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Globalization? Trade War? A Counterbalance Perspective

The embrace of globalization and protectionism among economies has ebbed and flowed over the past few decades. These fluctuations call for quantitative analytics to help countries improve their trade policies. Changing attitudes about globalization also imply that the best trade policies may vary over time and be country-specific. We argue that the imports and exports of all economies constitute a counterbalanced network where conflict and cooperation are two sides of the same coin. Quantitative competitiveness is then formulated for each country using a network counterbalance equilibrium. A country could improve its relative strength in the network by embracing globalization, protectionism, trade collaboration, or conflict. This paper presents the necessary conditions for globalization and trade wars, evaluates their side effects, derives national bargaining powers, identifies appropriate targets for conflict or collaboration, and recommends fair resolutions for trade conflicts. Data and events from the past twenty years support these conditions.


Introduction
Anti-globalization and trade wars are two levels of economic conflict -the latter directly involves only two trading partners, while the former represents one country's disintegration from the global economy. Nevertheless, both could adversely affect global economic activities. This paper formulates simple but decisive rules to mitigate either type of conflict using a game-theoretic approach. We focus on import and export activities, ignoring political, territorial, ideological, military, and cultural conflicts, though they all affect trade behaviors and volumes. From this perspective, the primary purpose of engaging in trade wars or globalization is to improve a country's competitiveness and maintain its trade balance in the trading system.
To analytically examine trade wars and globalization, we face several daunting challenges. First, there are always conflict and cooperation in international trade. While the trade of goods and services results from collaboration, a country's growing strength could mean the weakening position of its trading partners.
Thus, a pure cooperative or a pure non-cooperative study is too limited to capture the complexity of trading relationships. The second challenge is to adequately construct a high-level description of the interactive trading system because exports and imports already reflect comparative advantages of specific goods and services. Primarily, we need to quantitatively formulate the economies' goals and objectives in conflicting and cooperative situations. Next, policy variables, such as tariff levels and import quotas, should link to the economic goals. After that, countries apply optimization methods to choose appropriate policies to achieve their economic objectives. Finally, it is effortful to bring data into the analytics to explain past events and find optimal solutions for the near future. The solutions could identify the best trading partner, the optimal target in trade friction, a fair resolution to a conflict, or the optimal level of globalization.
We measure the relative strengths of the countries in the global trading system by the solution to a counterbalance equilibrium, defined by Hu and Shapley (2003). The solution, called authority distribution, integrates cooperation and competition inside the global economic system. The data-driven distribution is not a static utility function subjectively assigned by an expert. It represents the trading system more than a pure cooperative or non-cooperative solution. Moreover, it measures the competitive advantages of the countries and their long-term influences on the whole system. Additionally, the Matthew effect in the distribution is a driving force for competition and collaboration. The objective of maximizing national competitive strength is the purview for government policymakers rather than trading companies. The interests of trading companies only sometimes align with those of policymakers, who also consider the side effects of trade, such as the shrinkage of the domestic labor markets.
When starting a trade war against a trading partner, a country raises tariffs or creates other trade barriers to increase its competitiveness. However, the result de-pends on the strength of the counterpart's reaction. We find that a simple threshold -the ratio of their authority -determines whether the country should engage in a conflict or cooperation. On the other hand, the reaction should also benchmark their GDP ratio to avoid trade deficits. Using the threshold and the benchmark ratios, we derive a Nash bargaining solution for countries with different preferences between competitiveness and trade balance. In the literature on trade wars (e.g., Grossman and Helpman, 1995;Harrison and Rutström, 1991;Ossa, 2014), in general, a payoff function or welfare function is set up to find optimal solutions (such as tariff levels) from the equilibrium or maximum-valued point. One advantage of our solution is that the objective function is not explicitly expressed but implied from the counterbalance equilibrium. Also, we use fewer assumptions to accomplish this.
A country with no specific targets to confront could choose with further globalization or protectionism. There are many debates over economic globalization in the literature (e.g., Stiglitz, 2002 and2017). Different viewpoints, assumptions, and considerations could reach opposite conclusions about globalization for a country. Our data-driven approach ignores arguments from both sides of the debate about labor markets, production chains, environmental issues, and national security. Trade policymakers may have carefully weighed these considerations; thus, the trading data may already reflect them. Hence, a country could engage in either globalization or protectionism, depending on its particular situation at a specific time. By analyzing the ComTrade data of the United Nations (2021) for 2000 through 2019, we find that globalization was a better choice for economic superpowers in 2000. However, some protectionism could have been a better choice for the USA and the United Kingdom in 2017 but worse for China, Germany, and Japan. Data from 2017 also show that trade wars could be an option for the USA to maintain its competitiveness. However, adverse side effects on third parties were also tangible.
In the following exposition, we first introduce the competitiveness objective in Section 2. Then, Section 3 formulates a necessary condition for a country to launch a trade war or form a trade partnership to reach its objective. After that, a bargaining solution is derived in Section 4 to resolve conflicts or share profits.
