Xi Jinping vs Donald Trump: Game Theoretic Take on the Trade War

Introduction & Overview

The trade dispute between the United States and China is an issue of prominence in contemporary affairs. The established power, the United States, seeks to influence the behavior of the rising power, China, and conform its actions to fit within the generally accepted framework of international economics. Specifically, the United States wants China to (1) stop stealing intellectual property, (2) stop subsidizing its domestic and state owned industries, and (3) play ‘fairly’ on trade in general, especially in regards to its exchange rate policies. At the helm of the two countries are Donald Trump and Xi Jinping, respectively.

Since Deng Xiaoping brought capitalism to China, via his historic period of opening up and reform, China has been integrating with the global, rules based international economic system that the US spearheaded after WW2. This continued under Deng’s successor, Jiang Zemin, with China’s eventual accession into the World Trade Organization. China has become the world’s manufacturer and is now a critical piece in the supply chain of the global economy. However, many observers (many of whom are objective) have noted that China, despite its WTO entrance, does not quite follow the rules of the road.11‘2018 Report to Congress On China’s WTO Compliance.’ United States Trade Representative. February 2019. https://ustr.gov/sites/default/files/2018-USTR-Report-to-Congress-on-China%27s-WTO-Compliance.pdf Forced technology transfers, intellectual property theft, exchange rate manipulation, and massive subsidies to domestic industries are just some of the more blatant tools China has used. It is in this context that upon his election in 2016, Donald Trump vociferously criticized China (going back to the 1990s, however, Trump’s stance against China is one of only a few stances he has held consistently).

Donald Trump wants to nudge China on the trade issue. Prima facie, Trump views the massive trade imbalance (the US imports $400 billion more than it exports) between the two countries as evidence of unfair play.22“Trade in Goods with China.” US Census Bureau. https://www.census.gov/foreign-trade/balance/c5700.html While few economists agree with him on this point, many economists and objective observers do agree with Trump’s general stance that China has played unfairly. Trump is now exerting pressure on Xi to make decisive changes to the way China handles its trading relations, with massive implications for the global economy at stake.

Donald Trump and Xi Jinping are therefore currently engaged in a game. These two players represent not only themselves, but also their countries, and their constituencies. The game in this paper simplifies the situation by assuming that, ultimately, it is Trump and Xi who are the only players and the ones responsible for making the decisions for their countries. The way in which this paper models the game is as a signal game. A signal game is a sequential game wherein the first player’s actions give information regarding who they are to the second player. In this game, then, player one is Xi Jinping, and he may be one of two types: either the type who is willing to reform Chinese trade practices, or the type that is not willing to reform Chinese trade practices.

The reason I chose to differentiate Xi’s type based on whether he is truly willing to reform or not willing to reform China’s economic and trade practices is rooted in studies of Chinese contemporary politics. In China, a major area of focus is on the implementation of policy and reforms. Often times Beijing (and the Politburo / Central Committee of the Chinese Community Party) will agree to changes in principle, but does not actually have the political will or capacity to do the heavy lifting to implement changes. Indeed, contrary to popular perception, the Chinese policy makers at the center cannot exert pure top-down control. As an old aphorism goes, the center has their policies, and the peripheries (i.e. local governments and local branches of bureaucratic institutions) have their countermeasures. Implementing changes is often fraught with difficulty in China, as Michael Oksenberg and Kenneth Lieberthal33Lampton, David M. & Kenneth G. Lieberthal. “Bureacracy, Politics, and Decisoin Making in Post-Mao China” University of California Press. 1992. https://publishing.cdlib.org/ucpressebooks/view?docId=ft0k40035t;chunk.id=0;doc.view=print explained with their term ‘Fragmented Authoritarianism’ and which Andrew Mertha updated with ‘Fragmented Authoritarianism 2.0.’44Mertha, Andrew. “Fragmented Authoritarianism 2.0”: Political Pluralization in the Chinese Policy Process. China Quarterly. 2009. Due to the lack of formal institutionalization in China, there are many conflicting lines of authority. Fractured authoritarianism relates to the idea that there are multiple different arenas wherein decisions actually get implemented and thus policies rarely end up being implemented as desired. For the center (i.e. the Politburo) to effectively be able to implement a policy it must be extremely dedicated and have immense political will. For, as Mertha explains, “policy made at the center becomes increasingly malleable to the parochial organizational and political goals of various vertical agencies and spatial regions charged with enforcing that policy.” This has likely been a major reason why reform on intellectual property protection and other issues important to the US are still ongoing problems, despite ostensible commitment from China’s central policy makers. Thus Xi Jinping is either one of two types: someone who is willing to actually commit to implementing reforms or someone who is not willing to implement reform.

