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Volume 16: Game Theory and the Nuclear Age: Part II

June 28, 2017



How About a Nice Game of Chess?


“Gentlemen, you can’t fight in here! This is the War Room!” – President Merkin Muffley, Dr. Strangelove


The 1960s were perhaps the most dangerous period in our history. At the height of the Cold War, the United States and Soviet Union fought for influence on five continents. The Berlin Crisis, Bay of Pigs, and Cuban Missile Crisis flowed to escalation in Vietnam and the bellicose leaderships of Richard Nixon and Leonid Brezhnev.


The world of the Sixties existed every day in the shadow of nuclear weapons. Increasingly powerful and reliable, ready to launch from airplanes, missile silos and submarines. By the end of the decade, the superpowers felt obligated to possess nearly 40,000 bombs, enough to destroy the world many times over.


Somehow – and sometimes despite our best efforts – there was no nuclear holocaust. This is the story of how we avoided it and of whether we’ll be able to in the future.


  • What is Mutually Assured Destruction?

  • What are the problems with MAD?

  • Will this game work in the modern nuclear paradigm?


Your author cautions you (same as in Volume 14):


Nuclear weapons are terrible things. This is true in all of the word’s meanings. Terrible: extremely bad, as in “a terrible movie.” Terrible: formidable in nature, as in “a terrible responsibility.” Terrible: extreme or great, as in “a terrible disappointment.” Nuclear weapons are not to be trifled with, joked about or handled except with extreme care.


Which is all to say – this Volume will prominently feature death, destruction, fallout, nuclear winters and the end of the human race. We will flippantly talk about the gruesome deaths of millions and even billions. We will explicitly do this through the lens of a game, winning and losing, keeping score of the destruction. We will not modify our usual style of writing or the ongoing attempts at mild humor. We are not focused on the morality of these weapons, their development or their use. Except in the paragraphs immediately following.


The previous five hundred years have been a period of steady human advancement with no major steps back. It was inevitable that we would learn about the power of the atom. Every man, woman and child is innately aware of this power. It stares you in the face every day, from sunrise to sunset. We were going to learn about radioactivity, fission and fusion and chain reactions. It necessarily follows that we would try to harness this power. Even with knowledge of their destructive potential, we were eventually going to develop nuclear weapons capabilities. They are a by-product of these five centuries of discoveries about the Universe we live in.


Morality comes only after development: how can these weapons be controlled to prevent their use? The two decades after the Trinity Test were not just a scramble to build bigger bombs, but also to develop these controls. Game theory contains the most effective controls we’ve found. And we can only learn the game theory of nuclear weapons by repeatedly simulating the destruction of the world.

Only by understanding these weapons can we prevent their use.





What is Mutually Assured Destruction?


We left off in Volume 14 discussing the classic problems of game theory and this is where we will pick up. Recall that in game theory, Players (two of them, in our examples) make decisions from a set of choices, each called a “Strategy.” All the Players try to maximize their “Utility,” a generic measure of value, usually represented by a single number. Players in game theory are always assumed to act rationally. In some games, Players are allowed to communicate before choosing their Strategies. If you need a longer refresher on game theory basics, I encourage you to look back at Part I.


In game theory, Global Thermonuclear War is closest to Chicken, so we will start there. In classical Chicken, two drivers are headed towards a cliff and each must decide to Swerve First or Swerve Last. If both Swerve First, the game is a draw. If both Swerve Last, they drive off the cliff and are killed.(1) Table 1 shows the outcomes for our first game of Chicken:




The most important feature of Chicken is that the “Crash” outcome is far worse than the “Lose” outcome. This makes sense: the slight embarrassment you feel from swerving First is a better than the death you feel from driving off the cliff. Therefore, if you think that your opponent will swerve Last, you should swerve First. Even if you aren’t certain what he(2) will do, a significant likelihood of his choosing Swerve Last should make you cautious. Using the game theoretic terminology, the Nash Equilibriums of Chicken are where exactly one player swerves.(3)


Strategic Deterrence has been an element of warfare for most of history, long before the term was coined. The idea is simple: your nation-state tells all the others that, if attacked, you will respond disproportionally – usually against their homeland as well as their military forces. Knowing the losses they would experience, anyone thinking of attacking you will think twice. If this potential response is deadly enough, it will not be in their best interests to begin a war. Therefore, if they are rational Players, they will not attack you and peace is preserved.


