Shall we play a game?

Imagine that a crisp $100 bill lies on a table between us. We both want it, of course, but there’s no chance of splitting it — our wallets are empty. So we vie for it according to a few simple rules. We’ll each write down a secret number — between 0 and 100 — and stick that number in an envelope. When we’re both done, we’ll open the envelopes. Whichever of us wrote down the higher number pockets the $100. But here’s the catch: There’s a percentage chance that we’ll each have to burn $10,000 of our own money, and that chance is equal to the lower of the two numbers.

So, for example, if you wrote down 10 and I wrote down 20, I’d win the $100 … but then we’d both run a 10 percent risk of losing $10,000. This is a competition in which, no matter what, we both end up paying a price — the risk of disaster.

What number would you write down? I’ll reveal mine later, but for now try your hand against a fellow FiveThirtyEight reader who has already played.

Now imagine that you’re playing the same game, but for much more than $100. You’re a head of state facing off against another, and the risk you run is a small chance of nuclear war. The $100 prize becomes the concession of some international demand — a piece of disputed territory, say — while the $10,000 potential cost becomes untold death and destruction, nuclear winter and the very fate of our species and planet.

How would you play the game then?

We aren’t the only ones playing a version of this game. In July, North Korea conducted a series of intercontinental ballistic missile tests. One missile flew for about 45 minutes and, if it were fired on a standard trajectory, could have reached Chicago. In August, The Washington Post, citing U.S. intelligence officials, reported that North Korea had successfully miniaturized a nuclear warhead that could fit inside these ballistic missiles. Later that day, President Trump responded to the news: “They will be met with fire, fury and frankly power the likes of which this world has never seen before” should they threaten the United States. He also tweeted that “military solutions” were “locked and loaded.” North Korea quickly responded in turn, saying it was drafting plans to fire its missiles into waters near Guam, the U.S. territory in the western Pacific, creating an “enveloping fire” around the island. The country fired a missile again in late August, sending it soaring over Japan, prompting citizens there to take cover. Two days later, the U.S. and South Korea flew warplanes on bombing drills. Two days after that, on Sunday, North Korea tested what it said was a hydrogen bomb.

It’s hard to know how to make sense of all this, aside from harboring a low-grade existential dread. North Korea’s motivations and nuclear capabilities are shrouded inside the hermit kingdom. America’s true strategy is also shrouded, in part because of Trump’s long history of bluster.

But there is one way to try and decipher the motivations and actions of the last few weeks: game theory.

For decades, social scientists have tried to dissect nuclear standoffs by treating the saber-rattling as a game. Game theory applies to all sorts of situations — poker, our $100 bill game, or even two countries quarreling beneath the specter of nuclear Armageddon. A game theorist identifies the essential components of each — what motivations drive each player, and how those motivations conflict or coordinate — turning each “game” into a mathematical object. That object can then be analyzed and “solved.” These solutions find points at which games are in balance — when no player profits from changing his strategy. And when it comes to nuclear war, when the strategies in question are whether or not to press the button, we do well as a planet to understand how well balanced we are.

This isn’t a new approach. People have been thinking game theoretically about military conflict since at least 1832, when a Prussian general, Carl von Clausewitz, wrote “On War” to discuss strategy, tactics and the role of information in battle. But the modern application of game theory to understand war came from Thomas Schelling. Schelling, who once worked in the Truman administration, became fascinated by nukes during the early part of the Cold War. He began to study the strategies surrounding them like the economist he was: What are the players’ incentives, how will they act and where will their actions lead? The result was “The Strategy of Conflict,” a 1960 book in which Schelling laid out what he’d found. Schelling was a consultant on the film “Dr. Strangelove,” Stanley Kubrick’s Cold War satire, and he later won a Nobel Prize (in economics, not peace) for “having enhanced our understanding of conflict and cooperation through game-theory analysis.”

A typical Schellingian finding: Imagine you’re Trump or Kim Jong Un, essentially playing a game of chicken. You’re driving at high speed directly toward your opponent who’s also racing toward you. Neither of you wants to chicken out and veer away, but neither wants to die, either. Your best strategy? Rip off your steering wheel, make sure your opponent knows you’ve done so, and hit the gas.

That’s the terrifying thing about game theory: Sometimes the most rational choice can feel like the most dangerous. And that’s a problem when there are nukes involved. In the old days, if my country had better archers than yours, you’d keep that in mind when you felt like going to war with me. But nuclear weapons don’t work like archers. They decouple raw military strength from a state’s ability to win a war. That’s why North Korea, a country smaller than Mississippi with a GDP roughly equal to Wyoming’s, gets to compete alongside a superpower like the U.S. “What matters is if they can launch ICBMs to destroy Los Angeles or Chicago or Washington or wherever,” James Fearon, a political scientist at Stanford, told me.

