Discussion Paper No. 281
Farsighted Stable Sets of Tariff GamesRyo Kawasaki
Takashi Sato
Shigeo Muto
April 2012TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY
KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANFarsighted Stable Sets of Tariff Games∗
Ryo Kawasaki†
Takashi Sato‡
Shigeo Muto§
April 17, 2012

Abstract
This article analyzes the tariff negotiation game between two countries when the
countries are sufficiently farsighted. It extends the research of Nakanishi (2000) andOladi (2005) for the tariff retaliation game in which countries take into account subsequence retaliations that may occur after their own retaliation. We show that whencountries are sufficiently farsighted, all farsighted stable sets of the tariff game are
singletons, which are Pareto efficient and strictly individually rational tariff profiles.
These results hold regardless of whether coalitional deviations are allowed or not.

∗The authors thank Keisuke Bando and Shin Kishimoto for helpful comments and suggestions.
Institute for International Education/Graduate School of Economics and Management, Tohoku University, 27-1 Kawauchi, Aoba-ku, Sendai, 980-8576, Japan; rkawasaki@econ.tohoku.ac.jp‡
Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8552, Japan§
Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8552, Japan; muto@soc.titech.ac.jp
†

1

1

Introduction

This article analyzes the tariff negotiation game between two countries when the countries
are sufficiently farsighted. Primary papers in the literature, such as Johnson (1953–
1954) envision a scenario in which countries choose an optimal tariff rate given thatthe other country does not change its tariff rate. Tower (1975) and Rodriguez (1974)
have carried this analysis over to the game in which countries, instead of choosing thetariff rates, choose export or import quotas. Although not explicit in their formulation,
their framework employs an equilibrium concept similar to that of Nash equilibrium. In
these models, each country successively chooses a tariff rate or a quota level under theassumption that the other country stays put.
However, when each country chooses such an optimal level, it does not take into
account the consequences of such actions that it triggers, including the possibility thatthe other country may retaliate in response. Recently, Nakanishi (1999), for the quota
game, and Oladi (2005) and Nakanishi (2000), for the tariff game, have applied the theoryof social situations of Greenberg (1990) to the export quota game and the tariff gamerespectively to capture this possibility in their model. However, the domination relationthat their findings are based on does not take into account the situation in which players
are not myopic. In this paper, we analyze the stable outcomes in tariff games whenplayers can sufficiently take into account the consequences of their deviations and are
only interested in the final outcomes as results of such deviations. To do so, we apply
the farsighted stable set to tariff games.
There has been a growing literature of the application of farsighted stable set of Chwe(1994). The starting point of the argument for the farsighted stable set start with theargument by Harsanyi (1974) and Chwe (1994) that the classic stable set of von Neumann
and Morgenstern (1953) uses a domination relation that is myopic. Attempting to take
into account sequences of deviations that may occur, Harsanyi (1974) and Chwe (1994)
define a domination relation, called indirect domination, which is then used to define
the farsighted stable set. This solution concept has been used in papers, including, tothe authors’ best knowledge, Suzuki and Muto (2005) and Kamijo and Muto (2010)
in sending a message that farsightedness is the key element in reaching Pareto efficient
outcomes. This message has to be taken cautiously since they allow coalitional deviations– that is, simultaneous deviation by multiple players. The juxtaposition of the resultsin Suzuki and Muto (2000), Masuda (2002), Nakanishi (2009), and Kawasaki and Muto
(2009) reveal that there is not a direct relationship between the efficiency of the results infarsighted stable sets and the rules of the game ascertaining the allowance of coalitional

2

deviations.
In light of the aforementioned papers in the literature, we analyze the farsighted stablesets of two different games of tariff games. In the first model, we allow for coalitional
deviations – simultaneous deviations made by both countries. The first model corresponds
to the rule of negotiating outlined in Oladi (2005) and the first model in Nakanishi(2000), both of which consider tariff retaliation games. In the second model, we disallow
coalitional deviations. This restriction can be interpreted as an alternating negotiation
game in which one player proposes one tariff, while in the next step, the other player can
respond. This model is closely related to Nakanishi (1999), which also restricts deviations
to those made by individual players in a quota retaliation game.
We show that in both games, the tariff choices by two countries that is Pareto efficientand strictly individually rational constitutes a singleton farsighted stable set. Moreover,
we can show that no other farsighted stable sets exist in these two games. Thus, the rulesof the game regarding coalitional deviations do not affect the outcome of the results,
although the proof of the statement is far more involved in the second game. Unlike
Nakanishi (2000), to achieve efficiency the only main addition to the original model in
Oladi (2005) that is used is that countries are sufficiently farsighted, and in addition,
outcomes that are not individually rational are not supported by a farsighted stable set.
One possible criticism to this approach is that it requires the players to be able toforesee events multiple steps ahead. However, as will be apparent in the proofs of thestatements of this paper, we do not need to assume a substantial amount of farsightedness
to establish the results. All of the results hold when player can foresee at least four steps
ahead.
Our main focus of this paper is on tariff games, but we can easily use the same logic
employed in this paper to show a similar result for the export quota game.
The rest of the paper proceeds as follows. In the next section, we introduce twomodels of the tariff game as mentioned in the introduction. In section 3, we review theliterature on farsighted stable sets and provide key definitions and their properties. In
sections 4 and 5, we present the results for the two models. We conclude in section 6.