Signaling Game Model of Trade War: Xi vs Trump

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Modelling Assumptions

Several critical assumptions are made to arrive at this game. First, as already mentioned the massively simplifying assumption that Xi and Trump are the only two relevant actors here. Second, the probabilities that Xi is either the type that is willing or unwilling to engage in reform is assumed based upon information I will delve into. Third, the payoffs for this game are assumed, and I will explain them. Finally, the U’s are the assumed beliefs that Donald Trump has about how likely he is to be at that juncture of the game.

Assumption #1: Xi and Trump as actors

Using Xi and Trump as the two actors simplifies reality to an extreme degree. In reality, these deals are being negotiated by teams from both sides that are made up of diverse members. Furthermore, these teams are not simply acting unilaterally, nor are they simply acting on behalf of Trump and Xi respectively. The people negotiating the trade deal between the US and China must be informed the myriad of competing interests within the two countries. In China, for example, the negotiators must be responsive to the interests of provincial level governments, State Owned Enterprises, members of the Central Committee (200+ plus people), the Politburo Standing Committee (7-9 people), and of course Xi Jinping himself. Meanwhile, US negotiators must be responsive to business interests, congressional Republicans, and of course Trump. Thus in reality the game is more complex, with a multitude of actors impinging upon decision making.

Assumption #2 Probability of Xi’s Type

Nature makes a decisive first move in this game in determining Xi’s type. I have assigned a .7 probability that Xi is the type that is unwilling to reform Chinese economic and trade practices and a .3 probability to him being the type that is willing to reform. The reason for the weighting favoring him not being willing to reform is the evidence that has so far been presented of his actions, and from an analysis of the motivations of different actors in China. First and foremost, Xi does not want to seem weak by being seen as someone who is willing to reform at the behest of Trump’s demands. Indeed, the Trump trade deal is often compared to the 1846 Nanking treaty during the opium wars with Britain, insinuating that agreeing to Trump’s trade deal would be a humiliating blow for China, unjustly allowing the US to mettle in its fairs. But, even more importantly, many Chinese actors benefit substantially from the unfair practices. Most poignantly, Chinese SOEs and exporters who receive direct subsidies. Additionally, these Chinese economic actors also have a much easier time selling to the international market if the rmb (yuan, Chinese currency) is undervalued relative to other currencies. Meanwhile, Xi has made it clear in recent years that he views strong state champion State Owned Enteprises (SOEs) as critical for Chinese development. Toward this end he has preferentially funneled credit to them (over and above private enterprises), while the state subsidizes their activities internationally and domestically to a substantial degree. Thus Xi views China as necessarily benefitting from these non-market based practices, and is likely not inclined to reform these sorts of distortionary activities. Indeed, many established interests and members of his inner circle implore Xi not to deal with Trump, even castigating the ongoing discussions.

Assumption #3 Payoffs

Perhaps the most important assumptions in framing a game are the payoffs. In this case each player has eight possible payoffs. Payoffs depend upon whether Xi turns out to be the type of person who is willing to enact changes to Chinese trade and business practices or whether he is the type that is not willing to do so. In both cases though, player 2’s (Trump’s) options are still the same: either punish (with tariffs and the like) or acquiesce and do nothing.

_Willing to Reform_

In the willing to reform case, Xi faces payoffs [-4,-1;-2,1].

