Abstract

This essay is intended to provide some understanding of the China-United States trade war using game theory. It concludes that we should place more emphasis on cooperation-inducing institutions such as the World Trade Organization (WTO). No prior knowledge of game theory is required to understand this work; a GCSE Mathematics course should suffice. The following aspects of game theory are included in this essay: Nash equilibrium, Prisoner’s Dilemma, extensive form games, and backward induction.

Keywords:  Game theory, trade war, Nash equilibrium, extensive form games, backward induction.

Introduction

International cooperation is often called for by politicians and commentators alike. If countries stand to benefit from cooperating, should it not be rational for them to cooperate? In light of the China-United States trade war, such a claim has once again been put forward.

But what if this were not the case? What if there were a distinction between a socially desirable outcome and that resulting from the rationality of individuals? What if countries were simply confused? 

This essay analyses two distinct scenarios of the China-US trade war, with the aim of demonstrating that cooperation-inducing institutions such as the World Trade Organization (WTO) are paramount to solving both such scenarios. The body of this work is split into three sections; the fundamentals of game theory are first introduced, followed by a construction and analysis of two scenarios of the trade war.

1. Game Theory: The Fundamentals

Game theory is not about ‘playing games,’ It is about conflict resolution between rational but distrustful beings.” (Poundstone, 1992)

There are numerous ‘games’ in play around us, whether we actively participate in them or not. From poker to billion-dollar auctions, to the Cold War – games are truly everywhere. The purpose of game theory is to use mathematics to understand how and why we interact with each other the way we do. And it is not just humans that game theory can be applied to: it also finds its uses in evolutionary biology and computer science. 

This section will introduce three building blocks of game theory: players, payoffs, and the Nash equilibrium.

1.1. Players

Every game consists of at least two players. In game theory, there are two key assumptions we make about these players:

1.     The goal of each player is to maximise their gain. This can be thought of as each player acting in his or her self-interest.

2.     Both players know that the other player has the same goal. Thus, both know the other to be rational. (Nordstrom, 2020)

Hence, not all interactions may be modelled using game theory. Sometimes players may act irrationally (inconsistently). Sometimes players do not seek maximum gain: they may be satisfied as long as they achieve a certain absolute objective. For example, a student may be satisfied with achieving an absolute mark of eighty and not be incentivised to score any higher.

Other aspects such as instinct, habit, or laziness affect how players make decisions. 

Nonetheless, game theory can be incredibly insightful, as it is here. Using mathematics, we can logically explain why seemingly counter-intuitive actions, such as a trade war, are actually rational.

To circle back on our idea of players, each player must decide on an act based on their expected gains. These gains are expressed using some real number (e.g. +5, -3,) which is known as a payoff value. When an act corresponds to some payoff, the payoff is said to be the utility of the act. Note that for the same act, each player may assign a different utility to it. Hence, a player can be said to maximise their payoff by choosing the act with the highest utility.

1.2. Payoff Tables

When two players interact, their actions form a possible outcome. To express this outcome, we form a payoff table. As a simple illustration, the game Chicken is introduced here. In Chicken,

Alice and Bob are racing towards each other. They each would like to avoid a crash, and win. Whoever chickens out first, loses. Suppose Bob and Alice are both independently capable of choosing one of two actions: Speed or Slow. Alice’s options correspond to the rows of the payoff table, and Bob’s options correspond to the columns of the payoff table. The table below illustrates this (Figure 1a).  

Figure 1a: a payoff table[1]

If Alice and Bob both choose Speed, then they crash. Hence (Speed, Speed), which denotes (Alice’s choice, Bob’s choice), results in the lowest payoff for both individuals. Let us arbitrarily set the payoff vector of (Alice’s payoff, Bob’s payoff) for (Speed, Speed) as (−1, −1). If Alice chooses Speed and Bob chooses slows, then Alice wins, and Bob loses. They do not crash, but Bob chickens out first. Hence, both receive a payoff higher than (Speed, Speed); but since Alice wins, she receives a higher payoff than Bob. We set the payoff vector for (Speed, Slow) as (3,0). The converse applies if Bob chooses Speed and Alice chooses Slow. The payoff vector for (Slow, Speed) is (0,3). If Alice and Bob both choose Slow, then they do not crash, but no one wins. The payoff for playing Slow here is higher than when it was played in (Speed, Slow). Hence the payoff vector for (Slow, Slow) is (2,2). Figure 1b illustrates these values in the payoff table. For ease of interpretation, Alice’s payoffs are in the bottom-left corner of each box, and Bob’s payoffs are in the top-right corner of each box.  

Figure 1b: Chicken Payoff Table

So, what is the optimal response for Bob and Alice? The answer is the Nash equilibrium.

1.3. The Nash Equilibrium

The Nash equilibrium is perhaps the most important component to the whole of game theory. Discovered by John Nash, laureate of the Nobel Economics Prize and the Abel Prize, the Nash equilibrium is the basic building block of finding solutions to a game. The solution is the best reply a player can make given the decision of the other. For example, if Bob chooses Speed, since 0 > −1 and Alice wants to maximise her payoff, Alice chooses Slow. This outcome (0,3),

i.e., (Slow, Speed), is a Nash equilibrium, as when either player is given the other’s strategy,

i.e., Slow or Speed, neither wants to switch their strategy. For ease of visualisation, we circle her best reply and payoff given Bob’s strategy Speed (Figure 2a).

