Marty Weitzman’s Noah’s Ark Problem

Marty Weitzman passed away suddenly yesterday. He was on many people’s shortlist for the Nobel. His work is marked by high-theory applied to practical problems. The theory is always worked out in great generality and is difficult even for most economists. Weitzman wanted to be understood by more than a handful of theorists, however, and so he also went to great lengths to look for special cases or revealing metaphors. Thus, the typical Weitzman paper has a dense middle section of math but an introduction and conclusion of sparkling prose that can be understood and appreciated by anyone for its insights.

The Noah’s Ark Problem illustrates the model and is my favorite Weitzman paper. It has great sentences like these:

Noah knows that a flood is coming. There are n existing species/libraries, indexed i = 1, 2,… , n. Using the same notation as before, the set of all n species/libraries is denoted S. An Ark is available to help save some species/libraries. In a world of unlimited resources, the entire set S might be saved. Unfortunately, Noah’s Ark has a limited capacity of B. In the Bible, B is given as 300 x 50 x 30 = 450,000 cubits. More generally, B stands for the total size of the budget available for biodiversity preservation.

…If species/library i is boarded on the Ark, and thereby afforded some protection, its survival probability is enhanced to Pi. Essentially, boarding on the Ark is a metaphor for investing in a conservation project, like habitat protection, that improves survivability of a particular species/library. A particularly grim version of the Noah’s Ark Problem would make the choice a matter of life or death, meaning that Pi= 0 and Pi= 1. This specification is perhaps closest to the old testament version, so I am taking literary license here by extending the metaphor to less stark alternatives.

Weitzman first shows that the solution to this problem has a surprising property:

The solution of the Noah’s Ark Problem is always “extreme” in the following sense…In an optimal policy, the entire budget is spent on a favored subset of species/libraries that is afforded maximal protection. The less favored complementary subset is sacrificed to a level of minimal protection in order to free up to the extreme all possible scarce budget dollars to go into protecting the favored few.

Weitzman offers a stark example. Suppose there are two species with probabilities of survival of .99 and .01. For the same cost, we can raise the probability of either surviving by .01. What should we do?

We should save the first species and let the other one take its chances. The intuition comes from thinking about the species or libraries as having some unique features but also sharing some genes or books. When you invest in the first species you are saving the unique genes associated with that species and you are also increasing the probability of saving the genes that are shared by the two species. But when you put your investment in the second species you are essentially only increasing the probability of saving the unique aspects of species 2 because the shared aspects are likely saved anyway. Thus, on the margin you get less by investing in species 2 than by investing in species 1 even though it seems like you are saving the species that is likely to be saved anyway.

The math establishing the result is complex and, of course, there are caveats such as linearity assumptions which might reverse the example in a particular case but the thrust of the result is always operating: Putting all your eggs in one basket is a good idea when it comes to saving species.

Weitzman gets the math details right, of course!, but he knows that Noah isn’t a math geek.

Noah is a practical outdoors man. He needs robustness and rugged performance “in the field.” As he stands at the door of the ark, Noah desires to use a simple priority ranking list from which he can check off one species at a time for boarding. Noah wishes to have a robust rule….Can we help Noah? Is the concept of an ordinal ranking system sensible? Can there exist such a simple myopic boarding rule, which correctly prioritizes each species independent of the budget size? And if so, what is the actual formula that determines Noah’s ranking list for achieving an optimal ark-full of species?

So working the problem further, Weitzman shows that there is a relatively simple rule which is optimal to second-order, namely:

Where R is an index of priority. Higher R gets you on the ark, lower R bars entrance. D is a measure of a species distinctiveness–this could be measured, for example, by the nearest common ancestor metric. U is a measure of the special utility of a species beyond its diversity (Pandas are cute, goats are useful etc.) C is the cost of a project to increase the probability of survival and Delta P is the increase in the probability of survival so Delta P/C is the cost of increasing the probability of survival per dollar. Put simply we should invest our dollars where they have the most survival probability per dollar multiplied by a factor taking into account diversity and utility.

The rule is simple and sensible and and it has been used occasionally. Much more could be done, however, to optimize dollars spent on conservation and Weitzman’s rule gives us the necessary practical guidance. RIP.


If only a certain German government had been aware of this justification for its policy decisions ensuring that only those with the best chance of surviving would be allowed to survive, maybe they would not have felt the need to keep their actions to optimize the spending of Reichsmarks secret.

