Mea culpa: the link to the Python code for the deterrence simulation is broken, and I have misplaced the code. I will try to track it down and restore it soon.
Supplemental reading:
Fred Kaplan, The Wizards of Armageddon
Garrett M. Graff, Raven Rock: The Story of the U.S. Government's Secret Plans to Save Itself - While the Rest of Us Die
Fred Kaplan, The Bomb: Presidents, Generals, and the Secret History of Nuclear War
Michio Kaku & Daniel Axelrod, To Win a Nuclear War: The Pentagon's Secret War Plans
It was said that God, in order to test mankind which had become swelled with pride as in the time of Noah, had commanded the wise men of that age, among them the Blessed Leibowitz, to devise great engines of war such as had never before been upon the Earth, weapons of such might that they contained the very fires of Hell, and that God had suffered these magi to place the weapons in the hands of princes, and to say to each prince: “Only because the enemies have such a thing have we devised this for thee, in order that they may know that thou hast it, and fear to strike. See to it, m’Lord, that thou fearest them as much as they shall now fear thee, that none may unleash this dread thing which we have wrought.”
Walter M. Miller, A Canticle for Leibowitz
There are many X's that can fill the proposition “We all owe our lives to X.” In the annals of science and technology, the list is long and the “all” is very broad - Louis Pasteur, Sir Alexander Fleming, all the obstetricians who made modern childbirth a dramatic but still ordinary event compared to the descent into the valley of the shadow of death it had been for almost all of human history. 1
Bernard Brodie (1910-1978), an unassuming American academic, a scholar of the history, strategy, and tactics of naval warfare, virtually unknown outside of the small, secret world he worked in, has a good claim to having saved the lives of billions of people in the latter half of the Twentieth Century. He, and his nuclear strategist colleagues at the RAND Corporation, did this by developing and promoting a strategic insight contrary to all established opinion about weapons and warfare: that the value of nuclear weapons lay not in their use, but in their not being used.
Two Schools of Thought, And a Third
Soon after 1945, two opposed schools of thought about the role of nuclear weapons in national defence arose in the United States. On one side were the generals. In their thinking, nuclear weapons had been decisive - the Japanese surrendered after two of their cities were annihilated by atomic bombs, and future opponents would capitulate as well when faced with the same doom.2
If the bomb had to be used, nuclear weapons were unquestionably economical. Earlier in 1945, Operation Meetinghouse, the firebombing of Tokyo, killed over 100,000 people, and destroyed over 40 square kilometres of the city. To accomplish this, the US Army Air Force deployed 279 B-29 Stratofortress bombers dropping 1,500 metric tons of incendiaries. To completely destroy Hiroshima, and kill nearly the same number (~70-135,000 estimated, both immediately and from radiation sickness afterwards), the Air Force sent one plane, the Enola Gay, dropping one bomb weighing just over 4 metric tons. That one bomb, Little Boy, released the equivalent of 15,000 metric tons of TNT. As many bombers as flew over Tokyo in Operation Meetinghouse, each armed with an atomic bomb, could devastate any country in a matter of hours.3
We can call this the Just A Big Bomb theory: the nuclear bomb, for all its power, technical complexity, and fearful fallout, was a weapon, and the ultimate purpose of a weapon was to be used to fight and win wars, and nuclear weapons would join the American arsenal alongside the torpedo, napalm, and the M1 rifle, ready for use.
Even understanding that the Soviets were weakened by the preceding Second World War, the Department of Defense’s first atomic war plans treated any war with Russia as a total war:
PINCHER (like almost all the plans that succeeded it) clearly indicated that the Soviet Union would try to achieve its goals short of war. Because of Soviet military inferiority, all these plans stressed that the Soviets were unlikely to initiate a war with the U.S. However, the Soviets, “although desiring to avoid a major conflict for the next several years,” might make a “miscalculation.” Although the precise nature of this “miscalculation” was kept vague in PINCHER and succeeding plans, it was clear that, in a conflict, the U.S. would respond from the start with a full-scale nuclear attack. For planning purposes, PINCHER hypothesized an atomic attack between the summers of 1946 and 1947. PINCHER rejected the idea of localizing the conflict with the Soviet Union. It grimly stated that “no war with the U.S.S.R. can be less than a total war, requiring the full utilization of the entire U.S. and allied war potential,” which included the atomic bomb.
