectAll("svg > g[aria-label='y-axis'] > text").style('font-size', '20px')
  return p
}

In examining the graph above, begin by making sure you understand a simple point: Regardless of the value of \delta, a higher seawall always leads to a lower probability of flooding. This is because a higher seawall stops all the rises in the river that would have been stopped by any smaller seawall.

Now notice a more subtle, but much more important, feature of the model: When the mayor increases the height of the seawall by 1 meter, the decrease in the probability that the city will flood she achieves by doing so varies depending on the height she starts from. For instance, suppose for a moment that \delta = \frac{1}{2}. Then by increasing the seawall’s height from 1 meter to 2 meters, the mayor reduces the probability that the city will flood by: 0.23-0.1 = 0.13 On the other hand, by increasing the height of the seawall from 2 meters to 3 meters, the mayor reduces the probability that the city will flood by: 0.1-0.03 = 0.07

More generally, and regardless of the value of the parameter \delta, the amount that the mayor reduces the probability that the city will flood by increasing the height of the seawall by 1 meter decreases in the starting height of the seawall. In short, the mayor faces diminishing marginal returns to the height of the seawall. Use this slider to set \delta to various heights…

viewof delta2 = Inputs.range([0, 1], {value: 0.25, step: 0.01})

…and see how much a 1 meter increase in the height of the seawall reduces the probability that the city will flood:

{
  let prob = (x,delta) => (delta**x)/(1 + delta + delta**2 + delta**3 + delta**4)
  let starts = [
    {
      start: '0 meters',
      decrease: prob(1,delta2)
    },
    {
      start: '1 meter',
      decrease: prob(2,delta2)
    },
    {
      start: '2 meters',
      decrease: prob(3,delta2) 
    },
    {
      start: '3 meters',
      decrease: prob(4, delta2)
    }
  ]
  let p = Plot.plot({
    marginLeft: 100,
    marginTop: 10,
    marginRight: 10,
    marginBottom: 60,
    width: width,
    height: 300,
    x: {domain: [0,0.5], label: 'Decrease in the probability that the city floods caused by 1-meter seawall increase', labelAnchor: 'center', labelArrow: 'none'},
    y: {domain: ["3 meters", "2 meters", "1 meter", "0 meters"], label: 'Starting height of seawall', labelAnchor: 'center'},
    marks: [
      Plot.barX(starts, {x: 'decrease', y: 'start', fill: 'lightblue'}),
      Plot.text(starts, {x: 'decrease', y: 'start', text: (d) => (d.decrease).toFixed(2), dx: 20, fontSize: 14}),
    ]
  })
  d3.select(p).selectAll("svg > g[aria-label='x-axis'] > text").style('font-size', '16px')
  d3.select(p).selectAll("svg > g[aria-label='y-axis'] > text").style('font-size', '16px')
  return p
}

This parameterized model of flooding generates two insights that could not be gained from a model in which the probabilities were specified as definite numbers. First, it generates an interesting implication that holds for a whole category of probabilistic models. Specifically, it implies that there are diminishing marginal returns to investments in flood protection for a whole range of distributions of flood risk – i.e. the full range described by varying \delta from 0 to 1. Second, it depicts how incentives to build flood control infrastructure might change in response to changes in flood risk. It does this by allowing us to compare the marginal decrease in flood risk achieved by increasing the height of the seawall for different values of \delta.

Pause and complete check of understanding 4 now!

Footnotes

A more realistic model would capture the fact that any level of flooding above the river’s normal level up to some maximum is possible. For instance, it might assume that the river could flood to any level between 0 and 3 meters above normal. A probabilistic model like that, in which the set of possible events amounts to all numbers between two limits, is called a “continuous” probability model. In contrast, a model in which the set of events can be described a list of distinct items, such as the levels 0, 1, 1.5, 2, 2.5, 3, is called a “discrete” probability model. Although discrete models can be simplistic, the analysis of continuous models requires knowledge of calculus. Thus these lessons will only use discrete models.↩︎
Calculations based on Bright Line Watch, 2022, “Bright Line Watch Survey Wave 17 Dataset”. https://brightlinewatch.org/survey-data-and-replication-material/↩︎