With a surplus of opinions regarding Trump’s looming tariffs, let’s clarify the discussion by viewing tariffs through a game-theoretic lens.
An Introduction to Game Theory
Game theory is the study of strategic interactions via mathematical models.
Game theory was initially created to address two-person zero-sum games. In these instances, one participant’s gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, however, in light of rising tensions between the U.S.S.R. and the U.S., game theory was extended to the study of non zero-sum games. In the modern era, it is used as a broad term for the study of rational decision making.
To see definitions of some of the game theory terminology, consult the glossary at the end of this article.
The Prisoner’s Dilemma
Two criminals are arrested and interrogated separately, and each has the choice to either cooperate (remain silent) or defect (confess and implicate the other). If both cooperate, they receive a moderate sentence, say one year in prison each. If one defects while the other cooperates, the defector goes free, and the cooperator receives a harsh sentence, say three years. If both defect, they both receive a relatively severe sentence, though not as bad as the cooperator’s outcome in the asymmetric case — say two years. We call it a dilemma because, while mutual cooperation yields the best collective outcome, individual rationality leads both players to defect, resulting in a worse outcome for both.
The repeated Prisoner’s Dilemma extends this scenario by allowing the same players to interact multiple times. In this version, players can adopt strategies that consider past behavior, such as tit-for-tat/copy-cat (cooperating initially and then mirroring the opponent’s previous move). Unlike the one-shot Prisoner’s Dilemma, where defection is a dominant strategy, the repetition introduces the possibility of cooperation. Players can punish defection in future rounds, so cooperation is a more viable long-term strategy, and the well-being of the individual converges with the well-being of the group.
The repeated Prisoner’s Dilemma is often split up into the certainly finite case and the uncertain/infinite case. In the finite case, players know exactly how many rounds there will be. In the uncertain or infinite case, players either don’t know specifically how many rounds there will be or assume the game will go on infinitely (although it always will reach an arbitrary stopping point that is unknown to the players).
Foreshadowing: How would your strategy change based off of whether or not you knew how many rounds there would be?
In a Veritasium video titled, “What Game Theory Tells Us About Life, the Universe, Everything,” the work of Dr. Robert Axelrod is cited in clarifying winning strategies for the repeated Prisoner’s Dilemma: “Be nice, be forgiving, but don’t be a pushover.” While there are some dissenting opinions as to whether or not forgiveness is actually a hallmark of a successful strategy, simulations generally show that the tit-for-tat/copycat strategy performs well. We’ll explore this further in the context of trade wars to see if this model can inform our discussion.
To explore this topic for yourself, I’d highly recommend this game by Nicky Case.
Nash Equilibrium
A Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to change their strategy given what the other players are doing.
From The Grand Strategy: Unveiling Game Theory in Chess and Economics:
This equilibrium can be seen in the “Sicilian Defense,” a common opening in chess. In this opening, Black opts for asymmetrical play to disrupt White’s control of the center. White, knowing Black’s strategy, may choose a setup like the Open Sicilian to counter this. Both players adhere to their strategies because they consider them the best response to their opponent’s approach. It’s a situation of mutual best response, much like a Nash Equilibrium.
We can also consider the following example given by Sethi, Weibull, and Silva:
“If Chrysler, Ford, and GM choose production levels for pickup trucks, a commodity whose market price depends on aggregate production, an equilibrium is an array of production levels, one for each firm, such that none can raise its profits by making a different choice.”
In the context of the (single-iteration) Prisoner’s Dilemma, Nash equilibrium occurs when both players choose to defect (betray). Even though mutual cooperation leads to a collectively better outcome, if one prisoner chooses mutual cooperation and the other does not, the prisoner who cooperated has a worse outcome. Therefore, since neither prisoner knows the opponent’s strategy, the rational choice for both prisoners is to defect.