-Michio Kaku & Daniel Axelrod, To Win a Nuclear War: The Pentagon's Secret War Plans
Such plans were typical from the Pentagon until the mid-1950s.
On the other side were the scientists at Los Alamos who had designed and built the first nuclear weapons. With the notable exception of Edward Teller, many of them were filled with guilt about what they had done. They wanted the terrible destruction at Hiroshima and Nagasaki to serve some higher end, to drive the states of the world towards a more peaceful, unified existence. Many of them dreamed that the tremendous power of nuclear energy would be entrusted to a transnational organization like the newly formed United Nations, there to be used in a cooperative spirit to benefit humanity through abundant clean energy. Indeed, during the Manhattan Project it had been a motivation of Leo Szilard to bring about such an impasse: a decisive point where humanity would have to choose between unity and peace, or disunity and complete destruction. They wanted, in other words, a world government.
A conceptual gap existed between these two points of view and thinkers like Bernard Brodie filled it. In so doing, they determined the shape of the Cold War - the strategies, the weapon systems, the whole of the arms race.
Brodie was well prepared to provide more realistic and yet more forward thinking ideas than either the generals or the scientists. In the 1930s, he had studied the then new subject of International Relations at the University of Chicago. As now, they thought about many subjects a bit differently in Chicago.
In the years between the two world wars, most American universities reflected the popular notion that America was a unique creation, something apart from and beyond the politics of its European ancestors. Political science was taught as if politics were composed of organization charts, venerable static institutions and the Articles of the Constitution. International relations was considered synonymous with international law, its curricula crowded with summaries of peace-treaty provisions and the message that peace and world order were essentially the products of well-drafted international legislation and mutual good will.
Against this rather bland and abstract tradition that contradicted everything that was going on in the world, the Chicago school splashed a bracing tonic of realism. Charles Merriam led the way with his provocative discourses on American politics, expounding on the now common but then utterly earthshaking thesis that the essence of all politics lay not in structures of organizations or the Bill of Rights or the Electoral College, but in power—who uses it, for what ends, in what political context, against whom.
-Fred Kaplan, The Wizards of Armageddon
Brodie and his fellow strategists brought the world the idea of nuclear deterrence.
The Theory of Deterrence
Deterrence theory is a bit more complicated than the slogan “Mutually Assured Destruction,” that became associated with it. The promise of retaliation is part of it, certainly, but to truly be deterrence a few more elements are needed:
Nuclear weapons must be withdrawn from general use. Adversaries must know that the weapons are special, held in reserve, and would not be used in a merely conventional conflict, but only in some special (if ambiguously defined) circumstances.4
The weapons must be secure. Not only must they have their own chain of custody and ownership, that chain must be totally separate from the rest of the armed forces. Even if some military units have ‘tactical’ nuclear weapons - missiles, bombs, and artillery shells - those weapons are stored separately. The enemy must be able to observe them being brought out of storage and made ready in a crisis.
The weapons must be protected. The natural strategy against a weapon whose destruction is so great and so rapid that there may be no nation left to organize a counterattack, is preemption. To strike first and so overwhelmingly that the enemy is disarmed before war can properly begin is a temptation. Because of this nuclear weapons need to be numerous enough, dispersed enough and deliverable by enough redundant means that an enemy could have no certainty of success in preemption. They must always fear a devastating, unrecoverable second strike.
And if that was the strategy - known more generally as Punishment Theory - the United States had pursued consistently, the number of nuclear weapons in the American arsenal would not have soared into the tens of thousands. In a strategic situation where nuclear weapons serve only as a check against other nuclear weapons, each side needs only, at most, a few hundred, and possibly only one deliverable platform: silent, deep running ballistic missile submarines, the epitome of a second strike weapon. If true deterrence had been the chosen strategy, Brodie and his fellow strategists like Thomas Schelling and Herman Kahn could have filed their classified reports, cleared out their desks, and gone back to academia.
The problem, of course, is the next question curious minds ask: “assuming we cannot stop a nuclear war and are forced to use our deterrent, how do we win?” It's even easier to ask this question, and examine all its many complications, if your patron explicitly asks for it.
Beyond Deterrence: Counterforce
But the princes, putting the words of their wise men to naught, thought each to himself: If I but strike quickly enough, and in secret, I shall destroy these others in their sleep, and there will be none to fight back; the earth shall be mine.