Admittedly, this is the sort of thing that Prof. Cowen seems more interested in.

You need to drop the cheer leading for the Nazis. It's never a good look.

Thank God Noah saved the koala, that's what I say. And the puffin, of course. Or did he just leave dicky-birds to fly around? It's decades since I looked at the OT.

P.S. You presumably meant the counterfeit Nobel.

You know who got off easy in the flood? The fish. Why no big holocaust for them?

Let's see... One myth used to prop up a theory/myth. What could go wrong?

The terms freshwater and saltwater don't only apply to economists.

The best bit was this: "The math establishing the result is complex and, of course, there are caveats such as linearity assumptions which might reverse the example in a particular case"

Go on: what you get out of a mathematical model depends on what you put into it? Unheard of!

And is what you put into it an authentic attempt to model reality or is it simply a way of ensuring that the resultant model can be solved? (Or, as in Climate Science, is it a way of ensuring you get the answer you want?)

In spite of which cavilling I must say that that sounds like a fascinating bit of work. Good for Weitzman!

Interesting. There are two instances of species that have been managed into decline in my area. The mountain Caribou, and the Kokanee in Kootenay Lake. It seems that in both cases it is an all or nothing proposition, and the series of half measures have no effect or make the situation worse. Both have solutions proposed that involve killing other species; bull trout who operate the Kokanee fry, and with the Caribou increase the harvest on all the other ungulates, then shoot the wolves. When the real problem is production of green energy, the rigorous control of water levels for hydroelectric generation, and the cris-crossing of Caribou habitat with power transmission lines creating access to the high country for predators.

Weitzman's recent research focused, not coincidentally, on the economics of catastrophes. Will we rely, should we rely, on mathematics to determine the allocation of resources in a world with increasing numbers and levels of catastrophes? What's the alternative? Chance, political power (i.e., bribes), intuition, sentimentality, or ethnicity, race, national identity, or other markers? The U.S. is about to be hit by a major hurricane. How should relief be allocated? An aside, I have a friend who has a Ph.D. in anthropology. The focus of his research and his sub-specialty is catastrophes and how to respond to them. I reside in a hurricane-prone area. What I have observed is the coordinated way first responders deal with a catastrophe, in particular what is given priority. I assume that much planning goes into it, and mathematics plays a part. My friend often advises government agencies on such questions. My observation: better to plan ahead (mathematics) than to respond after the catastrophe, when emotions rather than logic are likely to affect the responders actions.

"The U.S. is about to be hit by a major hurricane. How should relief be allocated?"
Isn't that what private insurance markets are for? (Or why should public monies be spent to afford relief to people who bought homes on the beach?)

Much of the damage from hurricanes is to public facilities, from roads and highways to public utilities. Just removing fallen trees etc. from the roads is a major undertaking, the trees often laying over power lines. Overloaded storm sewers are often the primary source of damage to homes and businesses as runoff overwhelms the system causing the flooding that is often incorrectly attributed to storm surge. One cannot imagine the scope of the damage from a major hurricane, and how it overwhelms local resources, unless one has experienced it directly.

"The U.S. is about to be hit by a major hurricane." I see this statement more often than I see the statement "US is hit by major hurricane" but I admit I don't know the ratio. What fraction of such predictions come good? 50%? Less?

P.S. Is a "major" hurricane a defined category or just a bit of journo-speak?

It depends on how many days out but at this point (within three days of landfall) the prediction is almost certain. I believe we're able to predict the landing point within 50 miles by 48 hours out.

Hurricane prediction is far better than it was 20 years ago, and 20 years ago was a huge leap from 20 years before that.

For the last 10 years, the US has averaged 2 hurricanes a year, apparently. ( Frequency seems to go in a 40- or 50-year cycle, with US currently on the downtrend after an intense 1990s-2000s.

Not sure what constitutes a "major" hurricane, there are five categories (1-5), and having experienced a Category 1 and Category 2 (the lowest levels) when I was a kid in coastal North Carolina, I can assure you even those lower-intensity levels are capable of causing severe and widespread damage.

Thanks, TN. If prediction is pretty good when the prediction is landfall within three days, when is it poor? Five days? Seven days? Do people bother with such predictions?

