Online Supplementary Notes for“Multilateral Trade Bargaining:

A First Look at theGATT Bargaining Records”

Kyle Bagwell Robert W. Staiger Ali Yurukoglu

August 2015

These notes utilize the trade model presented in our main paper (Bagwell,Staiger andYurukoglu, 2015) and develop our findings for bargaining amongthree countries (with one domestic country and two foreign countries) whenproposals must satisfy MFN and multilateral reciprocity and when countriesuse dominant strategies.1 The notes are organized into four sections, whichrespectively (i) construct a mechanism that maps tariff proposals satisfyingMFN and multilateral reciprocity into assigned tariffs, (ii) characterize dominantstrategy proposals for each country, (iii) examine the effi ciency properties of theresulting bargaining outcomes, and (iv) discuss extensions concerning multiplecountries, private information and alternative prioritization rules such as theprincipal supplier rule.2

1 Multilateral Reciprocity and the ConstructedMechanism

Setup For simplicity, we focus on the model with three countries. Let theinitial tariff vector be (τ0, τ

∗10 , τ

∗20 ), where as in the main paper and throughout

1The analysis below is related to that in Bagwell and Staiger (1999), but connects moredirectly to the GATT bargaining data since each country proposes a tariff for itself as well astariff(s) for each trading partner. Bagwell and Staiger (1999) assume that the home countryproposes its own tariff policy as well as trade shares from foreign countries and then use thisproposal and the reciprocity restriction to arrive at implied foreign tariffs. Similarly, theyassume that each foreign country proposes a tariff for itself and then use all foreign tariffproposals and the reciprocity restriction to arrive at implied home tariffs.

2We present our initial findings in a complete-information setting in order to utilize themodel and notation from the main paper. With this foundation in place, the extension toinclude private information is easily described. The foundation likewise facilitates discussionof the other extensions.

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these notes the home-country tariff satisfies MFN. Let the initial world price bepw0 ≡ p̃w(τ0, τ

∗10 , τ

∗20 ).

Strategies The game form involves simultaneous proposals by all three coun-tries, where each country can only make proposals concerning its own tariff andthat of its trading partner(s) and where each proposal if accepted must maintainthe world price. Formally, the respective strategy spaces are:

H’s strategy: A proposal Th ≡ (Thh , T∗1h , T ∗2h ) ∈ <3+ such that pw0 ≡ p̃w(Thh , T

∗1h , T ∗2h ).

∗1’s strategy: A proposal T∗1 ≡ (Th∗1, T∗1∗1 , T

∗2∗1 ) ∈ <3+ such that pw0 ≡ p̃w(Th∗1, T

∗1∗1 , T

∗2∗1 )

and T ∗2∗1 = τ∗20 .

∗2’s strategy: A proposal T∗2 ≡ (Th∗2, T∗1∗2 , T

∗2∗2 ) ∈ <3+ such that pw0 ≡ p̃w(Th∗2, T

∗1∗2 , T

∗2∗2 )

and T ∗1∗2 = τ∗10 .

Notice that the subscript here indicates the party who is making the proposalwhile the superscript indicates the party whose policy is being proposed. LetSh, S∗1 and S∗2 denote the respective strategy spaces so defined, and let S ≡Sh × S∗1 × S∗2.Thus, once the simultaneous proposals are made, we have the following ob-

jects:

Th ≡ (Thh , T∗1h , T ∗2h ), T∗1 ≡ (Th∗1, T

∗1∗1 , T

∗2∗1 ) and T∗2 ≡ (Th∗2, T

∗1∗2 , T

∗2∗2 ), where

pw0 ≡ p̃w(Thh , T∗1h , T ∗2h ) = p̃w(Th∗1, T

∗1∗1 , T

∗2∗1 ) = p̃w(Th∗2, T

∗1∗2 , T

∗2∗2 ),

T ∗2∗1 = τ∗20 and T ∗1∗2 = τ∗10 .

Equivalence Class For any foreign country, there will be tariff adjustmentsby home and the other foreign country that leave unaltered the world price (i.e.,that satisfy bilateral reciprocity) and that thus leave the former foreign countryindifferent. In recognition of such policies, we define an equivalence class oftariffs for each foreign country.

Definition 1 Given T∗1 ≡ (Th∗1, T∗1∗1 , T

∗2∗1 ) and pw0 ≡ p̃w(T∗1), we may define

an equivalence class for foreign country ∗1 as a tariff set

EC∗1(T∗1) ≡ {T̂∗1 ≡ (T̂h∗1, T̂∗1∗1 , T̂

∗2∗1 )}

that satisfies the requirements that (i) T̂ ∗1∗1 = T ∗1∗1 and (ii) p̃w(T̂∗1) = pw0 . Like-

wise, given T∗2 ≡ (Th∗2, T∗1∗2 , T

∗2∗2 ) and pw0 ≡ p̃w(T∗2), we may define an equiva-

lence class for foreign country ∗2 as a tariff set

EC∗2(T∗2) ≡ {T̂∗2 ≡ (T̂h∗2, T̂∗1∗2 , T̂

∗2∗2 )}

that satisfies the requirements that (i) T̂ ∗2∗2 = T ∗2∗2 and (ii) p̃w(T̂∗2) = pw0 .

