Scenarios and Optimal Response (tariffs wars)

In a simplified game-theoretic model of international trade policy, each country chooses between imposing tariffs or not imposing tariffs on imported goods. When a country unilaterally imposes high tariffs, the other country faces a strategic decision: whether to respond with tariffs (retaliation) or to continue free trade. The choice depends on the structure of the payoffs and on whether the game is played once (single-shot) or repeated over many periods.

This explanation is divided into four parts:

1. Payoff Structure
2. Normal-Form (Matrix) Representation
3. Extensive-Form (Sequential) Representation
4. Interpretation and Best Response

All numeric examples are illustrative do not represent exact empirical data and the game has been simplified. They are used to clarify the strategic reasoning.

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1. Payoff Structure

Consider two countries, Country A and Country B. Each has two strategies:

• Impose Tariffs (T)
• Do Not Impose Tariffs (N)

A typical assumption in simple trade-war models is that each country’s payoff is higher if it can impose tariffs while the other does not. Both countries imposing tariffs can lead to a lower payoff for each. Both countries refraining from tariffs can lead to a payoff that is higher than mutual tariff imposition but lower than unilaterally imposing tariffs.

This can create a situation like the Prisoner’s Dilemma (Osborne 2004).

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2. Normal-Form (Matrix) Representation

A normal-form representation uses a payoff matrix. Suppose Country A chooses rows and Country B chooses columns. Each cell of the matrix contains a pair (UA,UB)(U_A, U_B) representing payoffs to Country A and Country B, respectively.

Below is an illustrative payoff matrix:

	B: N (No Tariffs)	B: T (Tariffs)
A: N (No Tariffs)	(3,3)	(1, 4)
A: T (Tariffs)	(4,1)	(2, 2)

• If A and B both choose N, each gets a payoff of 3.
• If A chooses T while B chooses N, A gets 4 while B gets 1.
• If A chooses N while B chooses T, A gets 1 while B gets 4.
• If A and B both choose T, each gets 2.

In many theoretical models (Dixit et al. 2009), the payoffs are ordered such that each player prefers to impose tariffs when the other does not, but both prefer mutual free trade over mutual tariffs. However, mutual tariff imposition can emerge as a Nash equilibrium if unilateral deviation from mutual free trade is more profitable.

Best Responses in the Matrix

• If B fixes its strategy as N, then A's best response is T (since 4 > 3).
• If B fixes its strategy as T, then A's best response is T (since 2 > 1).

So A's best response to B imposing tariffs is to impose tariffs as well. Symmetrically, B will choose T whether A chooses N or T. Hence, the single-shot Nash equilibrium is (T,T)(T, T).

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3. Extensive-Form (Sequential) Representation

In some models, one country moves first (imposing tariffs), and the other country observes and responds. The extensive-form (game tree) can be outlined as follows:

                                                    Country A

                                           /                                \

                                       /                                       \

              Imposes Tariffs                                       No Tariffs

                    (T)                                                           (N)

                     |                                                                 |

                     |                                                                 |

            Country B (Response)                            Country B (Response)

              /                      \                                          /                   \

             /                          \                                    /                           \

   Imposes Tariffs            No Tariffs            Imposes Tariffs        No Tariffs

         (T)                                (N)                        (T)                         (N)

           |                                    |                             |                             |

     Payoffs                        Payoffs                     Payoffs                Payoffs

An illustrative payoff assignment (corresponding to the matrix above) is:

• If A chooses T and B then chooses T, payoffs are (2,2).
• If A chooses T and B chooses N, payoffs are (4,1).
• If A chooses N and B chooses T, payoffs are (1,4).
• If A chooses N and B chooses N, payoffs are (3,3).

In a single-shot sequential game, if A moves first and imposes tariffs, B will compare its payoffs from responding with tariffs versus no tariffs. From the matrix, if A chooses T, B obtains 2 by imposing tariffs and 1 by not imposing tariffs. Thus, B imposes tariffs as well.

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4. Interpretation and Best Response

Single-Shot Game

• If one country imposes tariffs, the other country’s best response is to impose tariffs. This outcome is explained by each country's incentive to avoid being the only one open to trade while the other restricts trade (Osborne 2004).

Repeated Game

• If trade policies are repeated over many periods, strategies can involve “punishment” and “reward.” A known strategy is “tit-for-tat,” where one country responds to the other’s action in the previous period. In a repeated setting, a country might refrain from retaliatory tariffs if cooperation leads to higher long-term payoffs. However, if a country persists in imposing tariffs, the other country may retaliate. In many repeated-game analyses, cooperation (mutual free trade) can be sustained under certain conditions, but if one country breaks the agreement, the other may respond with tariffs (Dixit et al. 2009).

Conclusion on Optimal Response

• In a single-shot setting, when a country imposes high tariffs, the standard best response (in a Prisoner’s Dilemma-type payoff structure) is to impose tariffs as well.
• In a repeated setting, the best response can depend on long-run strategies of cooperation and retaliation. A one-period deviation can be punished in subsequent periods, which might deter a country from imposing tariffs in the first place.

“In a trade setting with a Prisoner’s Dilemma payoff structure, each country’s dominant strategy is to impose tariffs, even though both would do better with mutual free trade.” Quoting Osborne (2004)

“In repeated interactions, the shadow of the future can support cooperative outcomes, provided that the threat of retaliation is credible and sufficiently severe.” Dixit, Skeath, and Reiley (2009)

In a repeated trade setting where each country can impose or refrain from tariffs every round, the tit-for-tat strategy provides a simple rule: “cooperate first, then do what your opponent did in the previous round.” This strategy can sustain cooperation (low tariffs) over time, as long as each country values the future enough to avoid a prolonged tariff war.

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After a country imposes high tariffs, the basic game-theoretic analysis shows that the other country’s optimal response in a single-shot game is also to impose tariffs. This is because, given the other country’s tariffs, imposing tariffs is typically better than maintaining free trade in that single period. However, in repeated interactions, a country may choose not to retaliate if there is a strategy supporting cooperation for higher long-term payoffs.

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References

• Osborne, M. (2004). An Introduction to Game Theory. Oxford University Press.
• Dixit, A., Skeath, S., & Reiley, D. (2009). Games of Strategy. W.W. Norton & Company.

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Still we assume the players of this game are rational….