In the finite repeated Prisoner’s Dilemma, a Nash equilibrium occurs when players chose a strategy known as ALLD — always defect. With no prior knowledge of your opponents’ strategy, a rational player would realize that the game iterates finitely many times, and since the last round carries no possibility of retaliation, the correct choice is to defect in that final round. If you know your opponents’ best choice is to defect in the last rough, it wouldn’t be rational to cooperate in the second-to-last round. If you know your opponents’ best choice is to defect in the second-to-last round, it wouldn’t be rational to cooperate in the third-to-last round… and so on, until you come to the conclusion to defect every time. If two players share this same strategy, they would both experience a worse — but crucially, stable — outcome.
For the repeated Prisoner’s Dilemma to allow for both equilibrium and cooperative behavior, the number of rounds must be uncertain. If this is the case, as follows from the Folk Theorem, there are actually infinitely many cases of cooperation that are also Nash equilibria!
Tariffs as a Strategic Game
Tariffs can be understood as strategic moves in a complex game of international trade. This game involves players (nations) leveraging tariffs to protect domestic industries, gain bargaining power, or retaliate against perceived unfair trade practices. In the case of the Trump administration’s recent tariff threats, they are intended to be a bargaining chip for practices extending beyond the realm of trade; both Canada and Mexico could face 25% tariffs if they fail to meet the administration’s demands for increased border security. In this case, tariffs are not merely economic instruments; they are geopolitical tools of coercion.
Historical Examples
Historically, tariffs have been central to trade wars, where countries engage in tit-for-tat increases in duties, often leading to economic disruptions.
One of the most famous historical examples of tariffs as a strategic game is the Smoot-Hawley Tariff Act of 1930 in the United States. During the Great Depression, the U.S. imposed steep tariffs on over 20,000 imported goods to protect domestic industries and assuage economic anxiety.
This move backfired as trading partners retaliated with their own tariffs, leading to a dramatic decline in global trade. In May 1930, Canada, the United States’ most loyal trading partner, retaliated with tariffs on 16 products that accounted altogether for around 30% of US exports to Canada. Canada also forged closer economic links with the British Empire, while France and Britain protested and developed new trade partners, and Germany developed a system of trade via clearing. Cuba, Mexico, France, Italy, Spain, Argentina, Australia, New Zealand, and Switzerland all enacted their own retaliatory tariffs against the United States.
Consequently, the depression worsened for workers, despite Smoot and Hawley’s promises of prosperity from high tariffs. By 1933, international trade had fallen by nearly 66%, exacerbating global economic downturn. This example highlights the risks of using tariffs as a unilateral strategy without considering the potential for retaliation.
Another notable example is the Chicken War between the U.S. and Europe in the 1960s. This tariff war highlights one of the shortcomings of the ordinarily-dominant tit-for-tat strategy in the repeated Prisoner’s Dilemma. When Europe imposed tariffs on American chickens, the U.S. retaliated with tariffs on European light trucks. This back-and-forth retaliation continued for years; neither side was willing to break the cycle.
The Chicken Tariff War illustrates how retaliatory tariffs can escalate disputes, creating a cycle of economic harm, which extended beyond increased prices for consumers — some of the tariffs imposed during the Chicken War are still in place today. Robert Z. Lawrence, professor of international trade and investment at Harvard, argues the tariffs on light trucks crippled the U.S. automobile industry via 40-years of insulation from competition.
For our final example from the 20th century, entangled with the Chicken War, we can analyze the U.S.-Japan trade war. This trade war primarily occurred in the 1980s and early 1990s and was driven by a growing trade imbalance, coupled with U.S. fears of Japanese economic dominance in key industries (cars, semiconductors, and electronics).
By the 1970s, Japan’s rapid export growth led to a significant trade surplus with the U.S., causing tensions as American industries struggled to compete. The U.S. accused Japan of unfair trade practices, including currency manipulation and government-backed industrial policies that favored domestic firms.
In response, the U.S. imposed tariffs, voluntary export restraints (VERs), and other trade restrictions, particularly targeting Japanese auto and electronics exports. One notable example was the 1981 agreement limiting Japanese car exports to the U.S., which led Japanese firms to open manufacturing plants in the U.S. to circumvent restrictions.
