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:
- Payoff Structure
- Normal-Form (Matrix)
Representation
- Extensive-Form (Sequential)
Representation
- 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.
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).
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).
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.
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.
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.
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.
Still we assume the players of this
game are rational….