Next, Section 5 discusses a similar condition for globalization or protectionism.
Then, in an empirical study of historical events, Section 6 examines the trading data from 2000 to 2019. Finally, we discuss several extensions and limitations of this framework in Section 7. The proofs are in the Appendix.

Competitive Advantage in the Trading System
In the multilateral trading system, unilateral action could achieve unpredictable or even opposite results. Trade war or antiglobalization is often framed in the hope of shielding domestic industries or labor markets, reducing trade deficits, or anti-dumping. These objectives, however, often result in more harm than gains (e.g., Chung, Lee, and Osang, 2016;Read, 2005). Realistically, an attainable goal thrives on mutual understanding of the strengths of the trading counterparts.
To correctly gauge the strengths of the trading partners, we describe their positioning with a square matrix. Assume there are n countries in the trading system, labeled as 1, 2, · · · , n. Let P i j be the fraction of country i's production or GDP, which exports to country j. The exports include both goods and services, and P ii is the non-exporting fraction. Clearly, all P i j ≥ 0 and n ∑ j=1 P i j = 1 for any i = 1, 2, · · · , n. We collectively write the n × n stochastic matrix P = [P i j ], 1 ≤ i, j ≤ n.
Thus, P i j measures the direct influence of country j on i's production; it is the pro rata power when country i makes its production decision. Inflows of financial assets, including foreign direct investments and remittances, are also influential in making domestic products. Nevertheless, they are primarily payments for current or future exports. In essence, trade is about exchanging goods and services; as the medium of exchange, monetary flow makes transactions possible in modern international trades.
The direct influences in matrix P automatically generate indirect ones in the closed global trading system. In a chain of production, even though the final product bears the mark of being made in one country, its components or parts may come from elsewhere. When China exports a smartphone to the USA, for example, the chip and design software of the phone could come from the USA and the camera from Korea. Japan could manufacture the camera screen using US patents. To integrate these spillovers, we should also consider indirect and longterm impacts. This leads to the powers of P, such as P 2 , P 3 , · · · , P ∞ . For example, P ik P k j is j's two-step influence on i, directly via any country k. In general, P t contains the t-step aggregated impacts of all supply chains that crisscross the globalized world. Thus, t must be large enough for the impacts to travel from the top of global economic food chains down to the very bottom. Under some general conditions, the rows of P ∞ are a constant vector π, and π uniquely exists in this well-connected trading system. Specifically, π j = lim t→∞ P t i j regardless of the choice of i. Therefore, π j measures country j's long-run influences on the whole trading system. The existence of π and the convergence of P t can be found in the theory of Markov chains (e.g., Karlin and Taylor, 2012).
The 1×n row vector π, called authority distribution in Hu and Shapley (2003), solves the following counterbalance equilibrium subject to the normalization conditions n ∑ i=1 π i = 1 and π i ≥ 0. The counterbalance in Eq. (1) implies π's mixed cooperative and non-cooperative properties. From the non-cooperative aspect, as n ∑ i=1 π i = 1, an increase of π j may imply a decrease of π i . So, in theory and in reality, there are n(n − 1)/2 trade conflicts, either small or large, in the trading system. The vast majority of them are minor trade disagreements or disputes, and so the "war" metaphor is hyperbolic in these cases.
But it is worth mitigating even minor disputes before they become more problematic. From the cooperative aspect, country i should help j improve π j whenever P ji > 0 because π j P ji is part of Also, we can iterate Eq. (1) to get π = πP = πP 2 = πP 3 = · · · . and π i = n ∑ j=1 π j P t ji for any t = 1, 2, 3, · · · . When t is large enough, all (P t ) ji > 0 and so i should help all j in the trading system, including the poorest and the least competitive countries, to enhance π j . Thus, i's trade-off is how much it should assist or contest j without sacrificing its own interests.
More specifically, the equilibrium has a dynamic mechanism. On the one hand, in the above inflow equation Eq.
(2), i absorbs authority from other countries through direct influences. It derives more authority from influential trading partners (i.e., π j is large) than from non-influential ones, other things being equal.
It also derives more authority from countries on which it has a sizeable direct influence (i.e., P ji is large), other things being equal. Thus, π has the Matthew effect of accumulated advantage. Consequently, powerful economies have substantial authority compared to less-powerful ones, revealing a strong positive correlation between π i and i's GDP. For example, the USA's π i could be much higher than Japan's, compared to their GDP ratio. The amplification comes from the USA's relative advantages over other countries, compared to Japan's. As a consequence of the Matthew effect, an economic superpower would be expected to make all efforts to maintain its superpower status when facing challenges from emerging competitors; maintaining its advantage in π could avoid the domino effect. As Copeland (2014) says, "when expectations [of the future trade environment] turn negative, leaders are likely to fear a loss of access to raw materials and markets, giving them an incentive to initiate crises to protect their commercial interests." Also, small countries tend to fear the emergence of a neighboring superpower.