Despite trade disputes, the U.S.-Japan trade war led to structural changes in both economies. Japan responded by diversifying its economy, investing heavily in research and development. The U.S., meanwhile, pressured Japan to sign the 1985 Plaza Accord, which led to a sharp appreciation of the yen, making Japanese exports more expensive and less competitive. This currency adjustment contributed to Japan’s subsequent stagnation in the 1990s. Over time, tensions eased as China emerged as a larger trade competitor to the U.S. This conflict sheds light on the challenges of balancing protectionism with free-market competition.
Protectionism in the Trump Era
More recently, the U.S.-China trade war under the first Trump administration illustrated the modern implementation of protectionism as part of a strategic game.
The U.S. imposed tariffs on Chinese goods to address intellectual property theft and reduce the trade deficit, prompting China to retaliate with tariffs on American agricultural and industrial products. This escalation disrupted global supply chains and forced companies to rethink their reliance on Chinese production. While the Biden administration placed an emphasis on multilateral cooperation with China, most Trump-era tariffs were maintained. Further, Biden’s semiconductor export controls were a significant escalation in the trade conflict.
Tariffs are not economic measures that exist in a vacuum; they can be instruments of geopolitical strategy. When used as part of a broader negotiation framework, they can compel concessions or protect vital industries. However, they carry significant risks, including retaliation, economic inefficiency, and global instability. In short, the strategic game of tariffs requires careful calculation.
Modeling Trade Wars
If we model a trade war as a competitive game, each player (country) chooses a tariff level to maximize its own payoff, given the tariff levels chosen by other players.
Let’s make this model more robust by creating a Payoff Matrix and looking for Nash equilibria.
In order to get values for our Payoff Matrix, we could start by conducting a combined analysis of factors like GDP, inflation, wages, industrial production, and manufacturing demand, each with its own weight corresponding to its perceived importance in the overall national economy.
Here’s an example of how we could use these factors to create a risk score in Python:
import numpy as np
# Goal: Calculate an economic risk-assessment score based on indicators
def calc_risk_score(gdp_growth, inflation, wages, indust_prod, manufact_demand):
# Define *arbitrary* weights for each factor based on economic impact
weights = {
'gdp_growth': -0.4, # Negative weight: higher GDP growth reduces risk
'inflation': 0.3, # Higher inflation increases risk
'wages': -0.2, # Higher wages reduce risk
'indust_prod': -0.3, # Increased production lowers risk
'manufact_demand': -0.3 # Higher demand lowers risk
}
# Normalize inputs
gdp_norm = np.clip(gdp_growth/5, -1, 1)
inflation_norm = np.clip(inflation/5, -1, 1)
wages_norm = np.clip(wages/5, -1, 1)
indust_norm = np.clip(indust_prod/5, -1, 1)
demand_norm = np.clip(manufact_demand/5, -1, 1)
# Compute risk score as a linear combination of factors
risk_score = (
weights['gdp_growth'] * gdp_norm +
weights['inflation'] * inflation_norm +
weights['wages'] * wages_norm +
weights['industrial_production'] * industrial_norm +
weights['manufact_demand'] * demand_norm
)
# Scale risk score to a 0-100 range
risk_score_scaled = np.clip((risk_score + 1) * 50, 0, 100)
return risk_score_scaled
# Example -- values in hypothetical percentages
gdp_growth = 2.5
inflation = 3.0
wages = 1.5
industrial_production = -0.5
manufacturing_demand = 2.0
risk_score = calc_risk_score(gdp_growth, inflation, wages, indust_prod, manufact_demand)
print(f"Risk Score: {risk_score:.2f}")Note that the weights in this example are arbitrary; they’re just meant to illustrate how this score might be calculated — how do we know how heavily one economic indicator should be weighted compared to another? To compound this problem, it’s hard to pinpoint exact time intervals during which our economic indicators are under the influence of a trade policy. So, there’s no hard-and-fast rule to determine what those percent change values should be. To see how I attempted to find solutions to these problems using methods like Principal Component Analysis, check out my Python notebook: Creating Economic Indicators Using Machine Learning.