2

Two Models of the Tariff Game

In this section, we introduce the tariff game. The main components of the game follow those in Nakanishi (2000). First, we introduce some basic notations. Let G =(N, (Xi )i∈N , (Ui )i∈N ) be a game in strategic form where Xi is the set of strategies forplayer i ∈ N . In the tariff game, Xi is the set of tariffs from which a country can choose3

from. Ui is the payoff function for player i.
To incorporate farsightedness into this framework, Chwe (1994) defines the effectiveQ
ness relation →S for each S ⊆ N as a binary relation on X = i∈N Xi such that x →S ydenotes that players in S can realize the outcome y when x is the status quo. A concrete
definition of →S depends on the context of how the game is defined – including, forexample, whether coalitional deviations are allowed or not.1 The difference of the twomodels considered in this paper come from how this relation is defined.
In the first model, we do not impose any additional restrictions to the ones that are
implied by a game in strategic form. Therefore, in the first model, we havex →S y ⇔ xi = yi ∀i ∈ N \ S,where the latter condition is dropped if S = N . This model closely resembles the firstmodel considered in Nakanishi (2000) and Oladi (2005). In their model, they consider the
direct domination relation, one in which indirect domination holds where the length ofthe deviation is one. Oladi (2005) shows that the set of Pareto efficient tariffs is a stable
set defined by direct domination, but Nakanishi (2000) shows the existence of several
others and is able to eliminate these other solutions by restricting the deviation relationby what he calls the WTO tariff concession rules.
In the second model, we consider a situation resembling closely to an alternating offer
model. In this situationx →S y ⇔ |S| = 1 and xi = yi ∀i ∈ N \ S.
The above condition states that only one player can deviate at a time. Nakanishi (1999)
imposes this condition on the quota retaliation game. To reflect the alternating part, we
define an indirect domination relation with the effectiveness relation in this situation.
We now explain the main components of the tariff games themselves. These components pertain to both models; therefore, all the observations and properties introduced
here hold for the first model as well as the second model.
Let N = {1, 2} be the set of players, where in the tariff games, each country is aplayer. Throughout these two terms will be used interchangeably.
Xi represents the set of tariffs that country i can choose and is defined as Xi = (−1, t̄i ]where t̄i represents the highest tariff rate that is permitted. A negative tariff rate isdefined to be a subsidy from one country to the other; the value −1 is not included
1

The effectiveness relation is a simplified form of the inducement correspondence in the theory of social
situations. See Greenberg (1990) for details.

4

in the set, since the prices would be undefined. Although a negative tariff rate seems
impractical, we include the possibility here to make direct comparisons to Oladi (2005)
and Nakanishi (2000) both of which include them as well. All of our results hold (muchmore easily) if we restrict our attention to only nonnegative tariff rates. X = X1 × X2
represents the set of possible outcomes resulting from the choices of countries 1 and 2.
Throughout this paper we call an element t ∈ X a tariff profile or simply an outcome.
Following Nakanishi (2000), define X o as the set of tariff profiles at which there isno trade, because either country or both have set a relatively high tariff rate, thereby
discouraging trade. We assume that X o contains tariff profiles x such that xi = t̄i andxj > 0. We let Ui (x) = ūi for all outcomes x ∈ X o . Let X ∗ = X \ X o be the set of tariff
profiles at which there is a positive amount of trade.
We assume that the utility functions defined on the tariff profiles are continuous on
X; for each fixed xj , Ui is quasiconcave in xi ; and for each fixed xi , Ui is decreasing in
xj along X ∗ .2
A tariff profile x is said to be Pareto dominated by another tariff profile y if Ui (x) ≤Ui (y) for all i and Uj (x) < Uj (y) for some j. In that instance, we also say that y Pareto
dominates x, and we denote this by yP x. A tariff profile x is Pareto efficient if there
does not exist y such that yP x.
From Mayer (1981) and Nakanishi (2000), the set of Pareto efficient tariff combinations, denoted by E, is given by the union of the following sets.
E ∗ = {x ∈ X : (1 + x1 )(1 + x2 ) = 1}E1 = {x ∈ X : x1 = e11 , x2 ≤ e12 }E2 = {x ∈ X : x1 ≤ e21 , x2 = e22 }where ei = (ei1 , ei2 ) is such that eii = t̄i and eij is such that (1 + t̄i )(1 + eij ) = 1.
Also, for each x = (x1 , x2 ), define L(x) = {y ∈ X : y1 ≤ x1 , y2 ≤ x2 } and H(x) =S{y ∈ X : y1 ≥ x1 , y2 ≥ x2 } and for each subset A ⊂ X, L(A) = x∈A L(x) andS
H(A) = x∈A H(x).
In addition, we assume the following on the indifference curves of the two countries
in the tariff space:(A1) For x, y ∈ H(E ∗ ) ∩ X ∗ such that Ui (x) = Ui (y), if xi < yi , then Uj (x) > Uj (y)(j 6= i)2