On the other hand, in the dynamic mechanism, i also contributes to other countries as seen from the following outflow equation The larger P i j or π i , the more i contributes to j's authority π j , other things being equal. However, the outflow from i to j does not necessarily match the inflow, i.e., π i P i j = π j P ji in general. But with the involvement of third parties, the inflow and outflow dynamics finally meet at the equilibrium Eq. (1). For the same reason, it could be overly restrictive for all countries to balance their bilateral trades. For example, country i can still maintain an overall zero balance when it has a trade surplus with j and a trade deficit with k of the same amount.
The index π i measures country i's competitive advantage, referring to the ability to do better than its competitors in the trading system. The competitors include all other members in the system. On the contrary, comparative or absolute advantage is an economy's leverage over a specific trading partner on specific services or products. In the literature (e.g., Porter, 1985;Stutz and Warf, 2010), competitive advantages rest on the direct control of competitive factors of production, such as access to rare resources or advanced technology, cheap but highly skilled labor, or low cost of capital. These factors are assembled in the final products, either imported from the outside or manufactured domestically. However, the imported one carries the original producer's competitiveness and should be attributed to the original maker. In the above smartphone example, the competitive factors include the chips and patents from the USA, the cheap labor from China, the precision manufacturing in Japan, and the sophisticated cameras from Korea.
Various studies have also described qualitative strategies to enhance competition.
For example, Porter (1985) lists three generic strategies: cost leadership, differentiation, and focus. We quantify the competitiveness here and apply it later to trading strategies.
To better understand the quantitative competitiveness, let us make a strong but fair assumption for the time being only: no bilateral trade deficits. So the imports from i to j exactly equal those from j to i. But the matrix P is still non-symmetric because the productions are different across the countries. LetP be the transpose of P and writeP = (τ 1 , τ 2 , · · · , τ n ) where τ j is the jth column vector ofP. Theñ P i j = P ji is the exports from i to j, both in terms of j's GDP. In the limit matrix lim t→∞P t = (lim t→∞ P t ) , the ith row is a vector with all π i , which is the long-run impact on all countries due to i's exports. As fractions of country j's GDP, τ j except for its jth element is a measure of comparative advantages over country j because these are exported to, not manufactured by, j. Besides, all other countries also compete for the comparative advantages over j; a large value in τ j means a solid competitiveness. If we use country j as the reference to rank the competitiveness of all countries, then the competitiveness scores are the column vector τ j . Yet, the jth value is not excluded from τ j because domestic products also compete with country j's imports when its consumers make their consumption choices. This way, we define n sets of competitiveness scores, τ 1 , τ 2 , · · · , τ n . Now we let the non-negative column vector ρ = (ρ 1 , ρ 2 , · · · , ρ n ) with n ∑ j=1 ρ j = 1 be the system-wide competitiveness for all countries, and apply endogenous weighting to the ranking references to combine these scores τ 1 , · · · , τ n . In doing so, we believe that more weight should be placed on a more competitive reference (i.e., with large ρ j ), and less weight should be placed on a less competitive one (i.e., with small ρ j ). Thus, we have the weighted competitiveness scores n ∑ j=1 ρ j τ j =Pρ. As the weighted scores also quantify the competitiveness of the countries as references,Pρ should be a positive multiple of ρ. By Theorem 1, ρ i is just π i . In the endogenous weighting, world-wide consumers unconsciously participate in the ranking of competitiveness and all products are involved. Essentially, countries judge countries themselves without any external interventions from business interests, mainstream media, advertising, governments, or even military threats. In fact, endowing competitive countries with large weights accounts for the Matthew effect of accumulated advantage in π.
By the endogenous weighting, π amplifies real comparative advantages and offsets noisy ones. Similar to ρ, π endogenously weighs the rows of the direct influence matrix P. For example, if all elements in P have a white noise with zero mean and variance σ 2 , then, as the weighted value of the jth column of P, π j would have a variance smaller than σ 2 . However, if there is an expected δ > 0 escalation in the jth column of P, then π j is expected to rise δ without considering any spillover effects. In the spillover process, the other columns of P decrease and so the other values in π. In the end, therefore, π j would gain more than δ . The exact change of π j with respect to a small escalation on the jth column can be found using the method in Theorems 2 and 4. Clearly, the amplification is associated with the Matthew effect and domino effect. So, when dealing with fast growing country j, i should mitigate its conflicts with j before the counterpart becomes too strong to compromise.
The competitiveness index π is accumulated by absorbing all spillover effects through P ∞ . In fact, (P t ) i j already accesses all the countries by t −1 times because A global value chain may take only finite steps, but it may also contain indefinite domestic loops when we trace all added values of the final product. Again, in the smartphone example, the manufacturing labor's education, skills, and experience could ultimately achieve domestically over many years; but all contribute to the final product in some way. Also, many USA workers could develop the patents and design software domestically for a long period. These inner loops affect the other countries but not vice versa. This part of authority (i.e., π i P ii in Eq. (2)) is not distributed externally but remains internal as part of the country's competitive advantage. Thus, social infrastructure and political institutions, monetary and fiscal policy, microeconomic environments, and any quality determinants that help in doing business all contribute to the national competitiveness (e.g., Delgado et al., 2012). Of course, reserves of raw oil, natural gas, and rare minerals, which come not from external, also remain with π i P ii as a nation's competitive factors of production. Besides, there are millions of interlinked global value chains. As a result, P t becomes stabilized only if t is large enough.