Xi has a payoff of -4 if he agrees to the trade deal but Trump punishes him with tariffs anyway. The reason for this is that Xi has displeased his constituencies in China by giving into Trump’s demands (thus appearing weak), but then ends up facing punitive injuries from Trump anyway. This is a case of injury being added to insult. Xi has a payoff of 0 if he agrees to the trade deal and Trump acquiesces. The reason Xi receives 0 is that his domestic base is unhappy with him and SOEs and domestic Chinese producers face some short term harm as a result, however this is offset by the beneficial resumption of trade relations with the US as well as the Chinese economy conforming to a more successful long run development path, as Xi in this scenario is the type that will actually implement the reform. Xi has a payoff of -2 if he does not agree to the deal and Trump enacts punitive measures. This payoff occurs due to heavy reliance of the Chinese economy on exports, and the sizable amount that China exports to the US in particular. Though Xi does not face domestic backlash, and can indeed likely use this as nationalistic fodder for his base, the tariffs do cause real harm to the economy and thus threaten China’s economic stability and growth. Finally Xi, faces a payoff of 1 if he does not agree to the deal and Trump acquiesces. This would be Xi’s best case scenario as this type, as he is viewed as standing his ground against Trump, while not facing any punitive measures.

In the willing to reform case, Trump faces payoffs [-2, 3; 1,-1]

If Trump were punish Xi in this case when he agreed to the trade deal Trump would face a payoff of -2. This would simply be nonsensical for Trump, as he would be damaging his own credibility amongst his own supporters and internationally, while also inflicting real harm on the US economy and the farmers and rural folks who tend to support him. Trump faces a payoff of 3 if he acquiesces to Xi agreeing to the trade deal in this case. Trump comes off looking like someone who wrote ‘The Art of the Deal’ and Xi is actually willing to reform. Trump thus appears a savvy negotiator and is able to avert a punishment that would hurt both the US and China. Trump gets a payoff of 1 if he punishes Xi for not agreeing to the trade deal. The reason for this is that Trump earns credibility from his base for following through on his threat. Trump gets a payoff of -1 if he acquiesces to Xi not agreeing to the deal. Trump loses face amongst his constituents and may come across as weak, but this is offset to some degree by the fact that Xi is still willing to reform in this instance, and thus by acquiescing he may be salvaging beneficial relations down the line. Similarly, US consumers (who are also his supporters) will not face the downsides of higher prices from tariffs.

_Unwilling to Reform_

In the not willing to reform case Xi faces payoffs of [-3, -2; -1, 2]

Xi gets -3if he agrees to the trade deal but Trump punishes him, as this encapsulates insult to his reputation (for compromising), which is made worse by the fact that this reputational damage is unnecessary due to the fact that he never intended to pursue implementation of reforms. While, at the same time, the Chinese economy will be hurt by punitive tariffs. Xi gets -2 if he agrees to the trade deal and Trump acquiesces. Xi will not actually implement the reforms, so the economy will not benefit in the long run. Meanwhile his key constituents may lose faith in him and chastise him for having seemed to be soft against Trump and agreeing to the deal in principle. Xi gets -1 if Trump punishes him for not agreeing to the trade deal, due to the punitive economic costs on the Chinese economy, although he saves face within his country by snubbing Trump’s deal. Xi gets 2 if he does not agree to the trade deal and Trump acquiesces. Xi appears as the winner in this contest to the people in China.

In the not willing to reform case Trump faces payoffs of [1,-1; 2,-4]

This case is perhaps the most interesting for Trump’s payoffs, as Xi will not follow through on implement reforms even if he agrees to the trade deal. Thus if Xi agrees to the deal and Trump punishes he receives a payoff of 1. Trump hurts his credibility by punishing despite Xi agreeing to his terms, but actually acts correctly by being punitive as he understands Xi won’t reform. Trump gets a payoff of -1 if Xi agrees to the deal and Trump acquiesces. No resolution comes of this, and Trump comes off looking as though he got played, as Xi will not follow through on implementing the trade deal agreements, but the downside is offset slightly due to Trump’s ability to signal / pretend that he got what he wanted. Trump gets 2 for punishing Xi for not agreeing to the deal, as he is able to grandstand on the back of Xi’s obstinence, and there was no later possibility of salvaging relations, as Xi is the type that will not implement reforms. Finally, Trump gets -4if he acquiesces when Xi does not agree to the deal, as this is the worst of all possible situations for him, given the fact that he was purporting to be tough on China, but will come looking incredibly weak if he lets Xi walk away.