When both payoffs are circled in a box, we know that said outcome is a Nash equilibrium. Hence, from Figure 2b, we see two Nash equilibria in Chicken: (Slow, Speed), and (Speed, Slow). Here, “[t]he players have a joint interest in avoiding mutual disaster” (Binmore, 2007).

Figure 2: Chicken

There is actually a third Nash equilibrium. The previous two are known as pure strategy equilibria: players play one strategy only. The third equilibrium is a mixed equilibrium. This is when a player plays a mixture of strategies so the other is indifferent towards either of their strategy choices. The player is randomising. Note that this is also a perfectly consistent type of play. The opponent is now indifferent to any strategy he plays because he will receive the same payoff on average. A good way of understanding this is how we intuitively randomise our strategies in Rock-Paper-Scissors; we do not just play one move all the time: we keep our opponents guessing. Randomising can also be a Nash equilibrium! Here, perhaps Alice is unhappy that the Nash equilibrium (0,3) is being played. If Alice plays her mixed strategy, Bob is then indifferent towards any choice he plays, as his expected payoff from either choice is the same. The mixed strategy for Alice here is to play Speed half of the time and Slow the other half of the time, randomly (see Appendix 1). The game Chicken was chosen here because it demonstrates both pure and mixed equilibria.

To recap, we know that there are three Nash equilibria here: two pure and one mixed. This concludes the introduction to game theory fundamentals. In the next section, we shall construct and analyse our first model.

2. Scenario One: The Prisoner’s Dilemma

We may consider the trade war as a game between two players: The United States and China.

They may choose one of two actions: negotiate for free trade (N), or raise tariffs (R).

In our first scenario, we adopt the classical economic belief that, all else remaining the same, reducing imports and increasing exports results in higher GDP (Gross Domestic Product). This originates from the formula 𝐺𝐷𝑃 = 𝐶 + 𝐼 + 𝐺 + (𝑋 − 𝑀) , where 𝐶 denotes consumption, 𝐼 denotes gross investment, 𝐺 denotes government spending, and (𝑋 − 𝑀) denotes exports 𝑋 minus imports  𝑀. From this, it is therefore theoretically rational for a country to raise tariffs to reduce imports. When both countries adopt protectionist policies, both countries suffer. For example, the Smoot-Hawley tariffs of the 1930s played a major role in the Great Depression (Alfred, 1995). The same may be said for this trade war. Due to the recent trade war, American companies lost 1.7 trillion USD in value (Amiti, Kong, & Weinstein, 2020), and China suffered a 35 USD billion dollar loss in exports to the US market in 2019 (United Nations, 2019). Given the circumstances above, the payoff table has been constructed to reflect these relationships, as shown in Figure 3. Note that the numbers themselves are not particularly important, it is the relationship (difference) between the numbers that should be noted as expressing the conditions above.

Figure 3: An iteration of the Prisoner’s Dilemma

There is only one Nash equilibrium here (R, R): there is only one outcome where both payoffs are circled, and the mixed strategy equilibrium is invalid (for the country playing a mixed strategy, the probability of it playing both strategies must be positive; Appendix 1 provides the fundamentals of calculating a mixed equilibrium). It is never rational to play N here, as, if one plays N, the other will choose R to maximise their payoff. This model is an iteration of the famous Prisoner’s Dilemma, where although it may be beneficial for both players to cooperate, the result is that both players harm each other. Society tries to avoid such outcomes by cooperating and agreeing beforehand how they will play. For countries, this means institutions such as the World Trade Organization (WTO). This simple model is one possible explanation of the trade war. However, one of its assumptions is that the classical belief is actually correct.

What happens if the US benefits more from trade than tariffs? 

3. Model Two: Confusion

When countries change their stance on trade policy, their trading partners are not necessarily informed of these changes (Mann, 1987). This may result in two countries playing two different games, whilst each presumes the other to be playing the same one. This change may have been ongoing over many years, culminating in a sudden change of policy when there is a change in administration. Furthermore, our payoffs should be a function of both economic benefit and political benefit, particularly for the US.

We now refine our model with two payoff tables: one before, and one after 2017 (Figure 4). We make two changes to the US’s payoff in the pre-2017 table compared to our first model: the US’s payoff for (N, N) versus (R, N), and the US’s payoff for (N, R) versus (R, R). In both circumstances of the pre-2017 table, the US plays N whatever China plays. These changes have been made because we account for the moral benefits of the US administration upholding free trade (Woerth, 1995). Historically, political benefit has generally outweighed economic concerns in when the US conducts trade policy (Mann, 1987). These changes have not been applied to China because of its historically consistent focus on growing economic wealth above all else, as seen from its Five-Year Plans.