-Walter Miller, A Canticle for Leibowitz
The RAND Corporation was not academia: it was non-profit corporation fully funded, in its first decades, by the U.S. Air Force. Aside from the higher salaries, no teaching duties, and access to nuclear and other military secrets, RAND's nuclear strategists worked on the problems set for them by this master. This naturally made certain questions beyond the realm of consideration:
How do we persuade our adversaries to mutually limit our nuclear arsenals?
How do we move towards peaceful coexistence and cooperation?
What is the minimum number of nuclear weapons that will make deterrence feasible?
While I think better of the nuclear strategists than Fred Kaplan does - I don’t think trying to grapple with ‘unthinkable’ problems is necessarily a bad thing - here is where I acknowledge they were in a bind. An academic with a secure posting - not a certain thing to get even in the 1950s - could pronounce upon nuclear war but, with no access to secrets, be able to have only the most abstract and theoretical debates: to argue that nuclear war was bad, but have next to no real information about the weapons, the strategies to use them, and their likely effects. To work at RAND, on the other hand, was to have access to secret information, and even to the nuclear warriors themselves, but to be circumscribed in terms of what you could write about. RAND, to my knowledge, never wrote a report telling the Air Force they needed less nuclear weapons and less means to deliver them.
It’s easy to be cynical about this - the Air Force wanted higher budgets, more weapons, and thus more power over the other armed services - but one has to acknowledge the lack of strategic intelligence that the United States had at the time about its adversary. Aerial reconnaissance was difficult (the U-2 spy plane was only just coming into operation) and satellite reconnaissance was very much “on the drawing board” (indeed, thinking about the potential uses of earth-circling satellites was one of RAND’s first contracts). No one in the United States knew just how depleted fighting off the Wehrmacht had left the USSR - depleted of men, depleted of resources, and with an economy geared completely towards war production and that needed to be retooled for peacetime reconstruction. The secrecy of the USSR concealed great weakness, not strength, but going by the precautionary principle the United States took the bellicose rhetoric and clearly vast armies of the Communist world at face value.5
If deterrence were to fail, the analysts reasoned, the immediate goal of any nuclear weapons state would become immediate pre-emption.6 From this goal springs the strategy of Counterforce: using nuclear weapons to destroy nuclear weapons. When contrasted with its opposed strategy, Countervalue - an anodyne word for directly destroying cities, industries, and people - this sounds reasonable, even preferable.
The downside is that Counterforce creates a (perceived) need for new nuclear weapons, and it easily leads to arms races: building weapons to destroy weapons that were built to destroy weapons that were built to destroy … in game theory, popular with the RAND strategists, its a two-player zero-sum game. More weapons lead to a greater possibility of accidental war, and even greater pressure to preempt in a crisis. Worst of all, counterforce turns out to be countervalue one-step removed: many of the ‘strategic’ targets a counterforce strike would attack, both in the United States and the Soviet Union, are located near major urban centres, the citizens of which would, even if they escaped the initial thermal and overpressure damage, would be sickened and killed by the fallout.
The optimal solution to a two-player zero-sum game, of course, is cooperation to stop playing, but RAND’s analysts, as noted above, were not tasked to think about that. They gave much thought to how one would stop a nuclear war once one started, either accidentally or deliberately (spoiler alert: they could not come up with any strategy that was likely to work in the moment of crisis), and how to use deterrence to stop from having to fight a nuclear war
Theory in the Real World
There are two additional downside to the work RAND’s analysts performed, beyond the scope limitations of their efforts. Both of these are human problems.
Most basically, RAND’s ideas about crisis management, for all their sophistication and abstract interest, bear the marks of the environment they were developed in and the people who made them: air-conditioned offices with coffee and snacks, worked in by bright, highly-educated men and (later) women, reading books, giving presentations, and playing war games, then packing up at 5 p.m. to go home for the night.