For a detailed explanation of the assumptions of the utility functions and derivations from primitives
such as exports and imports, see Mayer (1981), Oladi (2005), and Nakanishi (2000).

5(A2) For x, y ∈ L(E ∗ ) such that Ui (x) = Ui (y), if xi < yi , then Uj (x) < Uj (y) (j 6= i).
The two conditions combined imply that i’s indifference curve and j’s indifference curve
can cross at most twice. If they do cross twice, they cross once in L(E ∗ ) and once inH(E ∗ ) ∩ X ∗ . These conditions are used mostly for Lemma 6.
Denote by mi , i’s maximin value of the game. That is, for j 6= i,
mi = max min Ui (xi , xj ).
xi ∈Xi xj ∈Xj

Let XiM denote set of xi ∈ Xi that solve the above problem. Each element in XiM is the
maximinimzer for player i. Based on the assumptions of the tariff game, we can showthat the maximin value of the game is ūi for player i and that XiM = [0, t̄i ].
Let X SI be the set of strictly individually rational strategy profiles, which is the setof x ∈ X such that for each i, ui (x) > mi for each i. In comparison, a strategy profile
is said to be individually rational if the previous inequality holds with a weak inequality
for each i. Thus, the set of strictly individually rational tariff combinations is given by
X SI = {x ∈ X : Ui (x1 , x2 ) > ūi ∀i ∈ N }.
We now introduce the reaction function φi (xj ) for each i and j 6= i defined in Nakanishi(2000). φi (xj ) assigns a tariff rate x∗i such thatUi (x∗i , xj ) ≥ Ui (xi , xj ) ∀xi ∈ Xi .
For xj 6= t̄j , the x∗i that satisfies the above condition is unique. However, when xj = t̄j ,for any xi ≥ 0, Ui (xi , t̄j ) = ūi , and for all xi < 0, Ui (xi , t̄j ) < ūi . By convention,
we set φi (t̄j ) = 0. Furthermore, we assume that φi is a decreasing function of xj andφi (0) < t̄i . These assumptions are the same as those made in Nakanishi (2000). Fromthese assumptions, we can deduce that these reaction functions cross at a point x∗ suchthat x∗i > 0 for both players, implying that there exists a Nash equilibrium in which bothcountries set a positive tariff rate. Note that for such x∗ , x∗ ∈/ E, and Ui (x∗ ) > ūi for alli.
Nakanishi (2000) then shows the following property of φi . This property is usedmostly in proving Lemma 6.
Proposition 1. Fix x̄j and let φi be i’s best reply function. Then, the following propertiesof φi hold.
1. If xi < yi ≤ φi (x̄j ), then Ui (xi , x̄j ) < Ui (yi , x̄j ).
6

2. If xi > yi ≥ φi (x̄j ) and (xi , x̄j ), (yi , x̄j ) ∈ X ∗ , then Ui (xi , x̄j ) < Ui (yi , x̄j ).
For a graphical depiction of the tariff game – such as indifference curves, reaction
curves, and the Pareto efficient region E ∗ – see Nakanishi (2000) and Oladi (2005).
We consider two different models of the above tariff negotiation game, where thedifference will be reflected in how the effectiveness relation →S is defined for each S ⊆ N .
The reason for doing so is to provide a unifying analysis based on the different rules ofdeviation imposed by Nakanishi (1999) and by Oladi (2005). In attempts to capture
retaliation more realistically, these two papers formulate their games on some contingent
threats situation as defined by Greenberg (1990). Nakanishi (1999) formulates the quota
retaliation game as an individual contingent threats situation of Greenberg (1990) that
does not permit coalitional deviations, while Oladi (2005) uses the coalitional contingent
threats situation of Greenberg (1990) which allows coalitional deviations. In their models,
how the effectiveness relation is defined plays a key role in their results.
Their results can be summarized in the following way. Define x to be directly dominated by y if there exists S ⊆ N such that x →S y and Ui (x) < Ui (y) for all i ∈ S. Thatis, x is directly dominated by y if there exists a coalition S that can enforce y from x andbenefit from doing so. Using this direct domination relation, one can define the stableset, first defined in von Neumann and Morgenstern (1953), of this game, along the lines
as outlined in Greenberg (1990).
Nakanishi (2000) shows that under this setup there exist multiple stable sets, most ofwhich involve inefficient outcomes. By imposing restrictions on the possible deviations
to emulate the situation as prescribed by the WTO rules, he shows that the set E is theunique stable set, restoring efficiency. On the other hand, E may include outcomes thatare not individually rational.
As Harsanyi (1974) and Chwe (1994) note, the direct domination relation is myopic
in that players do not take into account subsequent deviations that may occur after the
initial deviation. We wish to show that by using the domination relation that incorporates
farsightedness of the players, that efficient outcomes that are strictly individually rational
are stable, and those outcomes are the only ones that can be stable. In the next section,
we explain the indirect domination relation built off of the two situations introduced inthis section.