Weitzman’s research on climate change governance is priceless. I wish he had been awarded the Nobel Prize alongside Nordhaus.

Weitzman understood very little about climate change or its risks and uncritically followed the alarmists.

I tend to think the exact opposite. Weitzman understood that the important discussion is not about climate science itself but about probabilities and uncertainty. I derived two main conclusions from his papers.
"The probability of a disastrous collapse of planetary welfare from too much CO2 is non-negligible, even if this low probability is not objectively knowable.”

Demortuis Nil Nisi Bonum.

We simply do not realize that Climate Change killed us and we all are in Hell.

Live by the model die by the model.

The IPCC had a major error in its 2007 report that Weitzman continued to use after 2015: That is, a 10% chance of a >5 degree C increase by 2100, even after the IPCC corrected it in the 2014 report to much less than a 1% chance.

But Weitzman's analysis had other problems as well including assuming for the erroneous 10% chance that technology will hardly advance over the next 80 years.

His articles on geoengineering very much dealt with technology, but true to form, he didn't attempt to assess the technology itself, but rather addressed the governance challenges raised by it.

Weitzman's ignorance went much deeper as he was going beyond even the completely unrealistic, alarmist assumptions of RCP 8.5.

You are wrong. The global climate system is chock full of nonlinear relations that involve positive feedback, such as warming reducing the albedo of the earth's surface, which increases warming. This underpins that the probability distribution has "fat tail" kurtosis, unlike the Gaussian distributions assumed in the IPCC reports.

If the climate is so chalk full of feedback systems why have the climate models been way off for 30 years? Also, if you read the IPCC AR5, you will find they disagree with you. Do you even know what RCP 8.5 is?

Sorry, "zebra swallowtail," but you are dealing with someone who knew Marty well and has been working with climatologists off and on for nearly a half century. Your RCP 8.5 is still derived from a Gaussian distribution for its description as being extremely high. You picked the wrong person to argue with about this.

The model is nice, but if it is applied for real......Noah's ark is going to be filled mostly with plants, bacteria, algae and fungi. Large mammals are not that important.

Presumably, plants, bacteria, algae and fungi have a much higher probability of surviving a flood than large mammals do.

I had the opposite thought. The model seems fine -- its metric is completely subjective. ("U is a measure of the special utility of a species beyond its diversity (Pandas are cute, goats are useful etc.).") As such, it will happily conform to any priorities the person implementing it may have.

What I object to is the presentation -- as written up, this just assumes that "diversity" is a form of special utility. (If you look at the formula, diversity and utility are measured in the same units and are indistinguishable from each other.) That is in keeping with the times, but in my opinion mostly unjustified. There is a good form of this idea, though, which everyone already understands intuitively:

Suppose you're a peasant farmer choosing some animals to preserve. You use, on your simplified farm, cows, horses, and cats.

Cows are far and away the most useful of the three. They get the nod.

Horses are second most useful. However, many of the functions performed by horses could be performed, a little worse, by cows. Cats can patrol a granary to mitigate losses to rats. Cows and horses cannot do anything similar. Cats get the nod next.

Marty was my adviser my first year in grad school. He really cared a lot. I chose a somewhat unorthodox curriculum and he was almost tearing his hair out. Obviously it didn't affect him personally at all but he was genuinely concerned about my career. I would say he was a person who took other people's well-being very seriously.

Not many academics are that caring when it comes to their students' welfare. Very, very few professors have genuine human concern about students. Marty Weitzman was the rare exception. Prof. Michael Grossman, of NBER, New York, is another distinguished member of that noble circle of teachers.

". . . two species with probabilities of survival of .99 and .01. For the same cost, we can raise the probability of either surviving by .01 . . ."

I suppose this analysis was expanded to more common scenarios where eliminating the final 1% chance of failure of one species is tremendously more expensive than doubling a 1% chance of success to 2% for another species.

During Rachmaninoff’s Four String Quartet No. 1, Lisa began to roll her eyes, play with her hands and turn around in her seat. There were three men in black, two in the middle played a violin and viola, a celloist on the right end. On the left was a woman in a gold dress. She led, and from her bow, the other two pulled theirs, when she fingered decisively, the others moved their hands up the bridge. Two circles, two tangent lines, four tangent lines intersecting. The celloist, the oldest, matched her impassioned vogue. Across from him, she appeared robotic, as if when she swept her arms a string had pulled on her neck; in her youth she looked like a doll.

