## Sunday, 23 September 2018

### Is Trump vs. Xi a game of chicken, or a prisoners' dilemma?

There are several famous games that we use to teach game theory, one of which is the prisoners' dilemma (which I blogged on earlier in the week). Another is the game of chicken.

In the classic chicken game, two rivals line their cars up at opposite ends of the street. They then race directly towards each other. Each rival then has two options: (1) to swerve out of the way; or (2) to speed on. If one rival swerves and the other speeds on, the rival that sped on wins and the other driver looks foolish and loses some street cred. If both swerve, they both look a bit foolish. If both speed on, then there is a horrific accident and both may be severely injured or die. The game is presented in the payoff table below, for two drivers (Driver A and Driver B).

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 Driver A speeds ahead, Driver B's best response is to swerve (since a loss of face is better than dying in a fiery crash - maybe not immediately, but certainly in the long term) [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 Driver A swerves, Driver B's best response is to speed ahead (since winning is better than looking a little foolish);
3. If Driver B speeds ahead, Driver A's best response is to swerve (since, again, a loss of face is better than dying in a fiery crash);
4. If Driver B swerves, Driver A's best response is to speed ahead (since winning is better than looking a little foolish).
Notice that there are two Nash equilibriums in this game - where one driver swerves and the other speeds ahead. Both drivers prefer the outcome where they are the one speeding ahead though, so if both try to get that outcome, we end up with both drivers dying in a fiery crash. The chicken game suggests that both players, acting in their own selfish best interest (or not considering the response of the other driver), leads to the worst possible outcome.

Which brings me to this article by Ambrose Evans-Pritchard in the Telegraph UK (gated, but there is an ungated version here):
The US and China are on a combustible escalation path that can end only when there is economic blood on the floor and the political pain threshold of one side or the other has been hit.
Both think they can withstand the longer siege. Neither can retreat easily...
China has in any case stated already that it will match each round of US tariffs with a riposte in kind.
Beijing must carry out this threat or lose face, and Trump has already vowed to escalate further when it does.
It is very hard to see how asset markets priced for perfection can ignore this deranged game of chicken for much longer. The mystery is that they have not crumbled yet.
Is this really a game of chicken? It turns out that it depends on how you define the payoffs. There are two players in the game (the U.S. and China), and two strategies (enact tariffs or hold off - the equivalents of speeding ahead or swerving). So, we can represent the game easily in a payoff table.

First, let's consider the payoffs to each country. If one country enacts tariffs and the other holds off, both countries are worse off than the status quo (they both lose some gains from trade), but the country that holds off probably loses less (their exporters are a bit worse off) [*]. We'll say that the payoff to the country enacting the tariffs is "bad", but for the other country the payoff is just "not so bad". If both countries enact tariffs then all of the bad stuff happens (exporters are a bit worse off, and there are lost gains from trade). We'll say that payoff is "very bad" for both countries. If both countries hold off, then the status quo prevails - the payoff is "OK" for both countries. The game is presented in the payoff table below.

Again solving for Nash equilibrium, the best responses are:
1. If China enacts tariffs, the U.S.'s best response is to hold off (since "not so bad" is better than "very bad");
2. If China holds off, the U.S.'s best response is to hold off (since "OK" is better than "bad");
3. If the U.S. enacts tariffs, China's best response is to hold off (since "not so bad" is better than "very bad");
4. If the U.S. holds off, China's best response is to hold off (since "OK" is better than "bad").
Notice that the best response for both countries is to hold off, regardless of what the other country does. Holding off is a dominant strategy. There is one Nash equilibrium here, which is for both countries to hold off (the status quo). It is also a dominant strategy equilibrium (because both countries have a dominant strategy). The equilibrium outcome of this game is the best outcome overall.

Clearly, that isn't the game that is playing out at the moment though, so how are things different? The current game is not a game about trade, it is a game about political posturing. The players are not the U.S. and China, but Donald Trump and Xi Jinping. They want to look strong, and not appear weak (to each other, or to their respective peoples). So, the payoffs and the players are different. The game that is actually being played looks more like the payoff table below. If both hold off, the status quo prevails (the payoff is "OK" for both. If one of them enacts tariffs and the other holds off, whichever of them enacted tariffs appears strong, and the other appears weak. If both enact tariffs, the payoff is costly to the economy.

Again solving for Nash equilibrium, the best responses are:
1. If Xi enacts tariffs, Trump's best response is to enact tariffs (since "costly" is better than "weak");
2. If Xi holds off, Trump's best response is to enact tariffs (since "strong" is better than "OK");
3. If Trump enacts tariffs, Xi's best response is to enact tariffs (since "costly" is better than "weak");
4. If Trump holds off, Xi's best response is to enact tariffs (since "strong" is better than "OK").
Notice that the best response for both leaders is to enact tariffs, regardless of what the other leader does! Enacting tariffs is a dominant strategy, and both leaders enacting tariffs is both the only Nash equilibrium and a dominant strategy equilibrium. The single equilibrium is also unambiguously worse than one of the other outcomes - this is an example of the prisoners' dilemma. Notice that it is not a chicken game.

How could this become a chicken game? If costing the economy was worse than appearing weak, then that would change things around. In that case, the best response to the other leader enacting tariffs would be to hold off. There would be two Nash equilibriums - where one leader enacts tariffs and the other holds off. However, both would prefer to be the leader enacting the tariffs rather than the one holding off.

So, whether this game is a chicken game or a prisoners' dilemma depends on how you think each leader feels about appearing weak. It seems to me that both want to avoid that at all costs. In my mind, this is a prisoners' dilemma, not a chicken game.

The repeated prisoners' dilemma can be solved for the optimal outcome (both holding off), but this requires cooperation between the two leaders. In order for this cooperation to arise, each leader must trust the other (because enacting tariffs is still a dominant strategy). If we want global trade to survive this showdown, somehow we need these leaders to develop a trusting relationship. It's a pity that Trump will not be at the APEC leaders meeting - it seems like a group hug is in order!

*****

[*] This might sound surprising. For the country imposing tariffs, tariffs lead to a deadweight loss (lost wellbeing). They make domestic sellers better off, but make domestic consumers worse off by more than the gain to domestic sellers. In contrast, the market in the country that holds off has no deadweight loss. The exporting firms in that country will be able to export a bit less, but that probably doesn't have as big of a negative impact as the tariffs do on the country that imposed them.