Therefore, we use π i as an indispensable objective for country i when making trade and industrial policies. It quantifies both the breadth and depth of the economy, and incorporates both cooperation and non-cooperation with others. It is endogenously extracted from all firms' comparative advantages of all exported products and services. As each country has its unique location, natural and human resources, legislation and economic systems, a single utility function, which presumes one stereotype for all economies, may be inadequate to capture this diversity. In our study, we recognize that each country is different, as is its trading stance with partners. Each adds inclusiveness value to the global economy.

A Necessary Condition for a Bilateral Trade War
Say, country i considers a trade war against its trading partner j. Gaming on the matrix P, i's action would change the element P ji -the portion of j's GDP exported to i. Raising tariffs or decreasing an import quota upon j, for example, reduces imports from country j and, consequently, decreases P ji . We assume that the bilateral trade conflict involves the two counter parties without directly sacrificing or benefiting P ik , P ki , P jk , or P k j for any third party k. But the change on P ji would eventually ramify over the global value chains in multiple steps and third parties would get involved indirectly.
As a reprisal, country j could sooner or later react by increasing or decreasing P i j . Without the reprisal, the war dies out at inception; no reaction or over-reaction to the hostile change on P ji may also hurt π j . After the Trump administration began setting tariffs and other trade barriers on China in January 2018, for example,

China responded in April 2018 by imposing tariffs on 128 products imported from
America. Say, in retaliation for a ∆P ji change on P ji , country j changes P i j by λ ji ∆P ji , i.e., ∆P i j = λ ji ∆P ji . We require that P ji + ∆P ji ≥ 0 and P i j + ∆P i j ≥ 0 to ensure the non-negativity of the new P after the changes. Also, we assume that λ ji > 0 and λ i j = 1/λ ji , in veil of any first-mover advantage. Therefore, ignoring the temporally-lagged reactions on the long-run effects π, changing P ji by ∆P ji and changing P i j by λ ji ∆P ji have the same impact on π. Finally, to maintain the unit sum in each row of P, we need to change P j j by −∆P ji and P ii by −∆P i j ; so the changed exports will be consumed domestically. Again, we require P j j − ∆P ji ≥ 0 and P ii − ∆P i j ≥ 0.
The coefficient λ ji measures j's negotiation power with i in the trade friction.
Often it depends on many other exogenous factors from both sides, such as political considerations, anti-dumping protection, trade deficits, and specific trade commodities or services. In general, there is no specific formula for every contingency, and negotiations in practice could involve many rounds of bilateral talks.
For a specific context with a set of fair assumptions, however, many researchers in the literature have studied bilateral or multilateral negotiations (e.g., Bagwell, Staiger, and Yurukoglu, 2020;Schneider, 2005). In the following exposition, we first assume the coefficient is an exogenous constant in the potential trade war.
Then, Section 4 studies a generic but simple bargaining solution for most of the n(n − 1)/2 conflicts. The solution aims to mitigate the conflicts and fairly share the costs rather than eliminate them; it serves the diverse interests of the countries.
We introduce a few notations for the next three theorems and their corollaries.
Let I n be the n × n identity matrix and let π −i be the transpose of π with π i removed. The square matrix Z i is the transpose of P with its ith row and ith column removed. Also, the column vector α i extracts the ith row from P and then drops its ith element. As usual, 1 n is the n × 1 column vector with all ones. For any j = 1, · · · , n, the n × 1 vector e j has one for its jth element and zeros elsewhere.
Finally, the vector γ ji takes the values of e j and then removes its ith element. So, like α i and π −i , γ ji is also a non-negative (n − 1) × 1 column vector.
Theorem 2. With the above setting and notations, for any j = i, and By varying j, we could apply Eq.
(3) to identify country i's trading collabora-tors and competitors for given values of λ ji . Since has all non-negative elements, dπ i dP ji has an opposite sign of λ ji π i − π j . Therefore, if λ ji > π j /π i , then j is a potential target for conflict with i because a negative dP ji brings a positive dπ i . Similarly, the condition λ ji < π j /π i implies that j is a potential target for further collaboration. Whenever λ ji < π j /π i and dP ji > 0, the larger λ ji , the slower π i grows and π j shrinks. In either case, i could also increase π i by adjusting the sign of dP ji . Without the sign change, the first-order condition λ ji = π j /π i either maximizes or minimizes π i . If j insists on λ ji = π j /π i , then any small change on P ji would have no significant effects on both π i and π j regardless of the sign of dP ji .
