Here we go again: Donald Trump is preparing to move back into the Oval Office and once more boasting how “unpredictable” he is, convinced that posing as a “madman” makes him strong. That strength, he and his minions insist, will cow adversaries from Russia to China, Iran or North Korea; it will even discipline America’s friends if they are misbehaving. Strength through madness, and peace through strength.

Asked whether he would use military force against China if it threatened Taiwan, Trump replies: “I wouldn’t have to because he respects me and he knows I’m f***ing crazy.” He, in this case, is Chinese president Xi Jinping.

Trump makes the same assumption about Vladimir Putin. The Russian president would never have dared to invade Ukraine if Trump had stayed in the White House, the president-elect is convinced. That’s because he recalls warning Putin that “‘you’re going to be hit so hard, and I’m going to take those f***ing domes right off your head.’ Because, you know, he lives under the domes.” Peace is nigh, Trump implies.

So it goes, right through the gamut of policies and geographies, even vis-à-vis America’s best friends. In his first term, while reviewing trade with South Korea, Trump instructed his negotiator to “tell them, if they don’t give the concessions now, this crazy guy will pull out of the deal”. This time, Trump is preaching even more fire and brimstone, in the form of ruinous tariffs on imports from Canada, Mexico, China and other places. No matter that economists warn of huge damage to American consumers and businesses. Trump is bonkers, after all, and the other guys will fold before he even has to do anything.

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This so-called “madman theory” of conflict and competition has a long pedigree. Machiavelli was probably not the first to notice that “at times it is a very wise thing to simulate madness”. The nuclear age added frisson. In 1959, Daniel Ellsberg, a strategist later famous for releasing the Pentagon Papers, argued that a “blackmailer” could coerce opponents by acting “convincingly mad”. Thomas Schelling, who later won a Nobel Prize for game theory, also concluded that in specific circumstances feigning insanity can be advantageous.

The first — and until Trump, the only — president to pick up the idea explicitly was Richard Nixon. “I call it the Madman Theory,” he told his Chief of Staff. “I want the North Vietnamese to believe that I’ve reached the point that I might do anything to stop the war. We’ll just slip the word to them that ‘for God’s sake, you know Nixon is obsessed about communism. We can’t restrain him when he is angry — and he has his hand on the nuclear button’.”

With Nixon as with Trump, though, there was always an obvious irony: both talked rationally about being irrational, which hardly added credibility. That’s probably one reason why their theory has mostly failed, at least so far. When Nixon launched a global nuclear alert in 1969, neither the North Vietnamese nor the Soviets batted an eye, and the war went on. After Trump first threatened North Korea with “fire and fury like the world has never seen” and then exchanged “love letters” with its dictator, Kim Jong Un accelerated rather than halted his nuclear-weapons programme.

Prompted by Trump’s first term in office, academics took another look at the madman theory. The gist was that the theory “doesn’t work” or is a “myth”. Some analysts, though, now believe that both the first generation of analysts — Ellsberg and Schelling — and the second missed a nuance that could help us think about a second Trump term.

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Let’s frame the madman theory with a simple thought experiment in game theory, called chicken. (It’s one variant of another game theory called Prisoner’s dilemma.) In this model of conflict, two drivers race their cars towards one another. The one who swerves first loses; the other one wins. If both swerve, it’s a tie; if neither swerves, they crash. (Or you could add a twist, as in that scene with James Dean in Rebel Without a Cause.) For Trump facing off against Putin, Xi or Kim, say, the maths would look as follows.

One way to win would indeed be to pretend insanity. Trump could, for example, rip off the steering wheel and throw it out of the window. Even then, he still won’t know if his opponent is rational or mad. And if he survives this game of chicken, he will have to play more rounds against the other adversaries who are watching.

This means that his success — his strength — depends on how the world interprets Trump’s craziness shtick. Enter Roseanne McManus at Penn State, who has qualified madman theory by distinguishing between four types of madness. This gets complicated — and is highly subjective — so I gave her a call and together we simplified as follows.

Some leaders take huge but still predictable risks only in certain contexts and for specific ends. McManus calls them “fanatics”. As an example she gives Adolf Hitler during the crisis of 1938, when the Brits backed down, concluding that the Sudetenland was worth immeasurably more to him than to them. Predictable fanaticism can be effective.

A leader who is predictably extreme in any confrontation, by dint of disposition, is a “megalomaniac” in McManus’s nomenclature. She names the late Iraqi dictator Saddam Hussein and points out that this rarely ends well for the megalomaniac — because enemies, in this case the US, will conclude, correctly or not, that they have no choice but to go to war sooner or later anyway.

A leader who flares up only in some situations but then becomes totally unpredictable is a “hothead”. McManus points to Soviet leader Nikita Khrushchev during the Berlin Crisis of the early Cold War. That type could win some games of chicken but might also lose (Khrushchev eventually backed down).

And then there are the “wild cards”, people who are genuinely unpredictable all the time because that’s their nature. McManus mentions Colonel Muammar Gaddafi, the colourful former ruler of Libya, who met a particularly gruesome end. This kind of crazy is a recipe for disaster.

What about today’s leaders? Putin, even when he rattles his nuclear sabre, seems to be a fanatic: extreme but still predictable because he is focused on not losing in Ukraine —as opposed to Syria, say. The same probably applies to Xi, who wouldn’t swerve in a game of chicken, but only if the fight is over Taiwan. With both, there seems to be method in the madness, which according to McManus makes them strong in those specific situations.

North Korea’s Kim Jong Un, like his father and grandfather, is harder to read, but I see him as either a megalomaniac or a hothead — that is, as either predictably extreme in his enmity towards South Korea and the US or as unpredictable in any crisis. That makes the Korean peninsula more dangerous than headlines suggest.

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But what about Trump? He keeps emphasising his unpredictability, which suggests that it is dispositional; that would make him a wild card. Or he could be a hothead, since he also fancies himself as a gifted dealmaker with a transactional approach that discriminates between situations.

Either way, nothing in modern madman theory suggests that he will actually be “strong”. One problem with dispositional madness is that opponents conclude that they could never make it right for him, could never remove grievances even if they swerve in the first round of chicken. Kim seems to have inferred precisely that in 2019 after his summit with Trump in Hanoi; soon after, he ordered a buildout of his nuclear arsenal.

Another problem with wild-card or hothead madness is that America’s allies are also watching these rounds of chicken. If they can no longer rely on the US because it’s chronically unpredictable, they will make other arrangements for their security. South Korea, Japan, Poland or Germany may build their own nukes, and form axes or ententes with other powers, including those hostile to America.

This has punctured Trump’s credibility all along. John Bolton, one of Trump’s national security advisers last time, argues that Trump would not necessarily win many or any rounds of chicken because his strength and madness are just “bumper stickers”, when in fact he “has no history of principled behaviour, so he is simply threatening unpredictably”.

Trump could of course fix that by rationally laying out his objectives, allegiances and red lines so that, in time, friends would again learn to trust the US and foes would fear crossing it. Then again, he would no longer be unpredictable then, no longer a madman, or even a strongman.

• Andreas Kluth is a Bloomberg Opinion columnist covering US diplomacy, national security and geopolitics