Strategic Deterrence only works if the threat is credible. Napoleon couldn’t keep England off the European continent because he couldn’t credibly threaten a crossing of the English Channel.(4) Imperial Japan was forced to open its borders to Matthew Perry’s Black Ships. Japan’s Navy of the 1850s was archaic, making Perry’s threat credible. In World War II, strategic bombing took deterrence to new heights. The firebombing of Dresden on February 14/15, 1945 killed at least 22,700 people, rendered at least 100,000 homeless and destroyed the entire city.  The Allies lost only eight of the more than 2,000 warplanes sent against Dresden.(5)


But the disparate damage caused by raids in 1945 obscures the experience in World War II up to that point. When Germany (or, for that matter, England) had active fighter defenses, the bombers did not always get through.(6) In the British Bomber Command, 55,573 of around 125,000 total aircrew were killed by the end of the war. The American Eighth Air Force, also primarily tasked with bombing Nazi territory, suffered 47,000 casualties, 26,000 fatal.(7) It was not unusual for almost 10% of the crews to be lost in a single raid. In addition, the bombing campaign involved massive investments in ground crews, war materiel and critical aviation fuel. In short, it isn’t clear that Allied bombing campaign was actually successful, given the cost.(8) Conventional bombing would not be a credible strategic deterrence in the post-war era.


The limits of this threat were known by the Superpowers. Counterintuitively, the prevention of war relied on the ability to create credible deterrence. This need for credibility led to the development of Inter-Continental Ballistic Missiles, or ICBMs. ICBMs could be stationed in North Dakota and still reach central Russia. Flying at many multiples of the speed of sound, and with a tiny cross section, ICBMs could not be destroyed in flight.(9) To increase threat credibility, each ICBM came to contain Multiple Independent Re-Entry Vehicles, or “MIRVs.” Above the top of the atmosphere, an ICBM could release perhaps a dozen MIRVs, along with decoys and who knows what else. Each individual MIRV carried massive destructive power, enough to destroy a city.(10) The difficulty of intercepting the incoming attack thus became an order of magnitude greater.


Land-based ICBMs were, however, still vulnerable to attack. If you know where your opponent kept their missiles, you might try to destroy them using your ICBMs, a so-called “first strike.” So, everybody started looking for hiding places. Maybe keep them underground, or put them on trucks so they are mobile. But, the best hiding spot is below water; nuclear submarines are the key to credible deterrence. By 1961, ICBMs were on both American and Soviet submarines. Destroying one submarine was challenging enough; destroying a fleet impossible. Simultaneously destroying your opponent’s bomber force, submarine force and land-based ICBMs would be far too risky to try. Hence, the threat was completely credible. This arrangement became known as the Nuclear Triad.




By 1965, both sides had thousands of missiles arranged in their Nuclear Triads. A successful attack from the other side would be certain to draw a response. This is called “second strike” capability. American planners considered that nuclear weapons equivalent to 400 million tons of TNT would be enough to destroy the Soviet Union’s capability to make war; the Soviet Union probably had a similar number in mind.(11) Given that each side possessed around 10,000 million tons of bombs by the end of the 1960s,(12) any second strike would be on a massive scale. Both sides had a highly credible threat; a successful nuclear strike would return a strike in kind. This is Mutually Assured Destruction, also known by its acronym, “MAD.” Game theory tells us MAD should prevent nuclear war. In theory.


What are the problems with MAD?


We’ve talked a lot about the “D”, Destruction. There is little doubt that both Superpowers had the ability to annihilate the other in a successful strike, hence the “M”, Mutual. Which leaves us with the trickiest letter: “A” for Assured. To consider it, we’ll leave the historical-technical path and return to game theory.