So, when politicians favor a few industrial champions, they're basically right.

Central planners had it right all along.

You can make the same response to this that you do to more normal treatments of central planning: this approach explicitly assigns a purely subjective value to every [thing being preserved]. Most people will actually disagree about the values involved, making the rank ordering produced by any particular assignment into mostly nonsense.

Neat model. I wonder which species would end up being saved in this scenario. I’m also curious whether they could survive afterwards, with only a smattering of their ecosystem still available. That’s the challenge with an elegant model, it’s easy to leave out factors. I suppose in this case it would mostly only matter for predators, so maybe you would tweak the results to take the most flexible predators who can hunt a lot of different prey and control post-flood populations broadly.

Ahhh but we are assuming the Ark will be protected by God thru the flood. If you get your ass on the ark and you'll be golden. But what if the ark sinks?

Absent a golden pass from God, it would be prudent to have several arks rather than one. Or multiple libraries rather than one giant Library of Alexandria or preserve multiple wildlife environments around the world rather than one.

Now what happens if you have two arks and since animals are not evenly distributed, your 0.99 species is on one side of the world but not the other? I suspect the math would say put the 0.01 species on a few of the arks. Or in terms of libraries if we have 100,000 libraries it is not absolutely imperative all of them have 10 copies of every Harry Potter book, even though we'd like to see Harry Potter preserved.

"Absent a golden pass from God, it would be prudent to have several arks rather than one."

How do you know there weren't? This is similar to the Dolphin problem. IE Dolphins are known for helpfully pushing drowning humans to shore. On the other hand, if dolphins were randomly pushing humans both to and from shore, the reported effect wouldn't change. None of the drowning humans pushed away from shore are around to dispute the narrative.

Good point, but there is a limiting function here. God sent the flood because he was tired of bad humans and wanted a fresh start. You could say maybe he gave 100 different men instructions to build an ark and 99 of them sank leaving Noah to rewrite the story with him as the hero.

But if God has more and more people he can trust to build arks, then he has less and less reason to destroy the earth. While we don't know what God's utility function is here, there would be a limit to just how many arks he could have ordered before humanity ends up good enough to avoid the whole thing to begin with.

I wonder what Noah did with the unicorns mentioned in 9 different verses of the OT. In any case, the equation 300 x 50 x 30 = 450,000 cubits is wrong, since the cubit is not a cubic unit of volume, but of length. It should read 300 x 50 x 30 = 450,000 cubic cubits

The original article has a "3" behind "cubits," which didn't make it into the quote above in Alex's post. It means cubic cubits, the way M3 means cubic meters.

"Prices vs. Quantities" influenced me profoundly in many ways though my career as an economist. RIP.

I have posted on Econospeak about Marty and his work. I discussed several aspects of his work, but not this Noah's Ark matter. But then he wrote on many topics, including comparative economics issues such as Soviet planning methods and The Share Economy.

I admired Weitzman and his work, he had much the same attitude of "what cool things can we analyze using economics" that we see in the best microeconomics textbooks so he studied a range of topics, but used complex mathematical models to delve deeply into questions of real-world significance be it environmental policy or wages vs a share economy.

I'm not impressed by that Noah's Ark example though. Species don't exist in isolation, it takes a whole ecosystem. Eg. if you save polar bears but don't save seals and walruses, your polar bears are going to have a tough go. Ecologists seem to still be divided about whether we should eliminate mosquitoes, if we could. Maybe trout or bats would have trouble surviving. But if Noah held an election mosquitoes would probably come in last place on the "to save" list.

I was going to post what mkt42 posted. Namely, species don't exist in isolation; there are ecosystems and food chains to think about. And mkt42's example of the mosquitoes is perfect in illustrating that.

Marty taught a couple of my grad school classes but might have been most famous for his orientation talk that began "A graduate student leads a monastic existence."

His classes were only difficult in that he wanted students to understand the deeper logic of models and their assumptions, so that they would not fool themselves about any "magical" results. He put a big emphasis on understanding what the technicalities "meant" in terms of the economic problem at hand. So when he taught control theory he started with a simple problem about drinking a glass of water with discounting and showed how the most important result came from the oft-neglected transversality condition: "Drink all the water."

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