Why I think an invasion of Taiwan probably means WW3

Unfortunately, it's time for some game theory.

This is not a pleasant thing to have to write a post about. A decade ago, when I started blogging, I didn’t expect to be writing posts about the imminent possibility of World War 3. But, here we are. Top U.S. officials, including Secretary of State Anthony Blinken and Admiral Michael Gilday, have been saying recently that China could invade Taiwan within the next couple of years. I have no way of assessing how accurate their predictions are, but China’s recent 20th Party Congress has certainly featured more bellicose language toward Taiwan. And the long-term slowdown in China’s economic growth means that the value of waiting to conquer Taiwan — of “hide your strength and bide your time”, to use Deng’s phrase — has fallen, which is probably why Xi Jinping has been shifting the CCP’s goals from economic growth to “security”.

So this is certainly worth thinking about.

The question of whether China will attack Taiwan soon is beyond my ability to answer. But I do have some thoughts on the question of U.S. intervention in the event of an invasion. Right now I’m visiting Taiwan, and a number of people I talk to tell me that despite Biden’s assurances, they’re unsure if the U.S. would intervene on their behalf if China invaded. I’ve been trying to explain why I think that U.S. intervention — with an escalation to full U.S.-China war — is highly likely if China invades. In fact, I think this is likely even if the U.S. doesn’t actually intend to intervene.

To explain my thinking here, I’m going to try to use some simple game theory.

A simple “Taiwan invasion game”

Game theory can produce some truly amazing results when used in things like auctions, where the payoffs and the sequences of moves are well-understood. For things like war and diplomacy, however, game theory isn’t really a predictive tool — it’s more of a rhetorical device, for explaining and exploring concepts. So the little bit of baby game theory I’m about to show should not be taken as some kind of model that can be used to predict the quantitative likelihood of World War 3. It’s simply a way of explaining the mechanisms that I think would lead a Chinese invasion to spiral into a great-power conflict.

The scenario I’m thinking about looks like this:

In this simple three-move game, China first decides whether to launch an invasion. Then it decides whether to start that invasion with a surprise preemptive attack on U.S. bases in the region, in order to prevent the U.S. from intervening on Taiwan’s behalf. After that, the U.S. decides whether to fight China or to sit back and watch the invasion unfold.

The idea of a Chinese preemptive strike on U.S. bases — such as Guam’s Andersen Air Force Base is not something I just made up. I got the idea from a 2021 Foreign Affairs article by Oriana Skylar Mastro. This appears to be a key scenario in U.S. war gaming as well. And Chinese propaganda outlets have also put forward this idea. Pearl Harbor provides an obvious historical precedent.

The reason China might do this is to prevent America from acting as a spoiler in an invasion attempt. A Chinese invasion force would involve a lot of transports and other ships, which would be vulnerable to U.S. attack. If U.S. bases were destroyed or severely degraded in a preemptive surprise attack, the U.S.’ ability to halt the invasion would be significantly reduced. The U.S. could still fight, but it would have to pull in ships and planes and other resources from far away, and that would take a lot of time and risk.

So that’s the “game tree”. Obviously there are a lot of other things both China and the U.S. could do that aren’t included in this simple model — blockades, various force buildups, nuclear strikes, whatever. That’s just one reason why the game I drew above is an explanatory device rather than a predictive model.

Anyway, to complete the game, we now need some payoffs. These are basically measures of how much each country’s decision-makers would value each outcome, combined with probabilities for things that can’t be predicted in advance. In a well-defined situation like an auction, the payoffs are pretty easy to know — it’s just money. But in a war situation, the payoffs are much harder to anticipate — they depend on things like domestic political calculations, leaders’ feelings of personal honor and national destiny, and so on. Those aren’t easy to put into numbers. On top of that, the payoffs depend on probabilities — i.e., how likely the U.S. is to actually thwart China’s invasion. Even the best military analysts don’t know that for sure, and I’m not a military analyst at all.

Thus, the payoffs that I put into this game are going to be based on my own guesses and assumptions. In other words, I’m going to make them up.

So here are my basic guesses and assumptions:

1. If China successfully conquers Taiwan, its leaders would feel a huge sense of success, and win the adulation of their people. So this payoff is very positive. But if it tries to conquer Taiwan and is militarily defeated, its leaders would feel a catastrophic sense of failure, and would be in extreme danger of being removed from office. So this payoff is extremely negative. The payoff of simply not invading Taiwan, and putting the issue off for another decade, is somewhere in between these two extremes.

2. If the U.S. fights and defeats a Chinese invasion of Taiwan, its payoff will be positive, since it will have scored a victory over a major rival and proven, yet again, that the rules-based order that forbids territorial conquest is still in place. If the U.S. fights a Chinese invasion and loses, its payoff will be negative, since nobody likes to lose a war.