3

Farsighted Stable Sets in Strategic Form GamesA strategy profile x is said to be indirectly dominated by another strategy profile y,denoted by x  y if there is a sequence of strategy profiles a0 , a1 , · · · , am and coali7

tions (subsets of N ) S0 , S1 , · · · , Sm−1 with a0 = x and am = y such that for eachk = 0, 1, · · · , m − 1,
• ak →Sk ak+1 and• Ui (ak ) < Ui (y) for all i ∈ Sk .
The first condition states that y can be reached from x through a sequence of deviations.
The second condition states that each individual in each deviating coalition is made better
off in the final outcome y.
This definition of indirect domination can be applied to both models as is written
above. For example, in the first model, the definition accommodates deviations made by
two or more players at the same time, while in the second model, the first of the two
conditions above imply that only individual deviations are accounted for. However, tobe clear, we provide a definition and a separate notation for the second model.
Formally, a strategy profile x is said to be indirectly dominated through individual
deviations by another profile y, denoted by x I y if there exist sequences of strategy
profiles a0 , a1 , · · · , am and subsets of N denoted by i0 , i1 , · · · , im−1 with a0 = x and
am = y such that for each k = 0, 1, · · · , m − 1,
• ik 6= ik+1
• ak →ik+1 ak+1
• Uik (ak ) < Uik (y).
The first condition rules out consecutive deviations made by the same player. Thiscondition is not restrictive, since even if we were to allow such consecutive deviations, we
can surpress the sequence by making them into one single deviation.
This second definition is specifically defined to make a direct analogy to papers in theliterature that use noncooperative frameworks. In general, this restriction can lead to adifferent set of results as can be seen by comparing the results of Suzuki and Muto (2000)
for the two-player prisoners’ dilemma game and by comparing the results of Suzuki and
Muto (2005) and Nakanishi (2009) for prisoners’ dilemma games.
A farsighted stable set V with respect to the domination relation  is a subset of X
that satisfies the following two conditions.
• For every x, y ∈ V , x  y does not hold.
• For every x ∈/ V , there exists y ∈ V such that x  y.

8Similarly, we can define a farsighted stable set with respect to I by replacing each with I in the above definition.
As a general rule, Shino and Kawasaki (2012) show that farsighted stable sets ofstrategic form games only contain individually rational strategy profiles. This property
is not shared with its myopic counterpart, as can be seen by the results of Nakanishi(2000) for the tariff game. In fact, the results using the myopic stable set support tariffchoices in which one country subsidizes a substantial amount. Such a pair of tariffs is
Pareto optimal because one country benefits greatly from the subsidy of the other country,
while that country is hurt by it, especially if the other country is not subsidizing.
There is another property of the indirect domination relation pertaining to the idea ofthe maximin strategy. The result is first introduced in Suzuki (2002), and its proof follows
from the definition of the maximin strategy and the definition of indirect domination.
For completeness, we provide the proof here.
Lemma 1. Take any x such that u1 (x) ≤ m1 where m1 is the maximin value for playerM
1 and let xM
1 be player 1’s maximin strategy and x2 be player 2’s maximin strategy andM = (xM , xM )  x nor xM  x can hold. A similar
suppose xMI
1 6= x1 . Then, neither x
1
2

result holds for player 2.
Proof. We show the proof for the case of the indirect domination relation , as theproof for I holds by the same argument. Let x satisfy the conditions outlined in thestatement, and suppose by way of contradiction that xM  x were to hold. By definition,
there exists a sequence of coalitions S0 , S1 , · · · , SK−1 and xM = a0 , a1 , · · · , aK = x andoutcomes such that the definition of  is satisfied. Consider the first coalition in thesequence, S 0 . Because U1 (x) ≤ m1 ≤ U1 (xM ) hold by definition of the maximin strategy
and how x is defined, S 0 = {2} – that is, S0 cannot include player 1. Consider x1 and S1 .
1
Because x11 = xM
1 , we still have U1 (x) ≤ m1 ≤ U1 (x ), which implies by definition of that S1 = {2}. By continuing this argument, we arrive that player 1 is not included in anyof the coalitions S0 , · · · , SK−1 . On the other hand, because xM
1 6= x1 and by definition of→, there must exist some k such that Sk includes player 1, which is a contradiction.