In search for the best competitor and collaborator, we make the results comparable across countries using percentage changes in both π i and P ji . The limit of the percentage change of π i with respect to that of P ji is When the above ratio is larger than, say, .1 (or less than -.1), we say the derivative in Eq. (3) is significantly positive (or negative, respectively). If the ratio in Eq. (6) is .2, for example, then a 1% increase of P ji could roughly trigger a .2% increase in π i . Thus, a simple necessary condition for i to start a trade war against j (or form an economic cooperation with j) would be a significantly negative (or pos-itive, respectively) derivative in Eq.
as i's best trading partner and as i's best competitor. However, i may not be his best partner's best partner nor his best competitor's best competitor.
Eq. (5) is the side effects on all other countries due to i's change on P ji , including dπ j dP ji . With the perturbation on P ji , some countries gain and others lose their competitiveness π; the gain or loss may be even more substantial than dπ j dP ji . But π i and π j always move in opposite directions, as claimed by Corollary 1. Consequently, if the first-order condition λ ji = π j /π i maximizes π i , it minimizes π j , and vice versa. However, the movements are not of the same magnitude; so one movement could be significant while the other remains insignificant. The difference between the two movements is unevenly shared by third parties. The side effects may play a role in determining λ ji when i is much more powerful than j.
If λ ji = π j /π i maximizes π i , the conflict seems resolved. However, if λ ji = π j /π i maximizes π j instead, then in theory, i could extremely counteract and push λ ji to near 0. It could also push λ ji to near +∞ and dP ji < 0. By Eq. (5), this excessively non-cooperative counteraction could provoke outrage from third parties, and a compromised solution may be finally reached. Over the centuries, many norms and standards have been set up to tame trade bullies; today the World Trade Organization (WTO) is the global international organization dealing with the rules of trade between nations. In the next section, we study a cooperative solution for λ ji .

A Nash Bargaining Solution for λ ji
When other factors also play a role, the condition λ ji > π j /π i alone is not sufficient to determine a conflict to i. Among these factors, the most primary is the trade balance, either surplus or deficit. For a fixed time period, country i's trade surplus from j means j's deficit of the same magnitude from i, and vice versa. Domestic employment and national debt, for example, highly rely on balanced trade.
To maintain a zero trade deficit between these two countries, a benchmark choice of λ ji is g j /g i where g j is country j's production or GDP. If λ ji = g j /g i , then country i's changed imports from j, which are dP ji × g j , exactly offset country j's change of imports from i, which are λ ji dP ji × g i . For i to have a trade surplus with j, we need λ ji dP ji × g i > dP ji × g j . Thus, λ ji > g j /g i implies i's trade surplus if dP ji > 0; and λ ji < g j /g i also indicates i's surplus if dP ji < 0. However, in the case of dP ji < 0, both i and j lose exports even though i loses fewer. This could eventually adversely affect the production of both countries in the future, so most countries would avoid mutually-damaging stalemates. In general, the parameter λ ji is determined by both players; but the sign and size of dP ji is often decided by the one who has more bargaining power, especially if one has overwhelming power over its counterpart.
We have studied two ways to determine λ ji or λ i j to ease the trade tension.
On the one hand, if both sides aim for higher competitiveness, then λ ji = π j /π i is the steady-state solution for any minor change on dP ji . On the other hand, if they look for a balanced trade, then λ ji = g j /g i completely removes any further trade deficit in the future. Generally speaking, λ ji = π j /π i is a long-run target and would take a long time to achieve, due to π's involvement in all countries and all industries; λ ji = g j /g i could immediately resolve the trade deficit problem to relieve domestic problems such as unemployment and national debt. For example, the China-USA trade war started in this way: China had grown exponentially over four decades by exporting less competitive goods oversea while the USA lost much of its manufacturing industries but built up massive national debt, accumulated through wars on terror, significant tax cuts, and the Great Recession.
Nevertheless, when China wanted to build a global 5G network and make cuttingedge chips, the USA attempted to block the move and urged its allies to join the effort. In this example, the focus point is competitiveness rather than trade balance. Therefore, the conflict is expected to last a long time and π j /π i is a proper choice for λ ji . In general, λ ji = π j /π i is suitable for the rivalry between superpowers or emerging superpowers; but λ ji = g j /g i is a better choice for the conflict between small and less competitive economies.
If considering both trade balance and competitiveness, the countries may have to leverage between π j /π i and g j /g i . So we first let λ ji lie between and then extend the regime [p, q] to (0, ∞). In the trading system, some countries may prefer more or less to competitiveness while others more or less to improved trade balance. Their preferences would affect the choice of λ ji . If we ignore them or assume the countries are indifferent between competitiveness and trade balance, then the solution for λ ji could be the middle point, i.e., λ ji = g j /g i + π j /π i /2 and λ i j = g i /g j + π i /π j /2. However, this middle-point solution does not satisfy the identity condition λ ji λ i j = 1.
Whenever the countries have opposite priority between enhanced competitiveness and improved trade balance (either reducing deficit or increasing surplus), we have a bargaining problem. So one country prefers competitiveness to trade balance while the other prefers the opposite. We apply Nash (1950) bargaining to find a cooperative solution. The range for λ ji is [p, q] in the bargaining. We also have the following range for λ i j : since π i > 0, π j > 0, g i > 0, and g j > 0. In addition, the solution λ ji = p (or q) is equivalent to λ i j = 1/p (or 1/q, respectively).
