# A simplified nuclear game with Kim Jong-un

Despite its formal apparatus and cold reputation, game theory is in fact the systematic deployment of empathy. It's hard to overstate how powerful this can be, without or without mathematical machinery behind it, so let's take an informal look at a game-theoretical way of empathizing with somebody none of us would particularly want to, North Korea's Kim Jong-un.

First, a caveat: as I'm not trained in international politics, and this is an informal toy model rather than a proper analytical project, it'll be very oversimplified both in form and content. The main point is simply to show a quick example of how to think "game-theoretically" (in a handwavy, pre-mathematical sense) that for once isn't the Prisoner's Dilemma.

This particular game has two players, Kim and the US, and three possible outcomes: regime change, collapse, and status quo. We don't need to put specific values to each outcome to note that each player has clear preferences:

• For the US, collapse < status quo < regime change
• For Kim, collapse,regime change < status quo

(From Kim's point of view, a collapsing North Korea and one where he's no longer in charge are probably equivalent.)

Let's simplify the United States' possible moves to attempt regime change and do nothing. The latter results in the status quo with certainty, while the former might end up in a proper regime change with probability $p$, or in a more or less quick collapse with probability $1-p$. Therefore, the United States will attempt a regime change as soon as

There are multiple ways in which Kim's perceived risk can rise, even aside from direct threats. For example:

• Decreased rapport between the US and South Korea or China (the two major countries who would suffer the brunt of the costs of a collapse) decreases the cost of collapse in the US' strategic calculations, and hence makes a regime change attempt more likely.
• Every attempt of regime change by the US elsewhere in the world, and any expression of increased self-confidence in their ability to perform one, makes Kim's estimate of the US' estimate of $p$ that much higher, and hence a regime change attempt more likely.
• Any internal change in North Korea's politics risking Kim's control of the country, should it be found, will also raise $p$.
• For that matter, a sufficiently strong fall in their military capabilities would eventually have the same effect.

Kim most likely knows he can't actually defend himself from an attempted regime change (there's no repelled regime change attempt outcome), so his only shot at staying in power is to change the US' strategic calculus. Given how unlikely it seems to be that he can make the status quo more desirable, he has, from a strategic point of view, to make the cost of an attempted regime change high enough to deter one. That's what atomic bombs are for: you change the payout matrix, and you change the game equilibrium. Once you can blow up something in the United States, which of course has an extremely negative value for the US, then even if $p = 1$,

The unintended problem is that, by both signalling and action, Kim and his regime have convinced the world that they are not entirely rational in strategic terms. As Schelling noted, deterrence often requires convincing other players that you're "crazy enough to do it," but in Kim's case nobody feels entirely certain that he will only use a nuclear weapon in case of an attempted regime change, or exactly what he'd consider one, so, although possessing a nuclear weapon decreases the expected value of a regime change attempt, it also decreases the value of the status quo, making the net impact on the US' strategic calculus &mdahs; the real goal of North Korea's nuclear program — doubtful. It can, and perhaps has, set the system in a dangerous course: the US decries the country as dangerous, the probability of a regime change attempt grows, Kim tries to develop and demonstrate stronger nuclear capabilities, this makes the US posture harsher, etc.

In this toy model — and I emphasize it's one — any attempt to de-escalate has to being by acknowledging that Kim's preferences between outcomes are what they are. Sanctions that weaken the regime spur, rather than delay, nuclear development. Paradoxically and distastefully, what you want is to credibly commit to not attempting a regime change, which at this point can only be done by actively strengthening it. This is something that both China and South Korea seem acutely aware of: pressures on and threats to North Korea tend to be of the "annoying but not regime-threatening" kind, as anything stronger would be counterproductive and not credible, and their assistance to the country has nothing to do with ideological sympathy, and everything to do with keeping the country away from collapse.

But not everything is bleakly pragmatic in game theory, and more humane suggestions can be derived from the above analysis. E.g.,

• A Chinese offer to strengthen and modernize North Korea's nuclear command chain to avoid hasty or accidental deployments would raise a bit the value of the status quo without increasing the chance of a regime attempt, a mutual win that'd probably be accepted.
• Any form of humanitarian development, as long as it's not seen as threatening the regime, could be implemented if Kim can sell it internally as being his own accomplishment. That'd be very annoying to everybody else, but suggests that quality of life in North Korea (although not political freedom) can be improved in the short term.
• Credibly limited tit-for-tat counterattacks might, paradoxically, reinforce everybody's trust in mutual boundaries. So, if a North Korean hack against an US bank is retailed to by hitting Kim's own considerable financial resources in a way that is obviously designed to hurt him while also obviously designed to not impact his grip on power, that'd have a much higher chance of changing his behavior than threatening war.

To once again repeat my caveats, this is far from a proper analysis. To mention one of a multitude of disqualifying limitations, useful strategic analysis of this kind often involves scores of players (e.g., we'd have to look at internal politics in North and South Korea, China, Japan, and the United States, to begin with) with multiple, overlapping, multi-step games, and certainly more detailed and well-sourced domain information than what I've applied here. To derive real-world opinions or suggestions from it would be analytical malpractice.

The point of the article isn't to give yet another uninformed opinion on international politics, but rather to show how even a very primitive and only roughly formal analysis can help frame a discussion about a complex topic in a way that a more unstructured approach couldn't, specially when there are strong moral issues at play.

Sometimes emotions get in the way of understanding somebody else. Thankfully, we have maths to help with that.