4

When Coalitional Deviations are Allowed

Lemma 2. Let x ∈ X and y ∈ X SI such that for some i ∈ {1, 2}, Ui (x) < Ui (y). Then,x  y.

9

Proof. Let i be such that Ui (x) < Ui (y) and y ∈ X SI . Then, x  y via the sequence
x →i (t̄i , xj ) →j (t̄i , t̄j ) →N y.
To check that x  y indeed holds, note that Uj (t̄i , xj ) ≤ ūj < Uj (y), and the last step
follows from the fact that y ∈ X SI and Uk (t̄i , t̄j ) = ūk for each k = i, j.
The following corollary shows that any x that is Pareto efficient and strictly individually rational is a singleton farsighted stable set and immediately follows from the previous
lemma. Because X SI ∩ E 6= ∅, the corollary also establishes the existence of a farsighted
stable set.
Corollary 1. For any x ∈ X SI ∩ E, {x} is a farsighted stable set.
Proof. Take x ∈ X SI ∩ E. Internal stability of {x} is trivial since the set is a singleton,
so to show that this set satisfies external stability, take any x0 6= x. Because x ∈ E,
there exists i ∈ N such that Ui (x0 ) < Ui (x). By Lemma 2 and the fact that x ∈ X SI ,x0  x.
The next lemma shows that there cannot exist a farsighted stable set that includes
strictly individually rational outcomes that are not Pareto efficient.
Lemma 3. For any x ∈/ E and x ∈ X SI , there cannot exist a farsighted stable set V
such that x ∈ V .
Proof. Suppose that there exists a farsighted stable set V that includes such x. Becausex∈/ E, there exists y such that yP x, which implies that there exists some i ∈ N suchthat Ui (x) < Ui (y). Moreover, since ūj < Uj (x) ≤ Uj (y) for all j ∈ N , we have thaty ∈ X SI . Thus, by Lemma 2, x  y holds. Internal stability of V implies that y ∈/ V.
By external stability of V , there exists z ∈ V such that y  z. By definition of , y  xcannot hold, which then implies that x 6= z. By definition of  and the fact that yP x,
Uj (x) ≤ Uj (y) < Uj (z) holds for some j ∈ N . If z ∈ X SI , then Lemma 2 yields x  z,contradicting the internal stability of V . If not, then for some k, Uk (z) = ūk < Uk (x).
By Lemma 2, z  x, contradicting the internal stability of V . Therefore, no such V canexist.
Theorem 1. All farsighted stable sets V are of the form {x} where x ∈ X SI ∩ E.
Proof. By Lemma 3 and Corollary 1, if there were a farsighted stable set V different
from above, then V must contain an outcome y such that y ∈/ X SI . However, y must be10

individually rational (Lemma 2.1 of Shino and Kawasaki (2012)), implying that Ui (y) = ūifor some i and Uj (y) ≥ ūj . Let i = 1 for simplicity of argument. By the assumptions
made on the utility functions, there exists a Nash equilibrium x∗ = (x∗1 , x∗2 ) of the tariff
game and x∗ ∈/ E. Moreover, x∗1 > 0 and x∗2 > 0 hold, and each strategy is a maximin
strategy for players 1 and 2 respectively. Because φi is single-valued when xj > 0,
Ui (x∗i , x∗j ) > Ui (t̄i , x∗j ) = ūi for all i. This fact establishes (x∗1 , x∗2 ) ∈ X SI , which implies
by Lemma 3 that (x∗1 , x∗2 ) ∈/ V . On the other hand, there cannot exist x ∈ V such thatx∗  x for the following reasons:• If x1 6= x∗1 , then U1 (x) = ū1 implies that x∗  x cannot hold. (Lemma 1)• If x1 = x∗1 , then x∗2 6= x2 . However, because x∗ is a Nash equilibrium, U2 (x) ≤U2 (x∗ ). We also have U1 (x) = ū1 < U1 (x∗ ), so neither player can deviate in thefirst step of .
Therefore, V does not satisfy external stability, which is a contradiction.