We have two scenarios when i and j have opposite preferences on p and q.
In the first scenario, j prefers q to p while i prefers 1/p to 1/q. So u j (λ ji ) = λ ji could be the utility functions for j because u j (q) = q > p = u j (p). Similarly, In Eq. (8), λ ji − p and λ i j − 1 q are the excessive payoffs for country j and i, respectively. The bargaining solution maximizes the product of the excessive utilities.
In the left plot of Figure 1, j moves down from the left to the right along the curve while i from the bottom up to the top left along the same curve. When they meet at (λ * ji , λ * i j ), the area of the rectangle is maximized.
In the second scenario of opposite preferences, j prefers p to q while i prefers 1/q to 1/p. So u j (λ ji ) = q − λ ji and u i (λ i j ) = 1/p − λ i j could be the utility functions for j and i, respectively. The status quo is 0 for both i and j. The Figure 1: The Nash Bargaining Solution λ * for Eq. (8) and (9), respectively. maximization problem then becomes The right plot of Figure 1 illustrates this situation where j moves from q to p and i moves from 1/p to 1/q along the curve. They meet at the solution point which maximizes the area of the rectangle. In this utility function, λ ji seems to have no meaning of bargaining power. However, we can rephrase Eq. (9) by max p≤λ ji ≤q; λ ji λ i j =1 in which λ ji − q and λ i j − 1/p are the loss functions for j and i, respectively. To coerce an agreement formation, each country plays a threat strategy to maximize its counterpart's loss. The final outcome is Eq. (10). Clearly, a larger λ ji means less loss to j; thus, λ ji still preserves the meaning of bargaining power for j in Eq. (10) and, so, in Eq. (9).
Theorem 3 answers the maximization problems and the solution satisfies the identity λ * ji λ * i j = 1. The solution is the same for both preference scenarios. In Eq. (8), (9) and (10) lution λ * ji mitigates the Matthew effect in π j /π i . If g j < g i , for example, then π j /π i < g j /g i is more likely to happen than π j /π i > g j /g i , according to the effect. Since λ * ji is between g j /g i and π j /π i , the mitigation fills the gap between these two ratios without changing the likelihood.
Theorem 3. The Nash bargaining solution to Eq. (8), (9), or (10) is λ * ji def == π j g j π i g i for λ ji and λ * i j def == π i g i π j g j for λ i j . Therefore, country j's Nash bargaining power in the trading system is proportional to √ π j g j , regardless of its trading partners.
If we extend λ ji 's bargaining set from [p, q] to [cp, q/c] for any positive c ≤ q/p, the bargaining solution remains unchanged, according to Corollary 2. The bounds cp and q/c are specified such that λ i j 's bargaining set has a similar expansion from [1/q, 1/p] to [c/q, (1/p)/c)]. So both countries still have symmetrical status quos in the extension. Letting c → 0, the bargaining sets for both λ ji and λ i j become (0, ∞). In a real bargaining, however, non-economic factors (such as military power, population, culture, tradition, legislation, and location) and alternative optimal criteria could make bargaining solutions substantially deviate from λ * ji .
Additionally, the symmetry axiom in the Nash bargaining could also be compromised. In summary, when the preferences of i and j are known, we recommend using the following λ ji : π j /π i , if both prefer higher competitiveness; g j /g i , if both prefer better trade balance; π j g j π i g i , otherwise.
Corollary 2. For any positive c ≤ q/p, if we replace p with cp and q with q/c in Eq. (8), (9), or (10), the solution is still λ ji = λ * ji and λ i j = λ * i j .
The system-wide bargaining power √ π j g j holds the transitivity property and defines a total ordering for all countries. If i has a negotiation advantage over j and j over k (i.e., √ π i g i > √ π j g j and √ π j g j > √ π k g k ), then i necessarily has more bargaining power than k. Specifically, i's bargaining power over k equals the ratio of that of i over j and that of j over k. Other bargaining solutions in the literature generally do not have the properties of transitivity and total ordering. Based on √ π j g j , for example, a multinational alliance could be easily formed without any bilateral negotiations.
Furthermore, there are two equally-weighed and highly-correlated determinants for the bargaining power √ π j g j in international trade negotiations: the volume of production g j and the strength of competitiveness π j . So, a country's bargaining power declines when its companies rely on inputs for their production from its partner countries. The more the companies depend on a partner country, the less the country's power to coerce concessions from this partner. Conversely, the bargaining power increases when a country consumes more imported goods and services by influencing the production of its counterparts. We have normal-ized the production volumes in the counterbalance equilibrium of the integrated global value chains. The negotiation power combines the equilibrium point with the economy's breadth. However, the high correlation may imply the risk of a domino effect.