If you’re not familiar with Python or not interested in machine learning solutions, all you need to know is that we’ve found a reasonably reliable algorithm to produce risk scores, given two countries and reference intervals as inputs. The input values are pulled from the following historical examples of two countries defecting, two countries cooperating, and one country cooperating while the other defects:
intervals = {
# Both Cooperate
('Free Trade', 'Free Trade'): ('2010-01-01', '2017-12-31'),
# Both Defect
('Tariffs', 'Tariffs'): ('2018-01-01', '2023-12-31'),
# China defects, USA cooperates
('Free Trade', 'Tariffs'): ('2019-08-01', '2020-02-01'),
# USA defects, China cooperates
('Tariffs', 'Free Trade'): ('2018-03-18', '2018-06-30')
}Both Cooperate (Free Trade, Free Trade): January 1, 2010 — December 31, 2017
- July 9, 2013 — U.S.-China Strategic and Economic Dialogue: Both nations committed to trade liberalization and economic cooperation.
- June 7, 2016 — U.S.-China Bilateral Investment Treaty Talks: Progress was made toward mutual investment agreements without trade restrictions.
Both Defect (Tariffs, Tariffs): March 22, 2018 — December 31, 2023
- March 22, 2018 — U.S. Tariffs on $60 Billion of Chinese Goods: President Trump initiated tariffs citing unfair trade practices.
- April 2, 2018 — China Retaliatory Tariffs: China imposed tariffs of up to 25% on U.S. goods, marking the start of the trade war.
- May 10, 2019 — U.S. Raises Tariffs to 25% on $200 Billion of Chinese Goods: The trade war escalated further.
- August 23, 2019 — China Responds with New Tariffs: China imposed additional tariffs of 5%-10% on $75 billion of U.S. goods.
China Defects, USA Cooperates (Free Trade, Tariffs): August 1, 2019 — December 31, 2019
- August 1, 2019 — China Suspends U.S. Agricultural Purchases: China unilaterally halted purchases of U.S. farm goods despite U.S. trade negotiations.
USA Defects, China Cooperates (Tariffs, Free Trade): March 1, 2018 — June 30, 2018
- March 1, 2018 — U.S. Imposes Steel and Aluminum Tariffs: The U.S. placed a 25% tariff on steel and 10% on aluminum imports, affecting China. Initially, China did not impose retaliatory measures, urging negotiations instead.
- June 15, 2018 — China Retaliates with Tariffs: China eventually imposed its own tariffs, ending this interval.
Again, it’s hard to nail down exactly when our risk score is under the influence of a tariff, but this timeline should give us an approximate way to quantify the decision to cooperate (maintain free trade) or defect.
Unlike the traditional Prisoner’s Dilemma example, this Payoff Matrix is unbalanced. In standard symmetric games, strategies often mirror each other. But in unbalanced matrices, we can observe different risk-reward structures for each player. To determine a Nash equilibrium, we must first identify each player’s dominant strategy. If no dominant strategy exists, the next step is to compare best responses for each possible strategy combination, ensuring that neither player has an incentive to deviate unilaterally. Importantly, unbalanced matrices may lead to inequitable or counterintuitive equilibria, where one player disproportionately benefits while the other bears a higher cost.
Nash Equilibria in Trade Games
The concept of Nash equilibrium provides a framework for analyzing strategic interactions in tariff games by looking at rational solutions given limited competitor information. Recall from earlier that a Nash equilibrium occurs when each country’s tariff policy is a best response to the known policies of others, meaning no country can improve its economic outcome by unilaterally changing its strategy.
To find the Nash equilibrium in the above example of U.S.-China trade relations, we need to identify the strategy pair where both players are choosing their best responses at the same time. Looking at the different combinations, we see that (U.S. cooperates, China cooperates) doesn’t work because the U.S. has a better payoff from imposing tariffs. Similarly, (U.S. cooperates, China tariffs) isn’t an equilibrium since China would prefer to cooperate. The pair (U.S. tariffs, China cooperates) is the only one where both players are making their best choices: the U.S. prefers tariffs, and China prefers cooperation. Therefore, this is the Nash equilibrium.