5

When only Individual Deviations are Allowed

In this section, we show that for the negotiation game of alternating offers, we can obtain aparallel result – every farsighted stable set is a singleton comprised of a tariff combinationthat is Pareto efficient and strictly individually rational. Although, the result itself isthe same as before, the proof of this statement for this particular model is much more
involved, since the two indirect domination relations  and I are not equivalent.
To prove our preliminary results, we need to introduce the following subsets of X ∗ :• X − = {x ∈ X : xi < 0 and Ui (x) ≤ Ui (t̄i , xj ) for all i = 1, 2; j 6= i}• X + = X ∗ \ X − = {x ∈ X ∗ : xi ≥ 0 for some i} ∪ {x ∈ X ∗ : Ui (x) > Ui (t̄i , xj ) forsome i}• X −− = {x ∈ X : xi < 0 for all i}• X ++ = X ∗ \ X −− = {x ∈ X ∗ : xi ≥ 0 for some i}That is, X − is the set of tariff profiles at which both countries subsidize each otherand with the additional property that for every x ∈ X − and every i and yi ∈ Xi withyi ≤ xi , Ui (xi , xj ) ≥ Ui (yi , xj ), which follows from Proposition 1. X + = X ∗ \ X − is thecomplement of X − and is the set of tariff profiles where the ”usual” results hold (seeLemma 4). X −− is the set of tariff rates in which both countries are subsidizing, while11

X ++ denotes one in which at least one imposes a nonnegative tariff rate. Note that, by
how these sets are defined, X ++ ⊂ X + and X − ⊂ X −− .
For this game, the analogue of Lemma 2 may not hold over all of X ∗ . To see this,recall that in the proof of Lemma 2, the last step involved a simultaneous deviation byboth players. It is then not trivial to break this step into two separate steps and stillsatisfy the conditions necessary in establishing indirect dominance. The next two lemmas
show that the result still holds for X + , while the same may not hold for X − .
Lemma 4. For any x ∈ X and y ∈ X SI ∩ X + such that Ui (x) < Ui (y) for some i,x I y.
Proof. Without loss of generality let country 1 be such that U1 (t̄1 , y2 ) < U1 (y). Suppose
country 2 is such that U2 (x) < U2 (y). Then, consider the following sequence:x = (x1 , x2 ) →2 (x1 , t̄2 ) →1 (y1 , t̄2 ) →2 (y1 , y2 ) = y.
The above sequence establishes x I y because of the following inequalities:• U1 (x1 , t̄2 ) = ū1 < U1 (y) by the fact that y ∈ X SI ,• U2 (y1 , t̄2 ) < U2 (y) by definition.
If country 1 is such that U1 (x) < U1 (y), consider the following sequence:x = (x1 , x2 ) →1 (t̄1 , x2 ) →2 (t̄1 , t̄2 ) →1 (y1 , t̄2 ) →2 (y1 , y2 ) = y.
By the same logic as above, x I y holds under this case as well.
The following result is the analogue to Corollary 1 and is immediate from the previous
lemma.
Corollary 2. For any x ∈ X SI ∩ E, {x} is a farsighted stable set with respect to I .
Proof. Internal stability is trivially satisfied because the set is a singleton. To show
external stability, take any y 6= x. Because E ⊂ X + and from the fact that x ∈ E, there
exists i ∈ N such that Ui (y) < Ui (x). Then, by Lemma 4, y I x.
Next, we wish to derive a result parallel to Lemma 3. To do so, we need to show thatthe region X − does not interfere with the logic used in the previous game. The following
lemma shows that tariff profiles in X − cannot indirectly dominate those in X ++ . At thesame time, the lemma illustrates how Lemma 2 may not hold for this model.
12Lemma 5. For any y ∈ X − , if x I y holds, then x ∈ X −− . Equivalently, for anyy ∈ X − and x ∈ X ++ , x I y cannot hold.
Proof. Suppose x I y holds with y ∈ X − . Then, there exists a sequence of players
i0 , i1 , · · · , ir and sequence of tariff profiles x0 , · · · , xK+1 with x0 = x and xK+1 = y suchthat xk →ik xk+1 for all k = 0, 1, · · · , K and Uik (xk ) < Uik (y) for all k = 0, 1, · · · , K.
Without loss of generality, we can consider a simple sequence in which the sequence of
players is alternating such that ik 6= ik+1 for all k and iK = 1. We claim that xk1 < y1for each k = 0, 1, · · · , K and xk2 < y2 for each k = 0, 1, · · · , K − 1,3 which then impliesx1 = x01 < y1 < 0 and x2 = x02 < y2 < 0. To show these inequalities, we also claim thatfor each k = 0, 1, · · · , K and i = 1, 2, Ui (y) ≤ Ui (t̄i , xkj ) where j 6= i.
The proof is by backwards induction. Consider k = K, the last step in the sequence.
Without loss of generality, let iK = 1. Then, xK = (xK
1 , y2 ) by definition of →1 . Fromy ∈ X − , we know that y2 < 0. By definition of I , we must also have U1 (xK
1 , y2 ) <K
U1 (y1 , y2 ). We first show that xK
1 < y1 holds. If x1 ≥ y1 , we must haveU1 (xK
1 , y2 ) ≥ min{U1 (t̄1 , y2 ), U1 (y1 , y2 )} ≥ U1 (y),where the first inequality follows from the quasiconcavity of U1 , and the second inequality
follows from y ∈ X − . The above inequalities contradict the definition of I . Therefore,xK
1 < y1 < 0.
To show the last part, suppose that U1 (xK ) > Ui (t̄1 , xK
2 ). Then, the following inequality holds by the fact that y ∈ X − and that xK →1 y:
U1 (y) > U1 (t̄1 , xK2 ) = U1 (t̄1 , y2 ) ≥ U1 (y),which then contradicts the second condition in the definition of I . Therefore, we mustK
have U1 (y) ≤ U1 (t̄1 , xK
2 ). Similarly, if U2 (y) > U2 (x1 , t̄2 ), then the following inequalities