Economic Globalization
Globalization refers to removing the barriers to the flow of financial products, goods, technology, information, and jobs across national borders and cultures. In Gaming on the matrix P, if country i increases globalization, it shrinks its value of P ii . Otherwise, it increases P ii if it increases protectionism. As the actions have no specific target countries, we make the following assumption: whenever P has a change in its ith column, the other columns change proportionally to ensure each row has a unit sum. Say, i changes P ii by ∆P ii . This triggers changes on the whole matrix P, say ∆P, including three levels of fallout from the variation ∆P ii .
First, the other elements in the ith row respond proportionally to the change of Secondly, for any j = i, the change − P i j ∆P ii 1−P ii on P i j causes a change − λ i j P i j ∆P ii 1−P ii on P ji by country j's reprisal reaction. Therefore, the ith column of ∆P is Lastly, for any P st where s = i and t = i, its proportional response to Then ∆P = ∆P ii M. We also let M i be M with its ith column dropped.
Theorem 4. With the above setting and notations, we have and One decision rule could be like this: country i would engage in further globalization (or protectionism) if d log π i d log P ii is significantly negative (or positive, respectively) for all λ ji from Eq. (11). Otherwise, it could seek targets of trade wars or trade partnership. In between, it could also form preferential trade agreements (PTAs). Within an ideal PTA, the ebb and flow of λ j /λ i and g j /g i should be well balanced to prevent any major internal trade disputes. We may also choose λ ji = π j /π i because superpowers are generally the driving force for globalization or protectionism. In contrast to Eq. (11), without having specific knowledge about λ ji for all or most i and j, we suggest using the following rule of thumb in matrix M: π j /π i , if both π i and π j are large (say, > .01); g j /g i , if both g i and g j are small (say, < 500 billion $); π j g j π i g i , otherwise.
To maximize π i by adjusting P ii , the first order condition is necessarily zero in Eq. (12).
Nash equilibria exist in the constant-sum game in which the degrees of globalization act as the strategy variables and π as the utilities. However, the differentiation in Eq. (12) and (13) is based on the exogeneity assumption of all λ ji . Also the matrix P changes over time and contains sampling errors; both P and π evolute if we shock P ii . If we use Eq. (14) for all i and j, for example, Eq. (12) and (13) for all i define a dynamical system in which the first-order conditions make a steady-state solution of the system. The system describes the instant changes with respect to the diagonal of P; at the steady-state solution, the diagonal and the off-diagonal could substantially deviate from its original P. Given the complexity of the system, the solution is generally achieved by numerical algorithms through many tiny adjustments on π and P. In general, some would benefit from it while others would clearly not.

Empirical Study
This empirical analysis calculates π and its derivatives with respect to P ii or P their π i ratio was 3.388. Beyond these nine countries, many small-and medium-sized economies grew more competitive during these twenty years of globalization; their total competitiveness improved by about 20% from 2000 to 2019, as seen from Figure 3. Also, Figure 4 shows that competitiveness gradually became less correlated with GDP from 2000 to 2019.
A simple implication of Eq.
(2) is country j's direct contribution to π i in percentage, i.e., π j P ji /π i × 100%. Table 1 lists the contributions for 2017 and 2019. According to    the numbers on the diagonal, the USA was the most self-sufficient in feeding its π i ; the last column also shows that the USA was the largest contributor to the competitiveness of Canada, China, Germany, Japan, and the United Kingdom in 2019. For the same year, China was the largest contributor to Russia's π i and the second largest to those of Canada, Japan, and the USA. France, Germany, and Italy were highly reliant on each other's contributions, and their dependence on Russia was ten times higher than USA's dependence on Russia. Comparing the years before and after the beginning of the China-USA trade war, the mutual contributions between the two nations were reduced by 21% and 29%, respectively, in just two years. Next, to compare the bargaining power over time, we normalize country i's power by √ π i g i n ∑ j=1 √ π j g j such that the total normalized power sums to one in any year. Figure   Thirdly, Table 2  According to the same column, any trade friction against the USA would be a severe loss.   Table 3 lists the derivatives of log π with respect to log P ii for 2017 and 2000, respectively. From the diagonal of Table 3 calculated from Eq. (12) and (6) Table 4 is calculated from Eq. (5) and (6). When the USA reduced its imports from China by 1% in 2018, China's π j suffered a significant loss of .1086% while the USA's π i earned a nonsignificant gain of .0513%. Canada was a free rider and the other six countries were slightly negatively affected in both 2018 and 2019. Therefore, these countries would take no formal action to prevent the war. Also, based on the numbers in the table, Canada (or Russia) would be the best ally with the USA (or China, respectively) in this conflict. The second best would be the United Kingdom for the USA and Japan for China, respectively. In 2019, all the effects were slightly mitigated.
Probably realizing the mutually destructive consequences of the trade war, China and the USA agreed on a new trade deal in early 2020 (cf, USTR, 2020), concluding the first phase of a long-term battle.
Year  For any positive dP ji , therefore, any λ ji between π j /π i and g j /g i would not only make log π i log P ji < 0 but also create further trade deficits for the USA, which was three times more competitive. Thus, an economic conflict with a negative dP ji was very likely from the above analyses. In this lose-lose conflict, the Nash bargaining solution for the conflict was λ * ji = increased since λ * ji > π j /π i and dP ji < 0.