The final outcome is interesting because it suggests that the most stable situation occurs when the U.S. imposes tariffs while China maintains free trade. Even though it might seem unfair to China, it is the most stable scenario in terms of minimizing risk for both countries.
But aren’t trade wars more accurately modeled via the uncertain/infinite repeated Prisoner’s Dilemma?
In the repeated game with uncertain rounds, the following outcomes can be sustained as Nash Equilibria:
Mutual Cooperation:
- Players cooperate in every round.
- Supported by strategies like Tit-for-Tat or Permanent Retaliation/Grim Trigger, which punish defection.
- This is the most desirable outcome collectively and is sustainable if players value future payoffs over present payoffs.
Alternating Cooperation and Defection:
- Players alternate between cooperating and defecting in a coordinated manner.
- For example, one player cooperates while the other defects, and they switch roles in subsequent rounds. This can lead to payoffs averaged over time.
Partial Cooperation:
- Players cooperate some of the time and defect the rest.
- For example, they might cooperate 70% of the time and defect 30% of the time, leading to intermediate payoffs.
Punishment and Forgiveness:
- Players use strategies that punish defection but allow for forgiveness after a certain period.
- For example, a player might defect for a few rounds after the opponent defects but then return to cooperation.
In a repeated game, players can use strategies that condition their actions on the opponent’s past behavior. If a player defects, the opponent can retaliate in future rounds. This threat is what sustains cooperation and makes defection a non-dominant strategy in the long run.
In the context of tariff games, the repeated Prisoner’s Dilemma with uncertain rounds suggests that:
- Cooperation (Free Trade) can be sustained if countries value future trade relationships enough and use strategies that punish defection (e.g., imposing tariffs in response to tariffs).
- Defection (Mutual or One-Sided Tariffs) remain a stable but highly inefficient equilibrium.
Now, as a disclaimer, I’m a mathematician and data scientist — not an economist. There are likely many more nuanced ways to measure the costs and benefits of tariffs when it comes to national economies. However, the main point I hope to illustrate with this model is that the risk inherent in future punishment often outweighs the any benefit of defection.
Forgiveness = Weakness?
This threat of future punishment is part of what makes nice strategies successful.
But what about forgiveness?
Fellow Medium author Stuart Ferguson asserts that a complete (containing all possible strategies) simulation of the repeated Prisoner’s Dilemma best incentivizes a strategy known as permanent retaliation or Grim Trigger. In this strategy, you start by cooperating, but once an opponent defects, you defect for the rest of the game. While this runs perpendicular to common beliefs about the repeated Prisoner’s Dilemma, it brings up an interesting idea: in a trade war, are forgiving players more successful?
To draw from history, we can see the pros and cons of forgiveness by examining the Plaza Accord. Japan agreed to let the yen appreciate relative to the U.S. dollar to reduce its trade surplus (i.e. to make it cheaper for Japanese consumers to buy American goods). Even after the U.S. limited Japanese exports in 1981 and despite the complex geopolitical relationship forged in the aftermath of World War II, Japan chose cooperation in the form of forgiveness. The Plaza Accord helped stabilize global markets and reinforced the U.S.-Japan alliance. However, Japan’s economy stagnated due to the yen’s sharp appreciation, leading to the “Lost Decade.” Some argue the harsh terms contributed to Japan’s economic struggles.
To illustrate potential reprocussions of a lack of forgiveness, let’s revisit the Smoot-Hawley Tariff Act of 1930, which significantly raised tariffs on thousands of imported goods. Rather than continuing to cooperate, many major U.S. trade partners responded with retaliatory tariffs of their own. These countermeasures strangled international trade, exacerbated global economic downturn, and worsened diplomatic relations (at a time when nationalism was becoming more and more prevalent).