3. If China invades Taiwan without attacking U.S. bases and the U.S. simply sits back and watches it happen, its payoff will be somewhere between “fight and win” and “fight and lose”. U.S. influence and credibility will be diminished, but that isn’t as bad as losing a war. But if China launches a preemptive strike on U.S. bases — likely killing thousands of U.S. military personnel — and the U.S. still sits back and does nothing, the U.S.’ payoff will be extremely negative. A Pearl Harbor style attack would have the entire U.S. populace howling for blood. Failing to fight back after a Pearl Harbor or 9/11 attack would be so bad of an outcome for U.S. leaders that basically they will never do this.

4. If China takes out U.S. bases with a preemptive strike, the U.S. probability of defeating China’s invasion (in the case that the U.S. chooses to fight) will be much lower.

So with these assumptions, I’ll make up some numbers for the payoffs:

China’s payoffs:

• China succeeds in conquering Taiwan: 100

• China doesn’t try to conquer Taiwan: 0

• China tries to conquer Taiwan and fails: -100

U.S.’ payoffs:

• U.S. defeats invasion of Taiwan: 100

• China doesn’t attack U.S. bases and U.S. does nothing: 0

• U.S. tries to stop Chinese invasion of Taiwan and fails: -100

• China attacks U.S. bases and the U.S. does nothing: -200

Probabilities:

• Probability of Chinese victory if China attacks U.S. bases and U.S. fights: 80%

• Probability of Chinese victory if China doesn’t attack U.S. bases and U.S. fights: 20%

Using these payoffs and probabilities, we can calculate the expected payoffs (probabilities times payoff of the outcome) for each sequence of moves in the game. In particular:

Expected payoffs:

• China invades but doesn’t attack bases, US. fights:

  • China’s expected payoff = 20% * 100 + 80% * -100 = -60

  • U.S. expected payoff = 80% * 100 + 20% * -100 = 60

• China invades and attacks bases first, U.S. fights:

  • China’s expected payoff: 80% * 100 + 20% * -100 = 60

  • U.S. expected payoff: 20% * 100 + 80% * -100 = -60

All the other sequences of moves have payoffs that don’t depend on probabilities.

Therefore, we can write in the expected payoffs for this game:

In each parentheses, the first number is China’s expected payoff (since China moves first), and the second number is the U.S.’ expected payoff.

What would the equilibrium of this game be?

People who have taken a game theory class will immediately see that the way I’ve set this game up, what happens is that China attacks Taiwan AND attacks U.S. bases, and the U.S. tries to fight after being attacked — in other worlds, WW3.

Let me explain why that’s the outcome.

For games like this, the way we solve it is with backwards induction. In other words, first we think about the player who moves last, and what they’ll do in each case. Then we assume that the player who moves first will anticipate these moves, and make their own choices accordingly. In other words, we assume that everyone looks ahead and correctly anticipates their opponent’s reaction.

So let’s start at the end. Let’s look at the U.S. choice of “fight” or “don’t fight” in the case that China invades Taiwan. And let’s look at both cases — the case where China preemptively attacks U.S. bases, and the case where China doesn’t preemptively attack. I’ve written the U.S. expected payoffs in blue:

For each case, the U.S.’ payoff is higher if it fights. If China leaves U.S. bases intact, fighting has a higher payoff because the chance of winning is bigger than the chance of losing. And if China attacks U.S. bases, fighting has a higher payoff because although the U.S. will probably lose, caving in after a preemptive strike is even worse.

So in this game as I’ve set it up, if China invades, the U.S. will definitely fight.

Given that, we can work backward and look at China’s choice of whether to attack U.S. bases. I’ve written China’s expected payoffs in red:

Given that the U.S. definitely fights in either case, China’s best option, out of all the options available, is to invade Taiwan and attack U.S. bases. This gets it an expected payoff of 60, which is the highest expected payoff it can get.

So the equilibrium outcome of the game as I’ve set it up here is WW3.

Now, this doesn’t mean I predict that WW3 will happen. There are plenty of assumptions here that we could change! For example:

• If any fight between the U.S. and China will definitely escalate to mutually assured nuclear destruction, and if this is simply unthinkable for any U.S. politician, then there’s zero chance the U.S. will try to resist an invasion of Taiwan. In this case, China will just go ahead and invade, confident that the U.S. will not respond out of fear of nukes. In this case, we’re in for an era of nuclear-armed conquest that will be ended only by rampant nuclear proliferation.

• If the U.S. doesn’t actually have a >50% chance of defeating a Chinese invasion even if China leaves U.S. bases untouched, then the U.S. won’t resist a Chinese invasion as long as China refrains from a preemptive strike. In this case, China goes ahead with the invasion and the U.S. doesn’t fight.