Decisions about nuclear war, on the other hand, are not going to be made by people who have been fully rested and have, at best, professional or personal stress to contend with. They will not be made by people who have spent their whole careers pondering the many complexities and ramifications of different counterforce tactics, or the psychology of brinkmanship. A president and his advisors, to say nothing of the generals and officers tasked with executing any orders, will likely be facing ultimate, indeed existential, decisions with little sleep, possibly after days or weeks of an ongoing crisis. They’ll have conflicting and limited information, be fearful of both the other sides actions and of making mistakes. They’ll be thinking about their loved ones, and (if there's any time) how history will judge their decisions. They’ll face the overwhelming temptation to preempt, but also fears about what happens after the button is pushed. Their ability to think and act rationally, by any measure, will be next to nonexistent.
The other problem is that RAND’s theories were enablers. States and their leaders can use nuclear weapons without detonating the warheads. There is not a one-to-one connection here: presidents were not reading RAND studies of bargaining and blackmail when deciding how to respond to the Berlin Blockade or Chinese threats to Taiwan, for example. But the theories RAND developed shaped the thinking that advisors made available to leaders, and influenced their risk tolerance. Indeed, Kaku and Axelrod's book reveals dozens of incidents where the United States both planned and directly threatened to use nuclear war against both nuclear-armed and unarmed adversaries.
The Job I Would Have Wanted
Projecting oneself back into the past is a tricky business. Aside from any concerns about origin essentialism (could I have had parents different than the parents I in fact did have?) and personal identity (would I be the ‘same’ person if I was raised in a different time with different experiences, societal values, etc.?), its hard to avoid bringing your contemporary biases with you.
Consider it a modified version of Rawls’ “veil of ignorance” - where instead of total ignorance you have some idea of where you would end up - holding fixed your sex, physical characteristics, intelligence, beliefs and values. Many of these things, realistically (if it makes sense to speak of realism in a case like this), would be different. Most basically, far enough back into the past, you would very likely not have survived many of the childhood diseases you had or would have caught. So saying you would like to have been a knight, say, or a court page, is already making a huge number of assumptions.
But putting aside all those caveats, transporting myself back into the past and looking for work, I would have wanted to be a nuclear war planner, one of Kaplan’s “Wizards of Armageddon.”
I’m not proud of this but I think it’s a valuable bit of self-knowledge. I like abstract discussions, technical minutiae, logical arguments, and systems thinking. I think nuclear warfare is (intellectually!) highly interesting, and have enjoyed reading the (declassified) books and papers of the nuclear strategists. These were extremely smart people dealing with intractable problems. Their logic, given their assumptions, is compelling: counterforce looks like a saner, even more moral theory than countervalue; how to use or avoid blackmail is important for leaders to consider; tactical nuclear weapons are a valid means to check an opponent with massively greater manpower, not to mention many more tanks, planes, armored personnel carriers, and artillery pieces.
It strikes me, from reading Kaplan’s book and the works of the strategists themselves, as the ultimate academic paradise: an appreciative audience paying you to think about a subject, freedom to think about whatever you want related to it (within certain constraints - you wouldn’t get funding to think about how to stop the Cold War, for instance), having the opportunity to advise heads of state and military officers, and no teaching duties. Add to that a luxurious office and brilliant colleagues, and I can’t imagine a better job.
At the same time, I know that the atmosphere was not all fun and games. Looking back with hindsight bias, we know there was no nuclear war, and that the strategists’ plans never had to be tested in the real world. It certainly did not appear that way at the time. Daniel Ellsberg, recalling his time at RAND, wrote avout the long hours he and his colleagues worked during the early 1960s, convinced that the Russians were searching for a gap in America’s defense and that if they found it they could launch a surprise attack. From an outside perspective, with the benefit of hindsight, this was a projection by the analysts of the preferences of their patrons, the US Air Force and the Department of Defence. But no one could tell that at the time. Then there was the awful knowledge derived from being the first people to grapple with just how destructive thermonuclear weapons, then just being developed, would be:
Plesset’s damage circles showed that a five- or ten-megaton hydrogen bomb would kill people within 50 square miles of ground zero, and would severely burn people’s skin and topple buildings within 300 square miles. [During the Second World War], Hitch had dealt with bombing raids involving hundreds of airplanes, producing thousands—at the very most, tens of thousands—of civilian casualties. Laying Plesset’s circles on various maps revealed that a mere fifty-five H-bombs of twenty megatons each would completely wipe out the fifty largest cities of the Soviet Union, killing thirty-five million Russians, all in a matter of minutes.