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Thus, an equivalence class for foreign country ∗i maintains ∗i’s proposedtariff for itself and allows for alternative tariffs for home and foreign country∗j, where j 6= i, such that these alternative tariffs when joined with ∗i’s pro-posed tariff for itself serve to maintain the initial world price. Notice that anequivalence class is not empty, since T∗i ∈ EC∗i(T∗i). Next, we observe thatwe can reach any member of EC∗i(T∗i) by fixing T̂ ∗i∗i = T ∗i∗i and then allowingchanges from (Th∗i, T

∗j∗i ) to (T̂h∗i, T̂

∗j∗i ) that satisfy bilateral reciprocity between

home and foreign country ∗j and that thus maintain the initial world price. Fi-nally, let e∗i(T∗i) denote a representative member of EC∗i(T∗i). Since T∗i andany e∗i(T∗i) ∈ EC∗i(T∗i) generate the same world price pw0 and local price inforeign country ∗i (as ∗i’s own tariff is unaltered by requirement (i)), it followsthat T∗i and any e∗i(T∗i) ∈ EC∗i(T∗i) generate the same economic magnitudes(e.g., trade volume) and government welfare for foreign country ∗i.

Implied Import Volumes Each country’s proposal can be associated withan implied import volume for itself. We now introduce some convenient notationwith which to represent for each country the import volume that is implied byits proposal.

Definition 2 The home country’s proposal Th is associated with an impliedimport volume for home defined as

Mh ≡M(p(Thh , pw0 ), pw0 ).

Similarly, foreign country ∗i’s proposal T∗i is associated with an implied importvolume for foreign country ∗i defined as

M∗i ≡M∗i(p∗i(T ∗i∗i , pw0 ), pw0 ).

Notice that, given the initial world price, each country’s implied import volumedepends only on that country’s proposed tariff for its own imports. Notice alsothat, for any foreign country ∗i, any member of EC∗i(T∗i) entails the same tarifffor foreign country ∗i and the same world price, and so generates as well thesame implied import volume for foreign country ∗i.

Agreement We now define agreement between the three tariff proposals.

Definition 3 The proposals {Th, T∗1, T∗2} agree iff there exist e∗1(T∗1) ∈ EC∗1(T∗1)and e∗2(T∗2) ∈ EC∗2(T∗2) such that Th = e∗1(T∗1) = e∗2(T∗2).

Thus, home’s proposal must be the agreed-upon proposal, and it must be pos-sible to reach home’s proposal from each foreign country proposal.3

3An altnernative approach would be to define agreement utilizing an equivalence classnotion for the home country as well. Given Th ≡ (Thh , T ∗1h , T ∗2h ) and pw0 ≡ p̃w(Th), we coulddefine an equivalence class for the home country as a tariff set

ECh(Th) ≡ {T̂h ≡ (T̂hh , T̂ ∗1h , T̂ ∗2h )}

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Example 4 Suppose that the initial tariffs are (τ0, τ∗10 , τ

∗20 ) = (15, 15, 15) and

that the initial world price is pw0 = 1. Suppose further that home’s proposal isTh = (5, 10, 10), which means that home proposes to cut its tariff by 10 in ex-change for cuts of 5 by both of its trading partners. The underlying assumptionin the example is that the world price is preserved when one unit of home liberal-ization is balanced against one unit of liberalization from some foreign country.Now consider foreign country ∗1’s proposal. If agreement is to obtain, foreigncountry i’s proposal for its own tariff must match home’s proposal for foreigncountry i’s tariff. So, we require T ∗1∗1 = 10. Also, we know that foreign coun-try ∗1’s proposal for foreign country ∗2’s tariff simply entails leaving that tariffat its initial level, so we also require T ∗2∗1 = 15. We have left to specify foreigncountry ∗1’s proposed tariff for home, Th∗1. But in fact we have no remaining de-grees of freedom here, since foreign country ∗1’s proposed tariff for home is nowuniquely determined by the requirement that pw0 = p̃w(Th∗1, T

∗1∗1 = 10, T ∗2∗1 = 15).

In particular, we get that Th∗1 = 10 as then pw0 ≡ p̃w(10, 10, 15). Now, givenT∗1 = (10, 10, 15), we can reach Th = (5, 10, 10) by having home and foreigncountry ∗2 exchange reciprocal tariff cuts of 5 units, which preserves the worldprice and leaves foreign country ∗1 indifferent. In other words, Th ∈ EC∗1(T∗1).We thus conclude that Th = (5, 10, 10) and T∗1 = (10, 10, 15) agree in this sense.Using a similar argument for foreign country ∗2, we see that agreement impliesas well that T∗2 = (10, 15, 10). Further, it is now evident that the proposals{Th = (5, 10, 10), T∗1 = (10, 10, 15), T∗2 = (10, 15, 10)} agree. As this exampleillustrates, for a given home proposal, agreement uniquely determines the foreignproposals.

We may now report the following implication of agreement:

Lemma 5 The proposals {Th, T∗1, T∗2} agree only if pw0Mh = M∗1 +M∗2.