_Model Implications: Potential Perfect Bayesian Equilibria_

In arriving at a Perfect Bayesian Equilibrium (PBE) in a signaling game, the strategies pursued must be rational in light of the beliefs, while the strategies themselves must sequentially rational and the beliefs must be consistent. Thus payoffs are a function of player 2’s beliefs about how likely we are to arrive at particular junction in the game. I have chosen to model two potential scenarios, one wherein a separating equilibrium occurs and one wherein a pooling equilibrium occurs. A separating equilibrium occurs wherein player 1 pursues different strategies depending upon what type of player he is, and thus player 2 can update his beliefs depending upon what action he sees chosen. Meanwhile, a pooling equilibrium means that both types of player 1 pursue the same strategy and thus player 2 cannot update his beliefs based upon this action and must instead use the probabilities as they originally existed

Scenario 1: Separating Equilibria

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In this scenario we have a separating equilibrium, as Trump believes that if Xi does not agree to the deal then he is dealing with a Xi Jinping who is not willing to implement the trade deal’s reforms. On the other hand, if Xi does agree to the deal then Trump believes he is dealing with a Xi Jinping that is willing to reform. In order for this to be an equilibrium it must satisfy two conditions: it must be sequentially rational and the beliefs must be consistent.

As seen from the above payoffs, given Trump’s beliefs, Xi has no incentive to deviate from the strategies above, for if he did he would certainly be facing lower payoffs (-2 instead of -1 in the ‘willing to reform’ case and -2 instead of -1 in the ‘unwilling to reform’ case).

Meanwhile, using Bayes’ rule, we can check to see if Trump’s beliefs are consistent:

Bayes’ Rule:

P(unwilling to reform |not agree) = (P (not agree | unwilling to reform) * P(unwilling) ) / ( (P not agree | unwilling to reform) * P(unwilling) + P(not agree | willing to reform) * P(willing to reform) )

P(unwilling to reform | not agree) = (1)(.7) / ( (1)(.7) + (0)*(.3) = .7/.7 = 1

We can thus see that Trump’s beliefs in this scenario, using Bayes’ rule, are consistent. This means, then, that the Perfect Bayesian equilibrium (PBE) in the separating scenario is for Xi to play the strategy of ‘agreeing to the trade deal’ if he is the ‘willing to reform’ type, and for Trump to acquiesce. Meanwhile, Xi will play the ‘not agree to trade deal’ strategy if he is the ‘unwilling to reform’ type, and Trump will punish. Theoretically, then, if we were off these equilibrium strategies, Trump would respond in the same manner to both types of Xi (as depicted above).

Scenario 2: Pooling Equilibria

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In this scenario we have a pooling equilibrium, as both types of Xi Jinping follow the ‘do not agree’ to a trade deal strategy. In this equilibrium Trump cannot update his beliefs due to the fact that both types follow the same strategy. Thus Trump’s beliefs about Xi’s type for the equilibrium strategy mirror his original belief. Trump believes that Xi will be the unwilling to reform type playing the do not agree to trade deal strategy with .7 probability, while he believes he will be the willing to reform type playing the same strategy with .3 probability. In order for this equilibrium to be valid, however, it must satisfy the two conditions of sequential rationality and consistency.

Sequential rationality is followed when there is no incentive to deviate. In this way then, Trump’s beliefs in the off-equilibrium path must be as depicted above, i.e. Trump believes the probability Xi is the ‘willing to reform’ type playing ‘agree to trade deal’ is 28.5% or less, whereas the probability that he is the ‘unwilling to reform’ type playing ‘agree to trade deal’ is 71.5% or more (payoffs: Acquiesce: (3)(.285) – 1(.715) = .14 Punish: (-2)(.285) + 1(.715) = .145; if the probability is shifted even a half percent more in favor of acquiesce the PBE does not stand up to sequential rationality, as Xi has incentive to deviate due to Trump favoring acquiesce). This ensures a payoff structure wherein Trump will definitely choose to punish, which in turn means Xi has no incentive to deviate from equilibrium as he would be choosing a lower payoff.