Appendix: Does Deterrence Work? An Exercise in Probabilistic History
Activists who seek the abolition of nuclear weapons like to point out that deterrence is untested and untestable, that we have no reason to suspect that anything other than luck has prevented a nuclear war. In effect, they deny the deterring power of the bomb. They point out that just because we have not witnessed a nuclear war since 1945 is no reason to think that deterrence is what has protected us. If anything, the close calls like the Cuban Missile Crisis and the 1983 Able Archer war scare, not to mention dozens of near-miss near-launches because of misinterpreted signals, show that deterrence is unstable and could break down at any time. This is classic Problem of Induction stuff.
As much sympathy as I have for the nuclear abolitionists, and how much as I wish they were right, they are thinking about the situation in too limited a way. They think of deterrence as the only theory, believing that it blocked or superseded nuclear abolition. But this is, as we’ve seen above, inaccurate. Thinking back to Nassim Taleb’s call in Fooled By Randomness to think about not only the future but also the past probabilistically, to imagine how probable the present we currently live in is, let’s consider the options.
As I noted above, in 1945, immediately after the end of the Second World War, there were three theories about nuclear weapons:
(1) Just A Big Bomb
(2) International Control / Nuclear Abolition
(3) Deterrence
These options were not equiprobable (e.g. 1/3, 1/3, 1/3). As Kaplan points out, however desirable international peace might be, it required many countries to give up things without compensation, something no leaders, democratically elected or not, could sell to their own people. Deterrence is not ideal, but a “better than the most likely alternative” option.
I've made a simulation (the Python code for which is linked to below so you can play with them yourself if you would like). The simulation generate 25,000 ‘worlds’ and randomly assigns them a starting policy state: Deterrence, Abolition, or Conventional (i.e, treating the atomic bomb like any other weapon). The initial probability distributions of the starting states are as follows: Just a Bomb (45%), Deterrence (45%), and Abolition (10%).7
The simulation then steps forward year by year from 1945 to 2030. Each year, an outcome is selected for a probability distribution based on the starting policy state.
Conventional = Peace (85%), War (15%)
Deterrence = Peace (90%), War (10%)
Abolition = Peace (90%), switch to Deterrence (10%)*
*We’re not assuming Abolition stays abolition. States might choose to build nuclear arsenals to intimidate their neighbors, who would build arsenals in kind, bringing the world to Deterrence.
The above does not assume that “Deterrence Works” simpliciter, but that more probabilistic histories from 1945 get to 2030 without a nuclear war than under the Conventional policy
(recall the pressure of preemption, which is apt to make any crisis escalate into an existential, all-or-nothing one).
Peace is the default state, and if chosen the world ticks forward to another year. War is nuclear war, and that world's timeline is stopped and another world generated. At the end, the simulation provides an output of the results for further data processing (I could do it in numpy, but maybe when I’ve got more data to process).
Simulation Alpha (deterrencesimA.py) is a simple, linear simulation without feedback loopa, with no dependence of the future state upon anything other than the immediate past one - in other terms, a Markov Chain.
The Results
Here are the start and end distribution of strategies in Alpha.
All of the Abolition worlds switch to Deterrence at some point.
Let’s see how long the different worlds lasted. Here’s a table counting the ‘end decade’ of each world, when a nuclear war happened, categorized by starting (but not ending) strategy:
Yikes! It is total destruction across most possible worlds by the 1970s. It’s even more stark looking on this bar-chart.
So both conventional and deterrence strategies have very high attrition rates in their first 10 years. Abolition strategies switch to deterrence at a fairly steady rate across three decades, but are basically gone (as are most of the worlds…) by the 1980s. Deterrence is still ‘better’ (the worlds last for longer, which on some utilitarian calculus must count for something).
Let’s try that again, with some different values for the peace/war probability. We’ll set
Conventional = Peace (95%), War (5%)
Deterrence = Peace (99%), War (1%)
Abolition = Peace (90%), switch to Deterrence (10%)*
A much more dovish scenario. We could say that this distribution is biased more towards accidental than deliberate nuclear wars.
Here are the stats after a run with the new probability:
Here, more worlds make it through to the last decade. While conventional strategies still suffer very high attrition in their first decades, most worlds with deterrence make it through to the final decade.
Here is the chart, listing world attrition by by decade, categorized by starting strategy:
The big jump in abolition at the end are worlds that started out as abolition but switched to deterrence. If you’re interested, 1180 worlds that started as abolition and switched to deterrence made it through to the end, while 1317 had wars at some date.