Either you have nukes or you don’t. Either you use nukes or you don’t. It’s not a competition with arms or battlefields any more. It’s a competition in risk taking.

Fearon is the author of a 1995 paper called “Rationalist Explanations for War.” A modern classic in its field, it begins: “The central puzzle about war, and also the main reason why we study it, is that wars are costly but nonetheless wars recur.” In the paper, Fearon argues that there are two main reasons why wars break out. First, players have private information, and incentives to misrepresent that information. Second, the players have commitment problems.

Our $100 game, which Fearon teaches to his undergraduates, revolves around those two ideas. My private information is my appetite for risk. How much of it am I willing to take on to try and win the $100? You have no idea, and vice versa. And neither of us can really commit to a peaceful or bellicose strategy and make the other side believe it. The secret envelope and our unceasing self-interest stops that. That’s a commitment problem.

In our $100 game and in nuclear standoff, there’s no easy way to rip out the steering wheel.

Game theory suggests that nuclear standoffs have a pernicious consequence: posturing. What if, in our $100 game, we have a conversation before putting numbers in our envelopes? I tell you all about my recent skydiving trip, my fast motorcycle and my gambling jags in Vegas. “I’m a risk taker!” I swear.

But you quickly catch on to my tactics. Whether I’m a risk taker or not, I certainly have an interest in your thinking me one. What’s a small risk of burning $10,000 to a gambling, skydiving motorcyclist? If my rhetoric succeeds, and you believe I’m a “tough” type of player, you’ll be inclined to submit a very low number, ceding me the $100 but running a negligible risk of losing $10,000. But you also have the option to try and turn the tables on me. So you light up a cigarette, tell me about the bar brawl you escaped from last weekend and show me some trophies from your recent big-game-hunting safaris. But I’m onto you, too.

In our simple game, all of this risk-loving posturing is what game theorists call “cheap talk.” It’s to be taken with at least a grain of salt.

Ideally, we’d be able to prove to each other that our intentions are true — that we’ll commit to our stated intentions. I’d love to go first and publicly commit myself in a totally binding contract to write down “100.” If I could do this, you’d have no choice but to write down “0,” ceding me the $100 prize for free. (No reason to risk losing $10,000 if you know there’s no chance of winning $100.)

But our game doesn’t let me do that. Neither, in many cases, does the real world.

Trump and Kim are having this same conversation, and they seem to be expressing an appetite for risk. In mid-August, the New York Times reported there was talk at the White House of “preventive war.” On Sunday, from the White House lawn, Defense Secretary James Mattis addressed the North’s most recent bomb test. Any threat to the U.S. “will be met with a massive military response — a response both effective and overwhelming,” he said. “We are not looking to the total annihilation of a country, namely North Korea, but as I said we have many options to do so.”

To a theorist, tough talk could be the smart play. Trump’s painting himself into a corner, but that’s not necessarily a bad thing to do.

And the other player is surely listening from across the Pacific, and has responded with tough talk and actions of his own. So we arrive at the important question as we filter through one concerning headline after another: How cheap is the talk in the real world? Unlike in our simple $100 game, there may be some actual weight behind these words. By talking tough, the Trump administration knows, or should know, that there’s a higher chance of an accident — that some radarman in Pyongyang will sound the alarm at the first hint of a possible attack, Fearon said.

In many game theoretic models, nuclear war only ever happens by accident. No one ever just wakes up and decides to push the button. What incentive would they ever have? Robert Powell, a Berkeley political scientist who has written extensively on game theory and war, teaches his students a different model of nuclear brinkmanship than the one we played. In Powell’s version, there is a player named Nature. The Nature player is introduced into games as a sort of random number generator. It’s a chaotic actor, not a strategic one. It effectively flips a coin to decide its path — and when the other two actors are close enough to war, sometimes Nature nudges them toward — or over — the edge.

Take, for instance, a routine Air Force patrol flight that went wrong during the height of the Cold War that nearly devastated the U.S. In 1961, a B-52 bomber carrying two thermonuclear bombs broke up in mid-air, dropping one into a field in North Carolina — a “broken arrow.” Three of the bomb’s four failsafes failed or were destroyed. If the fourth hadn’t worked, vast swaths of North Carolina would have been destroyed and fallout would’ve blanketed the East Coast. Such could have been the cost of the standoff with the Soviets. (If you’re interested in everything that has to not go wrong when you maintain an arsenal of nuclear weapons, I’d suggest reading Michael Lewis’s recent Vanity Fair feature on the Department of Energy, which oversees the arsenal.)

This is in part why many commentators have been critical of Trump’s “fire and fury” comments. If the White House is vocal about its willingness to consider war, it could spark an actual war. It certainly sounds very dangerous.