Let’s go back to our game of Chicken, in Table 1 above. We can translate Swerve Last to Fire First and Swerve First to Fire Second – but there is a problem. In Table 1, the situation where you Swerve First is not such a terrible outcome. You are the Chicken, but you will survive. In Global Thermonuclear War, the situation where your opponent Fires First and you Fire Second is really bad. We need to modify the table; maybe to something like this:



We can quibble about the exact values here. For example, I assume you would prefer to be attacked by a second-strike rather than a first, but it could easily be the other way.(13) What really matters – and should make you feel a bit better – is that the outcome where neither side fires its Nukes is stable. Deterrence has worked. The prospect of nuclear war is so terrible that neither side will start one. This is what we hoped MAD would do, so QED.(14)


Unfortunately, there is a problem. In the real world, both sides have a third option, “do not retaliate.” There are many reasons why a Superpower might not actually conduct a second strike. They could be unable to or could choose not to. Maybe their weapons don’t work; after all, 400 million tons of TNT just exploded somewhere. Even launch detection was a real question in the 1960s. If you don’t know whether the other guy fired his missiles, how can you be sure you’ll strike back?


Let’s look at our game again, including the option not to retaliate.



Our first three-choice game, it looks confusing, but it’s not so difficult. The four outcomes in the bottom-right represent situations where nobody fires their bombs first. If nobody fires first, there is never a nuclear war and everybody wins. The other outcomes show what happens with the three possible responses to an attack. If you retaliate, you end up in the same situation as in Table 2. But if you hold your fire, it is worse. On the other hand, your opponent is even better than if the war never happened. If you get nuked and don’t fire back, the other guy dominates the world.


Fortunately, when both sides play “Retaliate” the game is still stable. However, the situation changes if your opponent plays Never. If you know he is doing this, you could fire your missiles immediately, winning the war. Recall that in game theory, players are always rational. In Table 3, Retaliate always performs as well or better than Never. Therefore, no player will ever choose Never and we are safe.(15)


But is our game truly descriptive of the real-world? Consider the situation where you are facing an imminent incoming strike and must decide what to do. The attack will destroy your entire country, probably killing half of your population and rendering your land uninhabitable for a long time. Given that this will happen no matter what – are you sure you would fire back? If you hold fire, your country will be destroyed, but your opponents’ will survive. Not a great situation, but a retaliation would kill hundreds of millions more and possibly result in the extinction of humanity. Not so easy to pull the trigger. If you think you might be better off not retaliating, it leads to a new game:



The key point to notice is that {First, Never} is now, unfortunately, a Nash Equilibrium, while neither {Retaliate, Never} nor {Never, Never} is stable. Therefore, game theory tells us that, if one player chooses Never, the other will immediately launch a full strike. This means that even if you aren’t sure you will Retaliate, you must make your opponent think you will. And, if you suspect your opponent of weakness, you must consider Firing First.


Let’s leave game theory and go back to the Sixties. Put yourself in the seat of President Kennedy or Chairman Khrushchev. In both countries, power to fire nuclear weapons sits with one person. If either Kennedy or Khrushchev chooses Fire First, no Congress, Presidium or military leadership can stop them.(16) And there is plenty of mutual distrust. Kennedy used the CIA to sponsor a coup in a Soviet-aligned nation ninety miles south of Florida. Khrushchev attempted to force the capitulation of West Berlin by provoking another crisis at that flashpoint. They sit, at the White House and at the Kremlin, buffered by military, political, domestic and foreign policy advisors, each with their own agenda. They have met in person only once, a tense one-day summit in Vienna. To keep the world safe (or at least stable), each must convince the other of their willingness to end the human race, should the situation arise.


Brinksmanship is the strategy of pushing conflicts to the edge of disaster without quite crossing the line into nuclear war. It became de facto American policy under President Eisenhower’s Secretary of State, John Foster Dulles, who said, “The ability to get to the verge without getting into the war is the necessary art.” President Truman intervened in the Korean War, but specifically prohibited the use of nuclear bombs. In 1961, American and Soviet tanks stared each other down over Checkpoint Charlie. A false move by a private manning a turret could have led to nuclear holocaust. And the most dangerous – ballistic missiles in Cuba were unacceptable to U.S. security interests. Kennedy had to find some way to eliminate this threat.




Brinksmanship was dangerous, but it did serve a purpose: communication. Defending your interests in the most far-flung parts of the world left little doubt that you would defend your own homeland to the hilt. If you are crazy enough to fly in every morsel of food and pound of coal to supply the entire city of Berlin for fifteen months, clearly you are not to be trifled with.(17) On the other hand, actual attacks directly on the personnel or materiel of the other Superpower were likely to lead to escalation. So, the Soviets shot down not a single one of the 200,000 highly vulnerable supply flights during the Berlin Airlift. Similarly, the American quarantine of Cuba went to great length to avoid firing a deadly shot.