Improving the Simulation
This crude simulation can obviously be improved in many ways. For example, it could allow switching from deterrence and conventional strategies to abolition, or conventional to deterrence strategies, rather than just a one-way jump from abolition to deterrence. I don’t include a switch from deterrence to conventional or abolition to conventional because that strikes me as an improbable switch - deterrence is ‘sticky’ as a concept. Witness the fact that we still have it 75 years after the end of the Second World War.
While I do not pretend this crude simulation proves that deterrence works, I hope it provides nuclear abolitionists with food for thought. In this world, our choice is not always between competing goods, but between greater and lesser evils.
An aside: there is a decent case to be made that obstetricians, as a profession, have saved more lives than all the other classes of physician put together. And they often save two at a time, both mother and child. Even just the avoiding of complications by simple procedures like regular hand washing and the availability of supplemental oxygen have saved uncountable infants, to say nothing of the premature children saved every day by advances in technology and technique.
Just a footnote for an enormous, endless discussion, that goes back and forth depending on the consensus about the effective/morality of nuclear deterrence: it is controversial whether the atomic bombs were the trigger for Japan’s surrender, or whether the declaration of war on Japan by the USSR, attacking Japan’s seized territory in Manchuria and China (its last source for the iron, coal, and other materials it needed to continue to wage war), was the deciding factor. Responsible historians - the ones I like, mostly - admit that the issue is a mystery, one that is worth discussing but not ultimately resolvable. There are too many causal factors to consider.
On a per unit basis, the Little Boy bomb cost $7.6 billion ($23 billion Manhattan project cost divided by three deliverable bombs - The Trinity Test device, Little Boy, and Fat Man), almost more than the full cost of researching, developing, and manufacturing the thousands of B-29 Stratofortresses used in the war. But mass production soon brought the per unit cost of the atomic bomb down significantly.
The United States, from 1945 until the date of this writing, has pursued a policy of ‘nuclear ambiguity' - while the nation would respond to an outright attack with nuclear weapons, it does not have a firm line separating conventional from nuclear war. The reason for this, the strategists argue, is that an adversary would be tempted in a crisis or conflict, to push right up to the edge of a clearly defined red line. Uncertainty, they claim, begets caution.
Indeed, the Kennedy campaign of 1960 alleged a ‘Missile Gap’ with the Soviet Union, claiming that the Russians were vastly ahead of the United States in deliverable nuclear weapons. Intelligence from the U-2 spy plane revealed this to be not true, but this was an earlier, more principled age, and the outgoing Eisenhower administration would not leak strategic intelligence to support Vice President Nixon’s bid for the presidency.
The U-2 intelligence was so closely held that it was not shared with RAND or even many senior officers in the Air Force. So the strategists and the planners continued their work believing they were facing a nearly equal, perhaps superior adversary.
Preemptive War is distinct, in Just War Theory, from Preventive War. In the latter, you attack an enemy perhaps months or years ahead of when you predict a conflict would happen. In the former, you launch the attack just before an enemy plans to attack you, to disrupt the attack. Preemption is, naturally, much more justifiable as a casus belli than Prevention.
The December 7th, 1941 attacks on Pearl Harbor are the best case study of this. Imperial Japan, fearing for its economy and strategic position after the United States embargoed oil and steel exports, launched a preventive war to conquer East Asia, with a strike at the U.S. Pacific Fleet to buy time.
If, on the other hand, the United States had detected the Japanese fleet steaming towards Pearl Harbor, getting ready to launch its torpedo planes, and chose to attack to disrupt the Japanese plans, that would have been preventive war.
Again, we can't just assume it was somehow just as likely as deterrence. One of the factors Kaplan notes, in discussing the Los Alamos scientists, was their assumption that international institutions were primarily engineering problems rather than tragic human ones.
Here I must pass on an acronym taught to me by a former government officials, describing the difficulty of working with PhD physicists turned startup CEOs: “PhD - Permanent Head Damage.” Apparently something about making advances in the abstract, math-heavy world of the strong and weak nuclear forces, electromagnetism, and gravity gives some (but not all) scientists a sense that all other problems must be easy to solve.
Footnote 6 you repeat "preventive" in both cases