• If losing a fight over Taiwan would be really, really, really bad for China — i.e., if China is very risk-averse — then the expected payoff to invading Taiwan is negative for China, whether or not it attacks U.S. bases first. The risk of losing to the U.S. even after a preemptive strike might be so great that it prevents China from invading Taiwan at all. In that case, we get peace. This is also the case if Taiwan gets enough weaponry to successfully resist a Chinese invasion on its own, as Ukraine has resisted Russia’s invasion. Note that in this setup, this is basically the only way we avoid any sort of war at all — making the risks of invasion too high for China no matter what it does.

So obviously anything could happen here. The game I drew above simply illustrates a basic principle that I think people aren’t thinking about: If China invades, it has a very strong incentive to attack U.S. bases first. And if China attacks U.S. bases, the U.S. is highly likely to respond by fighting China.

The purpose of drawing out this game, instead of just writing the paragraph I just wrote, is to make people think about the U.S. response in the two cases separately, instead of just asking “How likely is the U.S. to defend Taiwan?” Once we work back from the two cases separately, we see that the U.S.’ motivations change depending on what China does. If China doesn’t attack U.S. bases, the U.S. will be motivated to fight by the opportunity of victory. And if China does attack bases, the U.S. will be motivated to fight by anger and the fear of doing nothing. And China can anticipate these two motivations in these two cases, and make its own choice accordingly.

Uncertainty, guessing games, and miscalculations

A final note here: Game theory itself, or at least the simple kind you learn in an intro class, might not be a good way to analyze this situation. The real world is full of miscalculations and missteps and guessing. People don’t really perfectly anticipate what their opponent will do, or even what their opponent’s payoffs and possible choices are.

In reality, instead of trying to guess the U.S.’ payoffs, Chinese planners might simply assign a probability to the U.S. fighting or not fighting in each case. So for example, suppose China’s planners think the U.S. has only a 20% chance of fighting if China invades but doesn’t attack U.S. bases.

In that case, using all the other numbers in my original game, the expected payoff of this move for China is 80% * 100 + 20% * -60 = 68. This is greater than 0 (the payoff of not invading at all) and 60 (the payoff of preemptively attacking U.S. bases), so China goes ahead and invades but doesn’t attack U.S. bases first.

But suppose China is wrong, and the U.S. actually has an 80% chance of fighting back in this case. In that case, thanks purely to a Chinese miscalculation, we probably get WW3 by accident.

It’s also possible that China dramatically overestimates the U.S. likelihood of fighting even if its bases are not attacked. In this case, the outcome from the original game will happen, and China will attack U.S. bases, drawing a U.S. response that wouldn’t have happened otherwise. And again, leading to WW3 by accident.

Anyway, I expect that U.S. defense planners and their Chinese counterparts are gaming out far more sophisticated versions of these scenarios, with far better-informed probabilities. And I think those will still be subject to mistakes and miscalculations. In the past, humankind has often been very stupid about blundering into wars — Putin’s invasion of Ukraine being only the most recent example. So I think the people warning about a war over Taiwan are far from alarmist; there are lots of reasons to be worried here.

Update: At the beginning of this post, I wrote: “In fact, I think [a war between the U.S. and China] is likely [in the event of an invasion] even if the U.S. doesn’t actually intend to intervene.” I don’t think I was clear enough about what I meant by that, so let me explain.

Suppose China’s leaders believe that the payoffs and probabilities of the Taiwan invasion “game” are exactly what I wrote in my main example above. So they choose to A) invade, and B) attack U.S. bases first, causing WW3. But suppose the U.S.’ payoff to defeating China in a war over Taiwan is actually something like 20 instead of 100. This would mean that China blundered into WW3; if China hadn’t attacked U.S. bases, it could have taken Taiwan without opposition.

Alternatively, suppose that China’s payoff to being defeated by the U.S. in a war over Taiwan is really, really negative — say, -1000. And suppose that China treats U.S. intervention in the case of no preemptive strike as a probability rather than as a certainty. In that case, even just a small probability of U.S. intervention in the case of no preemptive strike would be a huge risk for China. So in that case they might attack U.S. bases just to be on the safe side, thus precipitating WW3 despite the fact that America probably wouldn’t have intervened if it hadn’t been attacked.

In other words, there are lots of different ways China could blunder into “Pearl Harbor 2.0”.

Update: Chris Blattman, economist, political scientist, and author of the book Why We Fight, has a post exploring the game theory of why the U.S. and China might want to fight about Taiwan in the first place. My post is much simpler in comparison — I was just focused on the question of whether a Taiwan invasion would spiral into WW3. So definitely check out Chris’ post.

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