Of course, any good brinksmanship player would play the game far from home. Both Superpowers had plenty of proxies around the world to use in these battles. Like most games of chess, it usually didn’t work out well for the pawns. Upwards of 2 million Vietnamese lost their lives during twenty years of American involvement in southeast Asia. Perhaps just as many Afghans were killed during nine years of Soviet incursion there; we don’t really know. Fully one third of the Afghan population became refugees, and the Taliban came to rule after the Soviets pulled out of the Bear Trap.(18)


Will this game work with in the modern nuclear paradigm?


Mutually-assured destruction worked when it was two superpowers facing off. Worked well enough, at least.(19) Never before in human history have nations possessing the best and the biggest weapons refrained from using them for such a time.


As we have seen, MAD requires any number of prerequisites to be effective. Each of the Players must have the ability to destroy their opponent. Each Player must know that his opponent is able – and willing – to produce a devastating retaliation. Having clear and consistent lines of communication is important; you need to convince the other guy that you are not playing the dreaded Never card. The world has changed after the fall of the Soviet Union. Do these prerequisites of MAD hold in the modern age?(20)




India and Pakistan


India and Pakistan have fought three major wars since their partition and independence in 1947.(21) India first conducted a successful nuclear test in 1974; Pakistan likely had a working weapon shortly after, although they didn’t test it until 1998. Since this time, two states with significant unresolved border issues along, and some slight differences with respect to religion, have had the Bomb. This is a not great situation.


The India/Pakistan nuclear game doesn’t look like the MAD we know and love. Both have around 100 total nuclear weapons, and they are fairly small (less than 1 megaton each). Therefore, neither country can credibly claim destructive power. As we’ve seen, this has the counterintuitive effect of making the weapons “easier” to use. India has a small nuclear triad, with a maximum of two nuclear submarines. Pakistan has no triad at all. Game theory says its lack of second-strike capacity also encourages a first-strike.(22) Oh, and bilateral communication between these long-time antagonists is strained at best.


There is some good news in South Asia. Both countries keep their weapons in a non-deployed state. Both countries have clear “no first strike” policies in place. Both have clear statements of purpose for their nuclear arsenals: “Minimum credible deterrence” for Pakistan and “Credible minimum deterrence” for India.(23) Both have stated and published fairly clear policies for what type of action would provoke a nuclear response. And, despite ongoing Kashmiri skirmishes, tensions between the two countries do seem to have calmed in the last two decades.(24) For me, the big risk is that, in a general conflict, battlefield commanders could be delegated nuclear response. In this situation, game theory is no help.




Israel’s nuclear arsenal is the world’s worst kept secret. Even Israel follows a policy of “nuclear ambiguity”; basically, if you ask Israel about its nuclear arsenal, they tell you how nice the beaches are in Tel Aviv. Israel has said it will not “introduce” nuclear weapons to the Middle East; I don’t know exactly what that word means.


It gets worse. As opposed to India and Pakistan, because Israel won’t even confirm it has the Bomb, it won’t say the conditions under which they will use it. Israel has historically had some disputes with its neighbors. It is entirely possible for one of them to accidentally stumble across an Israeli nuclear red line. Many believe that during the 1973 Yom Kippur War, Israel came close to using nuclear weapons against Egyptian and Syrian military headquarters. These were not located out in some remote desert location, rather Cairo and Damascus. And let’s not forget that Israel is one of the few countries that generally permits pre-emptive war.(25)


After the initial shock, Israeli forces quickly turned the tide of battle in 1973. The weapons went back into storage. Some believe that the deployment was a bluff – if so, it worked. The story is that, after satellites detected the deployment, the Soviet Union cautioned Egypt and the U.S. began a massive resupply. Today, Israel has signed peace treaties with Egypt and Jordan. The risk of conventional invasion from its direct neighbors is thus eliminated. Also, Israel is supremely concerned with its public image around the world. Outside of literal survival at stake, it’s hard to imagine a situation where Israel would launch nuclear weapons.


Unconventional weapons are different. Israel has said that it would respond to a weapon of mass destruction with everything in its arsenal. It has also reserved the right to strike against other nations’ nuclear development capacities. This could easily escalate.