Military solutions are now fully in place,locked and loaded,should North Korea act unwisely. Hopefully Kim Jong Un will find another path! — Donald J. Trump (@realDonaldTrump) August 11, 2017

But to a theorist, tough talk could be the smart play. Powell told me that Trump’s bluster could be useful, since there’s no other way for him to signal to the North Koreans how serious he is about stopping them from threatening the U.S. with a nuclear strike. “If you’re trying to signal credibility to the North Koreans, the way you would signal credibility is by making it more costly for you to back down.” Trump’s painting himself into a corner, but that’s not necessarily a bad thing to do.

The Trump-Kim standoff is particularly difficult to dissect because it’s unclear what each side is playing for.

Let’s return for a final time to the game we played earlier, and consider the Cuban missile crisis. The $100 prize for the Soviets was having its missiles in Cuba. The $100 prize for the Americans was the dismantling of these missiles and their return to the Soviet Union. The potential $10,000 penalty was all-out nuclear war between two superpowers. The U.S. won the $100 because it was “willing to bid higher in the currency of nuclear risk,” Fearon said.

But now consider the standoff with North Korea. As useful as the $100 game is, it may not be the right representation of this particular standoff. “I don’t know what the $100 is,” Fearon said, “There isn’t really a clear demand on the table.”

Instead, there are two sides whose goals don’t necessarily seem at odds with one another. The Trump administration wants not to be under the threat of nuclear attack and, like administrations before it, claims it’s not seeking to oust Kim Jong Un. Kim Jong Un, on the other hand, wants to be left alone and to be safe from American threats. But as Fearon wrote at The Washington Post’s Monkey Cage blog recently, North Korea sees right through the U.S.’s assurances it is not seeking regime change in the North. Obviously the U.S. wants regime change, and has no credible way to agree not to participate in this change if the opportunity should arise.

Another difference from our game: The stakes are different for each side. First, it’s possible that North Korea, run by a “madman,” is simply more inclined toward risk than the U.S., and therefore has an advantage in our nuclear auction — it’s willing to bid higher. Second, you can make a good case that the U.S. has more to lose, which in turn makes it less inclined toward risk.

North Korea has just launched another missile. Does this guy have anything better to do with his life? Hard to believe that South Korea….. — Donald J. Trump (@realDonaldTrump) July 4, 2017

All of this to say: While simple models like our $100 game may make nuclear war seem unlikely, we’re not living through simple times. Muddied goals, an actor with little to lose and the looming threat of an accident make things very, very complicated.

Game theory isn’t a perfect vessel. There are more than two players in today’s nuclear standoff, for example. China, South Korea, Japan and Russia all have their own $100 bills to gain. And, of course, the people involved in any game aren’t necessarily always rational. We often hear that Kim Jong Un, for example, is a madman — “We can’t let a madman with nuclear weapons let on the loose like that,” Trump once told the president of the Philippines.

But theorists do not begin their work from the premise that people or states are hyper-calculating rationality machines. Rather, they start from the reasonable notion that people respond to incentives. And madman or not, Kim will certainly respond to the disincentive of his country being blown to kingdom come.

Then there’s the concern that nuclear game theory suffers from a lack of data (thank God). There are only so many nuclear standoffs to go around, and only two nuclear bombs have ever been dropped in war. But there’s a counterpoint for that, too: While we may not have much data on actual nuclear catastrophes, nuclear weapons have been at the ready for over 70 years, and in this time historians have amassed troves of documents on nuclear decision-making and on the avoidance of nuclear war. Those documents are the data that one can test nuclear game theory against.

The Cuban missile crisis provides a useful example.

“If we have learned anything from this experience, it is that weakness, even only apparent weakness, invites Soviet transgression,” wrote Raymond Garthoff, a State Department intelligence analyst, in 1962. “At the same time, firmness in the last analysis will force the Soviets to back away from rash initiatives.”

“We are not struggling against imperialism in order to die,” Nikita Khrushchev, the Soviet leader, wrote to Fidel Castro the very next day.

People — Khrushchev and Kim included — respond to incentives.

Thinking about nuclear standoffs through the prism of game theory might bring some order to chaos, but it can’t divine how the conflict will end. “The lesson of these models is that you don’t get crises unless the outcome is unpredictable … but then you hear pundits making predictions about the outcome!” Powell said through an uncontrollable fit of laughter.

These recent events are interesting — and challenging — to map out game theoretically precisely because of their uncertainty. “You can’t make good predictions about what the other side will do,” Fearon said. “If you could, there wouldn’t be a crisis.”

But Powell remained optimistic, if optimistic is the right word: “I think it’s much more likely than not that we’re not going to go to war and the world’s going to accommodate a nuclear-armed North Korea that has the capability to strike the territory of United States,” he said. “Despite all the threats.”

I won’t dare make a prediction, either. But I can report that, at press time, Trump had unleashed no fire and North Korea hadn’t bombed Guam.

And I wrote down 0.1, by the way.