hold, which lead to a contradiction:
U2 (y) > U2 (xK
1 , t̄2 ) > U2 (y1 , t̄2 ) ≥ U2 (y).
Therefore, we must also have U2 (y) ≤ U2 (xK
1 , t̄2 ).
Now, suppose that xk1 < y1 and xk2 < y2 and Ui (y) ≤ Ui (t̄i , xkj ) holds for each i = 1, 2
and j 6= i. We first show that xk−1
< y1 and xk−1
< y2 . Without loss of generality,
1
2
suppose that ik−1 = 2. Then, xk−1
= xk1 < y1 holds immediately. If xk−1
≥ y2 , then we1
2
3

This inequality is not considered when K = 0.

13have the following inequalities:kk
U2 (xk−1 ) = U2 (xk1 , xk−12 ) ≥ min{U2 (x1 , t̄2 ), U2 (x1 , y2 )} ≥ U2 (y).

The first inequality holds since U2 is quasiconcave in its own argument. The second
inequality holds from the assumption that y ∈ X − (which implies U2 (xk1 , t̄2 ) ≥ U2 (y))
and that xk1 < y1 from the induction hypothesis and U2 being decreasing in the tariff
rate of country 1 imply U2 (xk1 , y2 ) ≥ U2 (y). On the other hand, the definition of I
implies that U2 (xk−1 ) < U2 (y), which leads to a contradiction. Therefore, we must havexk−1
< y2 .
2
To complete the induction argument, we need to show that Ui (y) ≤ Ui (t̄i , xjk−1 ) for
each i. First, consider i = 1 and suppose that U1 (y) > U1 (t̄1 , xk−1
2 ). Then, we have thefollowing contradictory set of inequalities
U1 (y) > U1 (t̄1 , xk−12 ) > U1 (t̄1 , y2 ) ≥ U1 (y),where the second inequality follows from the fact that U1 is decreasing in country 2’s
tariff and that xk−1 < y2 holds from the previous part. Thus, U1 (y) ≤ U1 (t̄1 , xk−1
2 ).
Similarly, for i = 2,U2 (y) > U2 (xk−1
1 , t̄2 ) > U2 (y1 , t̄2 ) ≥ U2 (y).
A symmetric argument can be used in the case that ik−1 = 1. Therefore, we have shownthat the claim holds.
Recall that the goal is to derive a result concerning the exclusion of Pareto inferiorand strictly individually rational outcomes from any farsighted stable sets. The next
result shows that we can take a tariff profile from the set X ++ that Pareto dominates
the inefficient ones without any loss of generality.
Lemma 6. For any x ∈/ E, there exists y ∈ X ++ such that yP x.4
Proof. Take x ∈/ E. Then, there exists y ∈ X such that yP x. If y ∈ X ++ , we aredone. If y ∈ X −− but y is such that Ui (y) ≤ ūi for each i, then we can then takez ∈ X SI ∩ X ∗ (⊂ X SI ∩ X ++ ) such that zP y, which in turn implies zP x. Thus, consider
the case in which y ∈ X −− . Because y ∈ X −− , and E ⊂ X ++ , y ∈ L(E ∗ ). Consider an4

According to the proof of the main theorem of Oladi (2005), each Pareto inefficient outcome is Pareto
dominated by some Pareto efficient tariff combination. However, this fact is stated without a proof in
Oladi (2005) and is claimed to follow from the definition of Pareto efficiency. For completeness, we provide
a proof of a weaker statement here, since this version suffices for the proof of the results that follow.