North Korea


When we look at the nuclear weapons on the Northern half of the Korean peninsula, we must give up game theory entirely. Without the assumption of rational Players, we are in a different realm, math is no help. In theory a nation should fear the stigma of making a nuclear sneak attack. In theory a nation should fear complete global isolation. In theory a nation should fear second-strike capacity. In theory a nation should fear invasion by a U.S.-led international coalition, bringing to bear the enormous capabilities of the advanced militaries of the world, hell-bent on deposing its leader, little-concerned with civilian casualties, willing to use all of its weapons including tactical nuclear devices and planning to establish some type of military government for an indefinite period while working to deprogram every one of the citizens from seventy years of devastating and devastatingly effective propaganda. On North Korea – I got nothing.



I’ve made it through two Volumes on Nuclear War without mentioning two of the great Cold War movies. Both are completely germane to our topic.


The key to maintaining a stable game of MAD involves convincing your opponent that, if push comes to shove, you will play Retaliate. You need to do this despite Retaliate not necessarily being an optimal strategy. As President or Chairman, you will have just minutes to make the decision, on your own. Will you really decide to destroy the world? Avoiding war depends on at least appearing as though you will. Wouldn’t it be helpful to credibly say the response will be automatic? If we detect an attack, we will automatically fire back, no chance for human “weakness.” This is the Doomsday Device from Dr. Strangelove, or the War Operation Process Response (WOPR) from WarGames. In theory, if you have a Doomsday Device or a WOPR, your opponent should never start the War.


But, as these movies showed, such devices have their own problems. To serve as a deterrent, you must tell the world about your Doomsday Device. If you announce it before it is operational, there is an incentive for your opponent to Fire First, before it comes online. If you turn it on today, planning to tell the world next week, you risk making a non-proportional response to a rogue attack. The world ends.(26) Perhaps your WOPR has a bit more nuance. Via machine learning, it might consider the optimal strategy, considering the outcomes before just firing back with everything you have. Then, you are still susceptible to a mistaken detection or incursion into your system. And you can never build in a manual override for your WOPR.(27) Any unilateral ability to turn it off will severely limit the credibility of your threat. A WOPR that can be turned off sounds like the kind of thing a “Never Fire Superpower” would own.(28)


How do we eliminate – or at least reduce – the threat of nuclear war? People smarter than I have been working on this since August 1945. And we haven’t had a nuclear war yet, so I suppose they have been successful. I can offer my own thoughts on where to go next.


Preventing proliferation is obviously critical. This involves a very-long-term game of carrots and sticks. The Non-Proliferation Treaty (“NPT”), while not perfect, has been largely successful. All nations but five are signatories. No signatory has developed nuclear capabilities. The NPT works via only trust: there is no way to enforce it and any state can leave at any time without penalty.(29) However, this does not make the treaty ineffective. If nuclear weapons are highly expensive to develop, and all your enemies, potential enemies, rivals, neighbors etc. are NPT signatories, it is rational for you to sign too.


The only nation to voluntarily eliminate its arsenal so far is South Africa.(30) Along with the end of the abhorrent system of apartheid, elimination of its nuclear arsenal did much to bring South Africa into the community of nations. The last twenty-five years have treated her far more kindly than if she hadn’t changed her ways. Sticks must also be imposed: economic sanctions, aid to neighbors, disengagement. The success of the recent treaty with Iran is far from certain, skepticism is fully warranted. But if it doesn’t work, what would have? Like we said in Part I, building nukes is just not that tough if you put in the effort. Prevention via technical complexity is a losing strategy.


Once we deal with proliferation, that leaves the stockpiles in the existing nuclear states. Here, there is again hope, if distant and disjointed. Bilateral and multi-lateral treaties over fifty years have lowered the ongoing threat. Nuclear testing frequency has dramatically fallen. The Comprehensive Nuclear-Test-Ban Treaty has not yet been ratified, but only three nations have tested weapons since it was agreed to in 1996, and only one nation since 1998. The United States and Russian Federation have around 7,500 nuclear weapons each in their current stockpiles. This is an enormous reduction from respective peaks of 30,000 and 37,000. Most of these remaining weapons are not deployed. Both former Superpowers plan to reduce by half again in the next decade, joined by reductions in the UK and China. For the first time in a half-century, the world should soon have fewer than 10,000 nuclear weapons. That’s a start.