We now compare the pre-2017 table to the post-2017 table. Just the US’s payoff from (N, R) has changed compared to pre-2017. It symbolises the growing economic burden on the US to maintain free trade and the shift towards protectionism in American politics. Economically, contrast the United States’ trade deficit with China the year before Trump’s election and the year before Obama’s election, we see an increase of 34.1% (United States Census Bureau, n.d.). Politically, the moral benefits of a free trade policy plausibly decreased between these two administrations: 65% of Americans viewed trade restrictions as desirable in 2016 compared to 40% in 2007 (PollingReport.com, n.d.). Crucially, the majority of Americans now viewed trade restrictions as desirable.

We may see that (N, R) is the pure Nash equilibrium before 2017. The administration sought to uphold free trade as a moral obligation, and the deficit upon election had yet to become large enough. But the solution to the post-2017 game is not actually (R, R). Why? The answer lies in introducing the element of time.

3.1. Extensive form games.

Order matters in many games we play. In games such as chess or poker, we play our move based on what our opponent(s) previously played, who played their move(s) based on what we had played, and so on. In our first model, the moves were assumed to be made simultaneously. This is unrealistic in real life. Mapping all our payoffs in a payoff table is known as the strategic form of the game. When we incorporate time into our game, the game is now known as the extensive form of the game. Supposing that China moves first[2], we denote our extensive form game using the decision tree shown below (Figure 5).

There are three layers to this diagram. China occupies the first node, which constitutes the first layer. The two United States nodes constitute the second layer, and the payoffs constitute the third layer. China moves first, then the United States. What moves they each make determine the payoff node. 

How do we find the solution here?

3.2. Backward Induction

The answer is a method known as backward induction. In backward induction, we start from the second-last nodes in our diagram (United States) and choose the action which maximises her payoff. When China plays N, the US (node on the left) will also play N to maximise her payoff. Similarly, when China plays R, the US (node on the right) will play R to maximise her payoff. These are indicated with the bold lines above the United States node in the diagram above. We now see that the US will always play N when China plays N, and R when China plays R.

We now conduct the same procedure for China. China now actually has two payoffs to consider: the payoff from (N, N) and the payoff from (R, R). China will now opt for N as her payoff, for her payoff from R is certainly lower than N (3 > 2). This is indicated with the bold line connecting China to the US in the diagram above.

Thus, by identifying the bold line which runs through the decision tree, we see the solution for our post-2017 game is (N, N), leading to the payoff node (5,3). 

But since China is necessarily confused in the change in US policy, it will choose to play according to the payoff table of pre-2017: by playing R. The US is therefore forced to respond with R as well. Thus, we see that this confusion leads to an undesirable trade war. Only until the US has played a sufficient number of times of R would it successfully signal to China that the game has changed. Here we see why we supposed China to move first. The purpose of the assumption is to enable us to examine how the change in US administration caused China to play in a “confused” manner.

We may combine these two games into one as seen in Figure 6, where P denotes the probability of China believing the US to be playing the pre-2017 game, and (1 − 𝑃)  denotes the probability of China believing the US to be playing the post-2017 game. As the US plays R more often, P decreases and (1 − 𝑃) increases. Thus, we see that the China-US trade war depends on both the magnitude of payoff change for the US in maintaining free trade, and the level of conviction China has on whether the payoffs for the US have indeed changed.  Here again, if trade negotiations were more accessible, the signalling of policy change would have been far more effective.

Conclusion

In this essay, we explored the outcomes of two simple models of the China-US trade war. The first took on classical economic beliefs and demonstrated the irrationality of cooperation under such beliefs. The second provided a more realistic model of confusion and payoffs from free trade, and used backward induction to show why even if cooperation is desirable, it does not necessarily happen. Both of these models showed the need for cooperation-inducing institutions such as the WTO mediating between countries.

Appendix

References

Alfred, E. E. (1995). Opening America's Market: U.S. Foreign Trade Policy Since 1776.

University of North Carolina Press.

Amiti, M., Kong, S. H., & Weinstein, D. (2020, May). The Effect of the U.S.-China Trade

War on U.S. Investment. The National Bureau of Economic Research.

Binmore, K. (2007). Game Theory: A Very Short Introduction. New York: Oxford University

Press.

Mann, C. L. (1987). Protection and Retaliation: Changing the 'Rules of the Game'. Brookings

Papers on Economic Activity, 311-335.

Nordstrom, J. (2020). Introduction to Game Theory: A Discovery Approach. 

PollingReport.com. (n.d.). International Trade/ Global Economy. Retrieved 07 03, 2020, from PollingReport.com: https://www.pollingreport.com/trade.htm

Poundstone, W. (1992). Prisoner's Dilemma. 

United Nations. (2019, November 06). Trade war leaves both US and China worse off.

Retrieved from United Nations Conference on Trade and Development: https://unctad.org/en/pages/newsdetails.aspx?OriginalVersionID=2226

United States Census Bureau. (n.d.). Foreign Trade Balance. Retrieved 07 01, 2020, from United States Census Bureau: https://www.census.gov/foreigntrade/balance/c5700.html

Woerth, D. E. (1995). International Aviation. In P. Capelli, Airline Labor Relations in the

Global Era (pp. 41-53). New York: Cornell University Press.