14

indifference curve for player 1 through y. This indifference curve must cross E ∗ at somepoint. Label this intersection as z. We claim that zP y which implies that z ∈ X ++ andzP x hold.
First, note that y1 < z1 . Otherwise, if y1 > z1 ,5 we must then have y2 ≤ z2 . To see
this, if y2 > z2 , thenU1 (y1 , y2 ) > U1 (z1 , y2 ) ≥ U1 (z1 , z2 ).
The first inequality follows from Proposition 1, and the second inequality follows from thefact that U1 is decreasing in player 2’s tariffs. These inequalities imply that U1 (y) 6= U1 (z),but y and z lie on the same indifference curve, which lead to a contradiction. Thus,y2 ≤ z2 .
Now, y1 > z1 and y2 ≤ z2 imply the following inequalities
U2 (y1 , y2 ) ≤ U2 (y1 , z2 ) < U2 (z1 , z2 ),where the first inequality follows from Proposition 1, and the second inequality follows
from U2 being a decreasing function of player 1’s tariffs. On the other hand, assumption(A2) implies that U2 (y) > U2 (z), which contradicts the above inequality. Thus, y1 < z1 .
Now, by (A2), U2 (y) < U2 (z). Thus, with U1 (y) = U1 (z) and U2 (y) < U2 (z), we canconclude that zP y.
Now, we finally have the following analogue of Lemma 3.
Lemma 7. For any x ∈ X SI with x ∈/ E, there does not exist a farsighted stable set V
with respect to I such that x ∈ V .
Proof. First, we show that there cannot exist a farsighted stable set V that includes anx with x ∈ X SI ∩ X + and x ∈/ E. Suppose that there exists a farsighted stable set V
such that x ∈ V . Because x is not Pareto efficient, there exists y such that Ui (x) < Ui (y)for i = 1, 2. Also, we can find such a y with y ∈ X ++ by Lemma 6. Because x ∈ X SI ,y ∈ X SI must hold as well. Then, by Lemma 4, x I y, which in turn implies y ∈/ V byinternal stability. Thus, by external stability of V , there exists z ∈ V such that y I z.
Moreover, by definition of I , y I x cannot hold, which implies that we must havex 6= z. By Lemma 5, z ∈ X + . By definition of I , there exists some j ∈ {1, 2} such that
Uj (x) < Uj (y) < Uj (z). If z ∈ X SI , then x I z. If z ∈/ X SI , then because x ∈ X SI ,there exists some j such that Uj (z) = ūj < Uj (x). By Lemma 4, since x ∈ X + ∩ X SI ,z I x. In both cases, internal stability of V is violated. Therefore, there cannot exist afarsighted stable set V such that x ∈ V .
5

y1 = z1 is ruled out since z ∈ X ++ and y ∈ X −− and U1 is strictly decreasing in player 2’s tariffs.

15

To complete the proof, consider now x ∈ X SI ∩ X − and x ∈/ E. Then, by Lemma 6,
there exists y ∈ X ++ that Pareto dominates x. By the same logic as above, y ∈/ V , which
implies that by external stability, there exists some z ∈ V such that y I z. However,by Lemma 5, z ∈ X + . If z ∈ X SI , then x I z, which contradicts the internal stabilityof V . If z ∈/ X SI , by the fact that V can only include individually rational outcomes,
for some i, we must have Ui (z) = ūi . However, since x ∈ X SI , Ui (z) < Ui (x). By thesame logic as before, there exists j with Uj (x) < Uj (z), Thus, j 6= i. Moreover, we havethat since z ∈ X + , Uj (zi , t̄j ) < Uj (z) because Ui (t̄i , zj ) ≥ ūi = Ui (z) holds. To see thelatter inequality, if zj ≥ 0, then Ui (t̄i , zj ) = ūi , which implies in turn that if zj < 0, thenUi (t̄i , zj ) > Ui (t̄i , 0) = ūi . Then, x I z through the following sequence, resulting in theviolation of internal stability:x = (xi , xj ) →j (xi , t̄j ) →i (zi , t̄j ) →j (zi , zj ) = z.
To check that x I z indeed holds, we need to only check that Ui (xi , t̄j ) < Ui (z) holds,
but this fact can be checked by the fact that Ui (xi , t̄j ) < Ui (0, t̄j ) = ūi = Ui (z) sincexi < 0.
We can now state the main result of this section. The proof of the theorem follows
from the same logic as Theorem 1 as Lemma 7 is the same as Lemma 3 and Corollary 2
is the same as Corollary 1.
Theorem 2. If V is a farsighted stable set with respect to I , then V = {x} for somex ∈ X SI ∩ E.

6

Concluding Remarks

We have shown in this paper that farsightedness can lead to efficiency in tariff games
regardless of how whether coalitional deviations are allowed or not. Moreover, we havemade no additional assumptions, including those to the effectiveness relation, to themodel of Nakanishi (2000) to obtain efficiency results. In addition, we have ruled out
outcomes that are not individually rational. We can exclude outcomes in which one
country is subsidizing too much, while under the basic WTO framework in Nakanishi(2000), this was not possible. We have seen how drastic the effect of the assumption offarsightedness alone has on the stability of the tariff choices of the countries.
While the main focus of this article was on tariff games, we can also consider the export
quota retaliation game of Nakanishi (1999). It can be easily shown that the arguments
used here can be applied almost in a straightforward manner to show a similar result.
16That is, in the quota game, if both countries are sufficiently farsighted, all farsighted
stable sets are singleton sets consisting of quota choices of both countries that are Pareto
efficient, which in this game also imply strict individual rationality. The results holdregardless of whether coalitional deviations are allowed or not.

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