Why tit-for-tat tariffs may not work against Trump

Last week, my ECONS101 class covered game theory. At the end of the final lecture, after we had been covering repeated games and tit-for-tat strategies, a really perceptive student asked me about Trump's tariffs. A lot of the rhetoric about tariffs has been posed in terms of tit-for-tat (see here and here, for example). The student's question got me thinking though, about why a tit-for-tat strategy may not work in this case.

Before we get that far though, we need to think about the tariff game, as outlined in the payoff table below. There are two players: USA and 'Other Country'. Each player has two strategies: high tariffs, or low tariffs (which includes no tariffs). The payoffs are expressed as "+" for good outcomes (and "++" is particularly good), and "--" for bad outcomes (and "--" is particularly bad), while zero is a neutral payoff. 

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 other country chooses high tariffs, USA's best response is to choose high tariffs (since "0" is better than "--" as a payoff) [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 other country chooses low tariffs, USA's best response is to choose high tariffs (since "++" is better than "+" as a payoff);
3. If USA chooses high tariffs, the other country's best response is to choose high tariffs (since "0" is better than "--" as a payoff); and
4. If USA chooses low tariffs, the other country's best response is to choose high tariffs (since "++" is better than "+" as a payoff).

Notice that USA chooses high tariffs no matter what the other country does, high tariffs is a dominant strategy for the USA. Similarly, since the other country chooses high tariffs no matter what the USA does, high tariffs is a dominant strategy for the other country. Both countries will choose to play their dominant strategy (because it is always better than the other strategy, not matter what the other country chooses to do). The outcome where both countries choose high tariffs is the Nash equilibrium in this game (it is also a dominant strategy equilibrium, because both countries have a dominant strategy).

However, I'm sure that you can clearly see that both countries would be better off with low tariffs (since the payoff for each country would be "+", instead of "0"). This game is an example of the prisoner's dilemma (it's a dilemma because, when both countries act in their own best interests, both are made worse off).

However, it is important to remember that this game is a repeated game. It is played more than once, with the same players, and the same strategy choices. When a game is repeated, then the outcome may differ from the equilibrium of the non-repeated game, because the players can learn to work together to obtain the best outcome.

In a repeated prisoners' dilemma game like this, each player can encourage the other to cooperate by using the tit-for-tat strategy. That strategy, identified by Robert Axelrod in the 1980s, works by initially cooperating (low tariffs), and then in each play of game after the first, the player does whatever the other player did last time. So, if the USA chooses high tariffs, then the other country should punish them and choose high tariffs in the next play of the game. And if the USA chooses low tariffs, then the other country should reward them and choose low tariffs in the next play of the game. The tit-for-tat strategy works because it encourages the other player to cooperate. And that is what many people have been expecting of Trump. If you punish the USA for high tariffs by setting your own high tariffs, eventually they will realise their error and start cooperating with low tariffs again.

But there is a problem. The whole edifice of the tit-for-tat strategy assumes that Trump knows that he is playing the game outlined above, where there is an unambiguously better outcome for both countries, if they both choose low tariffs. By his past statements, it is absolutely clear that Trump thinks that trade is a zero-sum game (for example, see here or here).

So, what does the game look like if you believe that trade is zero-sum? Instead of there being gains from the low-tariff/low-tariff outcome, the payoffs become zero, as shown in the payoff table below.

Solving this game for the Nash equilibrium, the best responses are:

1. If the other country chooses high tariffs, USA's best response is to choose high tariffs (since "0" is better than "--" as a payoff) [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 other country chooses low tariffs, USA's best response is to choose high tariffs (since "++" is better than "0" as a payoff);
3. If USA chooses high tariffs, the other country's best response is to choose high tariffs (since "0" is better than "--" as a payoff); and
4. If USA chooses low tariffs, the other country's best response is to choose high tariffs (since "++" is better than "0" as a payoff).

Notice that the game itself doesn't change. Imposing high tariffs is still a dominant strategy for both countries, and the outcome where both countries choose high tariffs is the only Nash equilibrium (and is also a dominant strategy equilibrium).

However, there is an important difference in this game when it is played as a repeated game. There is no incentive for players to cooperate. That's because cooperating results in a payoff of "0", just the same as not cooperating. And because of this, the tit-for-tat strategy would be pointless.

Which seems to be what other countries are finding, when dealing with Trump. Other countries may think they are playing the first game, where a tit-for-tat strategy may get Trump to reconsider. But if Trump thinks they are playing the second (zero-sum) game (and it seems that he does), then the tit-for-tat strategy is simply not going to work.

[HT: Sarah from my ECONS101 class]