Even assuming these goals are achieved, it just gets harder. The remaining number of weapons will still create a game of Mutually Assured Destruction, but just barely. MAD has worked so far; a nuclear-free world is better, but the path from here to there involves a period of high instability. I don’t know how to travel this road. I do know that doing so successfully will require an enormous number of baby steps, building more trust after each. I am hopeful. I think that every nuclear power would agree to a nuclear-free world – but this doesn’t make it so. The nuclear knot we’ve created will take a generation or more to untie. But I do believe it can happen.




(1) Classic, game-theoretical Chicken doesn’t properly handle the Rebel Without a Cause scenario, where a driver intends to swerve but isn’t able to.


(2) I think we can safely agree that only men are dumb enough to drive cars towards an oncoming cliff.


(3) Recall that a Nash Equilibrium is an outcome of a game where no Player can benefit from changing Strategies, keeping the other Player’s Strategy constant. In this sense, they are “stable,” a term which I’ll use interchangeably.


(4) Now I’m vastly oversimplifying the Napoleonic Wars too.


(5) Again, I’m not debating here the morality of the Dresden raid, or the even-more-deadly Tokyo raid, or the eventual dropping of the atomic bombs. There are subjects even I won’t wade into.


(6) This is a bastardization of the famous quote from Stanley Baldwin, British Prime Minister in the 1930s, “the bomber will always get through.” This belief was a key reason why Britain didn’t bother building up its air defenses until it was very nearly too late. If they would get through no matter what, why invest in defense. Neville Chamberlain is better known today, but he was playing a bad hand dealt by his immediate predecessor. In my opinion, Baldwin did more to lose the war for the Allies than any other leader. For more on bombers in World War II, see The Bombing War: Europe 1939-1945.


(7) Bomber Command and Eighth Air Force losses from Wikipedia. This web site has more information on the Eighth, including downloadable (but incomplete) by-raid data.


(8) At least until destruction of the German Luftwaffe reduced this “cost.”


(9) At least not with 1960s technology. Also: an ICBM costs far less than a bomber. Of course, it is also a single-use item.


(10) Again, the bad Google searches are piling up. Destructive power of Minuteman ICBM; how many nukes did we have in 1970; modern ICBM velocity; first ICBM with MIRVs.


(11) “How much nuclear weapons tonnage did the Soviet Union think it needed in 1965” returns zero results on Google, as you would expect.


(12) I’m actually not really able to find reliable data on total arsenal tonnage in the 1960s. I found a reasonably credible estimate that global nuclear weapons capacity was 6,400 megatons in 2009. Arsenals fifty years ago had many times more total warheads. I don’t have data on tonnage per weapon over time, but I’m confident both sides had 10,000-megaton arsenals at the height of the Cold War. If you don’t believe me and think 3,000 megatons each is a better estimate, the narrative doesn’t change. But why are you reading this in the first place if you aren’t going to believe me when I do highly uncertain extrapolations of the sizes of nuclear arsenals?


(13) Both this Volume and Volume 14 were reviewed by a nuclear engineer, i.e. a professional. His view is that all of the nuclear war scenarios should have more negative utilities than I’ve used.


(14) Quod Erat Demonstrandum, “thus it has been demonstrated.” Math people say this when they are pretending to be smart. Don’t worry, they aren’t nearly as bright as they think they are.


(15) Well, as safe as we can be in a world with Gigatons of nuclear weapons scattered around.


(16) This is still true today. To spell it out: Donald Trump, a (failed) real-estate developer and (successful) reality-TV star, well-known to have a short attention span and limited grasp of factual reality, has the power to destroy the world without approval from any member of the government or military.


(17) More than 2 million tons of supplies in total. Most of it was in fact coal.


(18) The Taliban were “our guys” at the time. Not the greatest decision we’ve ever made.


(19) Source: you are reading this in a currently-inhabitable world rather than some type of Mad Max post-apocalyptic  nuclear winter wasteland.


(20) Google searches for this section: India nuclear arsenal history; does India have a nuclear triad? Pakistan first nuclear test; Trump pre-emptive North Korea; Does Star Wars work. Seriously – if the FBI, CIA or NSA comes looking for me, I probably deserve it.