Instead of practicing forgiveness, countries continued to double down on protectionist policies. The economic distress fueled political extremism, particularly in Germany and Japan, where leaders blamed their struggles on Western economic policies. The failure to de-escalate the trade war and find cooperative economic solutions played a direct role in the rising tensions of the 1930s and the outbreak of World War II. In short, trade war escalation can spiral into undesirable outcomes for all players.
Although it seems part of the game is finding the optimal time and the optimal way to forgive, forgiveness is a trait that is ultimately rewarded in the game of trade.
Viewing Trade as a Cooperative Game
Holistically, nations maximize collective payoffs by working together rather than engaging in zero-sum competition. In such a framework, countries create efficiency gains that, in theory, should benefit all parties involved.
“A good man draws a circle around himself and cares for those within — his woman, his children. Other men draw a larger circle and bring within their brothers and sisters. But some men have a great destiny. They must draw around themselves a circle that includes many, many more.”
- 10,000 BC, Cliff Curtis (Tic’Tic)
For cooperation to be sustainable, the distribution of these payoffs must be equitable, ensuring that no participant has an incentive to deviate unilaterally. A key challenge in this model arises from labor issues in the Global South, where multinational corporations often exploit weak labor protections to minimize costs. While cooperative trade agreements can increase total surplus, the disproportionate concentration of profits in wealthier nations — paired with poor working conditions, wage suppression, and limited bargaining power in developing economies — undermines systemic stability.
Therefore, in my view, achieving a truly cooperative trade equilibrium requires mechanisms such as fair labor standards, stronger worker protections, and equitable wealth distribution.
Implications
With the impending tariffs of the second Trump administration, it is crucial to remember the cost of defecting against trading partners. We’ve seen through historical and game-theoretic examples that cooperative solutions are dominant in the long-term. In conclusion, the Trump administration must consider the risks inherent in harsh tariffs.
What are your thoughts on Trump’s tariff threats? Share your comments below.
Glossary
- Game: A mathematical model of a strategic interaction between multiple decision-makers. Each player selects a strategy from a set of possible actions, and the outcome for each player depends not only on their own choice but also on the choices made by others. The goal of a game is to determine the best strategies for players, often aiming to maximize their own payoff or minimize their risk. Games can vary in complexity, from simple one-time decisions to complex scenarios involving multiple stages and information, and they are used to model various real-world situations such as economics, politics, and social interactions.
- Players: Players are the decision-makers in a game. They can be individuals, groups, organizations, or any entities that have a set of possible actions and whose actions affect the outcome of the game. Each player aims to maximize their own payoff based on the strategies they choose.
- Strategies: A strategy is a complete plan of action that a player can take in a game, considering all possible situations that might arise. It defines what a player will do in every possible scenario. Strategies can be pure (a specific choice) or mixed (a probability distribution over possible choices).
- Payoffs: Payoffs represent the utility or benefit that a player receives from a particular outcome of the game. Payoffs are often represented as numerical values and are used to rank the desirability of different outcomes for each player. The goal of each player is typically to maximize their own payoff.
- Zero-Sum: A zero-sum game is one in which the total gains and losses of all players sum to zero. In other words, one player’s gain is exactly balanced by the losses of the other players. Poker is a classic example of a zero-sum game because the total amount of money won by some players equals the total amount lost by others.
- Cooperative: Cooperative game theory deals with situations where players can form binding agreements and coalitions to achieve better outcomes. In cooperative games, the focus is on how players can work together and how the resulting benefits can be distributed among them.
Sources and Further Reading
- The Evolution of Cooperation | Robert Axelrod
- A Non-cooperative Equilibrium for Supergames | James W. Friedman
- Tariffs and Game Theory | Cornell
- The game theory of Trump’s tariff threats | Nate Silver
- Talking Tariffs — the History of Global Trade | Adam Hamilton
- Dynamic Leveraging-Deleveraging Games | Minca, Wissels
- The Theory of Games and the Balance of Power | R. Harrison Wagner