Discussions
Economic globalization and trade wars have emerged as broad public interests because of global competition among countries, large-scale restructuring of labor markets, and international distribution of added value. However, we argue that imports and exports data could have already reflected these issues, even before policymakers became aware of the rapid changes. Therefore, we attempt to extract a country's competitive strength from the data, identify its best strategies for dealing with trade conflicts and cooperation, and determine if it should move more or less toward globalization.
The forgoing account, however, does not address several vital factors that deserve further investigation. First, counterbalance disequilibrium does exist when trades are imbalanced. For example, when country i uses borrowed money to consume j's exports, i's consumption still has power on j's production. However, the power comes not from i's production but from the lender of the money. If the lender happens to be j, then π i is overestimated and π j underestimated. Secondly, unlike an economic superpower in the trading network, an emerging economy may focus on its absolute growth of exports instead of competitive advantages, like China in the 1990s. Thus, maximizing π i may not be the first priority of an emerging economy. The time dimension, absent from the current research on counterbalance and bargaining, is required to formulate the growth. Thirdly, we have yet to include the capital, information, and technology sides of economic globalization. They interplay with goods and services, but the data are not as easily accessible as goods and services. Next, the integration of regional economies needs to be studied. Theorem 2 could be used to find the right partners in the integration. Also, the data contain gaps and uncertainty. There are a large number of missing values in the services data that need to be filled. In addition, country i's reported imports from j are often inconsistent with j's reported exports to i. Lastly, one could admit other variables to the productionside counterbalance Eq.
(1), such as labor mitigation, the environment, national security, and those from the consumption side.
One could extend the framework in several directions. First, a country could target a select group of partners without directly affecting others, rather than choosing a bilateral trade war or anti-globalization. For example, as Table 2 indicates, the USA could simultaneously raise the tariffs on imports from China, Germany, and Japan. Secondly, though the dynamical system of Eq.
(3)-(5) or Eq. (12)-(13) assumes a fixed P, we could add a time dimension and white noise to P. With the time horizon, we could discuss a trade war's effects on inflation, π i 's dependence on its historical values, and the improvement of trade deficits. The time dimension could also improve the forecast capability of the model. Next, we could let λ ji depend on other determinants besides the π i and GDP ratios. For example, an industrial analysis could shed light on the choice. As both Japan and South Korea are rivals for automobile and electronic exports, for example, π j /π i is proper for λ ji . In 2000, for another example, no industry could be a battlefield for the USA and China, so g j /g i seemed a natural choice for λ ji . Finally, in anticipation of the side effects of i's trade war against j, country j could seek cooperation with third parties, which would substantially suffer from the conflict. However, adding more countries to both sides might result in a world war of trade, just like the Cold War between 1947 and 1991.

Disclaimer
The views expressed herein are those of the author and should not be attributed to the IMF, its Executive Board, or its management. Declarations of interest: none. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

A1. Proof of Theorem 1
As introduced in Section 3, 1 n is the n × 1 column vector with all ones. We multiply 1 n on both left sides of the equationPρ = β ρ to get Thus, β = 1.

A2. Proof of Theorem 2
When we make a small perturbation ∆P to P in Eq. (1), the new authority distribution π + ∆π satisfies the counterbalance equation of π + ∆π = (π + ∆π)[P + ∆P] (A.1) subject to ∆P 1 n = 0 n and ∆π 1 n = 0 where 0 n is the n × 1 zero vector. After subtracting π = πP from Eq. (A.1), we get ∆π[I n − P] = π∆P + ∆π∆P and its first-order approximation is We let P ji have a small change ∆P ji and attempt to calculate its effect on π, including both π i and π −i . Accordingly, P have three other simultaneous changes: −∆P ji on P j j to offset the change on P ji in the same row; λ ji ∆P ji on P i j for country j's retaliation upon i's change at P ji ; and −λ ji ∆P ji on P ii to maintain the unit sum of the ith row. Therefore, dP dP ji is a zero n × n matrix except for: 1 at ( j, i); −1 at ( j, j); λ ji at (i, j); and −λ ji at (i, i).
Without loss of generality, we assume 1 ≤ i < j ≤ n. Dividing Eq. (A.2) by ∆P ji and letting ∆P ji → 0, we get the following equation for the derivative of π with respect to P ji : where the right-hand side has all zeros except the ith and the jth elements.

A5. Proof of Corollary 2
When we extend the bargaining set of λ ji from [p, q] to [cp, q/c], the corresponding range for λ i j then changes from [1/q, 1/p] to [c/q, 1/(cp)]. The bargaining solution to Eq. (8) is In the above, we need c > 0 and cp ≤ q/c, i.e., 0 < c ≤ q/p. Besides, for Eq. (9) or (10) after we replace p and q with cp and q/c, respectively, Finally, it is easy to check that √ pq ∈ [cp, q/c] for any 0 < c ≤ q/p.