The Perfect Bayesian Equilibrium (PBE) of this game, as displayed, is for both types of Xi to play the same strategy of ‘not agreeing to the trade deal’ while Trump will play ‘punish’ in both instances. And, for the off equilibrium path, given Trump’s beliefs, he will also play punish, thereby choosing to punish in every instance.

Thus, as we can see from the above examples, there can be multiple different equilibria in this game, dependent upon Trump’s beliefs. I have utilized and demonstrated what I believe to be two plausible extrapolations of Trump’s beliefs.

_Model vs Reality_

Although trade negotiations between Trump and Xi are still ongoing in the real world, we have seen a scenario playout that I believe is indicative of the separating equilibrium depicted above. In the last week, Xi has returned the trade deal in a manner that was denuded of any substance, which is effectively negating the trade deal.55Bradsher, Keith & Ana Swanson. “US-China Trade Talks Stumble on Beijing’s Spending at Home.” New York Times. May 12, 2019. https://www.nytimes.com/2019/05/12/business/china-trump-trade-subsidies.html To me, then, this is congruent with following the ‘no trade deal’ strategy. Given the interests within China pushing for this outcome, it is predictable that this would occur (I gave it .7 probability). Thus Trump sees Xi as the type who is not willing to reform Chinese economic and trade practices and has followed the path of punishment. Indeed, Trump increased tariffs on $200 billion worth of Chinese goods from 10% to 25% and is threatening to add tariffs to an additional $200 billion worth of Chinese goods. Although we cannot test the equilibria completely in the separating game scenario, due to the fact that China has not accepted the trade deal, I believe it is safe to assume Trump would believe Xi to be a sincere reformer, and would thus act according to the equilibrium scenario and acquiesce.

In the broad context of the China-US trade dispute, my model illuminates the reality of pressures facing both Trump and Xi. For Trump, he needs to appear as if he is following through on his campaign promises and being a President that is tough on trade, globalization, and on China for playing ‘unfairly.’ That’s why his payoffs are highest when he punishes China, with the sole exception being when he legitimately gets what he wants out of Xi (i.e. the willing to reform Xi playing the ‘agree to deal’ strategy, with Trump then playing ‘acquiesce’). Meanwhile, Xi’s payoffs are almost all negative as he is in a tough situation: either back down and accept a Trump trade deal and face domestic backlash, or stand his ground and face punitive tariffs. Xi’s best outcomes are thus off the equilibrium path (as I have modeled it), due to the fact they are only likely to occur if Trump ignores his own best interest and acquiesces in the face of Xi not agreeing to the trade deal, as this would allow Xi to both safe face domestically and avoid punitive tariffs. In the real world, then, my model predicts that Xi is most likely to not agree to the deal and that Trump will punish him (as seems to be currently happening). However, if on the off chance Xi actually agrees to the trade deal, Trump is likely to interpret that as a sign that he is the willing to reform type and acquiesce.

_Conclusion_

This paper has developed a signaling game model of the ongoing trade dispute between Xi Jinping and Donald Trump. The paper has discussed the underlying methodology of the model, and two possible Perfect Bayesian Equilibria of the game, one that results in separating equilibria and another that results in pooling equilibria. Ultimately, it seems to this author that the scenario most similar to reality is the separating equilibria form of the game, wherein if Xi Jinping is ‘willing to reform’ he will follow the ‘agree to deal’ strategy, and if Xi Jinping is ‘unwilling to reform’ he will follow the ‘not agree to deal’ strategy. Trump believes with probability equal to one that Xi is willing to reform if he employs the former strategy, and (also with probability equal to one) that Xi is not willing to reform if he employs the latter strategy; acquiescing and punishing accordingly.