In The Three Body Problem trilogy, Wallfacer Rey Diaz needed to better understand game theory

Regular readers of this blog may have noticed that I haven't posted a book review in a while. That's because I've been reading Cixin Liu's Three Body Problem trilogy (technically, the Remembrance of Earth's Past trilogy, an adaptation of which has just been released on Netflix as The Three Body Problem). I'm currently reading the second book, The Dark Forest.

Warning: Spoiler alert!

To give you some context, in the first book of the trilogy, Earth made contact with an alien civilisation, the Trisolarans. The Trisolaran fleet is currently on its way to Earth, in order to conquer us. Their homeworld is about to be destroyed, and their only hope of survival is to take over another planet. The fleet will take some 400 years to arrive, so Earth has some time to prepare. However, the Trisolarans have advanced technology, including deploying sophons, which are able to prevent Earth from conducting basic research in physics and other areas. So, Earth is stuck in a low-technology state, awaiting the arrival of the Trisolaran fleet. Even worse, the sophons can watch anything that happens on Earth and relay the information back to the Trisolarans, so Earth's preparations will be known to the Trisolaran fleet. To combat this, in the second book, Earth appoints four 'Wallfacers', who are given access to almost unlimited resources to execute plans that are known only to themselves, hidden from the rest of the Earth's population (and to the sophons, because the sophons can't read minds).

The second book of the trilogy is devoted to the Wallfacers and their plans (admittedly, I haven't finished reading it yet). I want to focus on the plans of Wallfacer Rey Diaz, whose plan involved planting large solar hydrogen bombs on Mercury, which when detonated would set off a chain reaction, destroying most of the solar system, including Earth. Diaz's plan was to negotiate with the Trisolaran fleet, warning them that if they didn't divert, Earth would be destroyed, sealing the fates of both the human and Trisolaran populations.

However, Wallfacer Rey Diaz's strategy is flawed. He needs to understand some basic game theory. To see why, consider the game shown in the payoff table below. The two players are Earth and the Trisolaran fleet (we'll assume that Diaz would choose strategy on behalf of Earth). Earth's two strategies are to blow up Mercury (detonate) or not. The Trisolaran fleet's two strategies are to continue to Earth, or divert. If Earth blows up Mercury, then Earth becomes extinct, regardless of what the Trisolaran fleet does. If Earth doesn't detonate Mercury, then Earth loses if the Trisolaran fleet continues, and wins if the Trisolaran fleet diverts. If the Trisolaran fleet diverts, they become extinct. If they continue to Earth, they become extinct if Earth blows up Mercury, but win if Earth does not.

To find the Nash equilibrium in this game, we use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium). In this game, the best responses are:

1. If the Trisolaran fleet continues to Earth, Earth's best response is to not detonate (since losing is a better payoff than extinction [*]) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
2. If the Trisolaran fleet diverts, Earth's best response is to not detonate (since winning is a better payoff than extinction);
3. If Earth chooses to detonate, the Trisolaran fleet's best response is either option (since both payoffs are the same - extinction - both are best responses); and
4. If Earth chooses not to detonate, the Trisolaran fleet's best response is to continue to Earth (since winning is a better payoff than extinction).

Note that Earth's best response is always to choose not to detonate. This is their dominant strategy. A player would always choose to play their dominant strategy, because choosing the other strategy makes them unambiguously worse off. And the Trisolarans would know this. This is what Wallfacer Rey Diaz gets wrong in his strategy. Earth won't blow Mercury up, and the Trisolarans know this, so there is no leverage for Earth in the negotiations.

The Trisolaran fleet has a weakly dominant strategy. Notice that continuing to Earth is always the Trisolaran fleet's best response. However, diverting is a best response if Earth chooses to detonate. So, continuing to Earth is not always better for the Trisolaran fleet, but it is never worse than the other strategy.

The single Nash equilibrium occurs where both players are playing a best response (where there are two ticks), which is where all Earth chooses not to detonate, and the Trisolaran fleet continues to Earth. It is little wonder then, that when Rey Diaz's strategy was revealed, the Earth governments were not happy. Not only was his strategy imperilling the Earth to the same extent as the Trisolarans, it was a strategy that simple game theory shows would not have succeeded.

*****

[*] You may wonder what the difference between losing and extinction is. Earth could lose, but some humans remain alive as slaves, or otherwise escape the planet before the Trisolarans arrive. It's not a great outcome, but better than extinction.