Proof: The home proposal Th implies the market-clearing world price of p̃w(Th) =pw0 and an implied import volume for home of Mh = M(p(Thh , p

w0 ), pw0 ). Under

trade balance for home and market clearing for good y, we have that

pw0Mh = M∗1(p∗1(T ∗1h , pw0 ), pw0 ) +M∗2(p∗2(T ∗2h , pw0 ), pw0 ). (1)

Suppose that the proposals agree. Then we know that Th = e∗1(T∗1) = e∗2(T∗2)for some e∗1(T∗1) ∈ EC∗1(T∗1) and e∗2(T∗2) ∈ EC∗2(T∗2). Thus, for i = 1, 2,

that satisfies the requirements that (i) T̂hh = Thh and (ii) p̃w(T̂h) = pw0 . Then, we coulddefine an alternative notion for agreement and say that the proposals {Th, T∗1, T∗2} agree iffthere exist eh(Th) ∈ ECh(Th), e∗1(T∗1) ∈ EC∗1(T∗1) and e∗2(T∗2) ∈ EC∗2(T∗2) such thateh(Th) = e∗1(T∗1) = e∗2(T∗2). This alternative definition is too demanding, however, and issatisfied only by the initial tariff vector, (τ0, τ∗10 , τ

∗20 ). This follows because the alternative

definition of agreement requires that each country’s proposal for its own tariff is agreed upon;further, each foreign country is restricted to propose the initial tariff for the other foreigncountry. Thus, under the alternative definition, agreement is feasible only if each foreigncountry proposes its initial tariff for itself. Since the initial world price must be maintained,the home country must propose its initial tariff for itself as well.

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the home tariff proposal for the tariff of foreign country ∗i equals the tariffproposal that foreign country ∗i makes for its own tariff: T ∗ih = T ∗i∗i . It nowfollows from (1) that

pw0Mh = M∗1(p∗1(T ∗1∗1 , pw0 ), pw0 ) +M∗2(p∗2(T ∗2∗2 , p

w0 ), pw0 ).

Referring again to the definition of implied import volumes, we now see thatpw0Mh = M∗1 +M∗2, which completes the proof.

The lemma does not hold in the other direction. Suppose that pw0Mh =M∗1 + M∗2. In terms of our example, this might occur if the proposals are{Th = (5, 10, 10), T∗1 = (12, 12, 15), T∗2 = (8, 15, 8)} so that foreign country∗1 proposes a 3 unit liberalization while foreign country ∗2 proposes a 7 unitliberalization. The low value for M∗1 is then offset by a high value for M∗2.Each foreign proposal preserves the world price and is feasible. While it seemsthat these proposals could satisfy pw0Mh = M∗1 + M∗2, there is no possibilitythat they agree since each foreign country proposes a different tariff for itselfthan the home country proposes for that country.

Mechanism We define amechanism as a pair (S, g(·)), where S is the strategyspace as defined above and g is an outcome function that maps from a vector oftariff proposals, (Th, T∗1, T∗2), to a vector of tariffs, (τ , τ∗1, τ∗2). Given a vectorof tariff proposals, a mechanism thus assigns ( or selects) a tariff vector forapplication. We impose two baseline rules (or requirements) for the mechanismthat we construct. First, if the tariff proposals agree, then we require that themechanism assign the home tariff proposal. We know from Lemma 5 that thissituation of agreement can arise only if the home country and foreign countries(in aggregate) propose the same value of import volumes. Second, if the tariffproposals do not agree, then we require that the mechanism assigns a tariffvector that maximizes trade volume valued at world prices (with good y asthe numeraire) while not forcing any country to import more than its impliedimport volume and while preserving the initial world price. As we will see,under disagreement, the baseline rules do not uniquely determine the outcomefunction, and so we will add further rules below to ensure a unique mapping forour constructed mechanism. For now, we note that there are three ways thatdisagreement may occur: the home country may be on the long side (pw0Mh >M∗1 +M∗2), the home country may be on the short side (pw0Mh < M∗1 +M∗2),or agreement may fail (as illustrated above) even though the aggregate importvolumes match (pw0Mh = M∗1+M∗2). We address each of these cases and definecorresponding assignment rules. Then, in the next section, we characterizedominant strategies for countries when the resulting constructed mechanism isused.

Home Long: We begin with the case in which the tariff proposals are suchthat the home country is on the long side:

pw0Mh > M∗1 +M∗2. (2)

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In this case, given the tariff proposals, we maximize the value of trade volumewhile respecting implied import volume limits and maintaining the initial worldprice by assigning the tariff vector

(τ̃ , T ∗1∗1 , T∗2∗2 ) (3)

where τ̃ = τ̃(T ∗1∗1 , T∗2∗2 ) is defined to satisfy

p̃w(τ̃ , T ∗1∗1 , T∗2∗2 ) = pw0 . (4)

Notice here that, conditional on (2) holding, the assignment rule uses only theforeign proposals.The assigned tariff vector clearly maintains the initial world price. It also

satisfies the constraint that implied import volumes are not exceeded. To seethis, observe that foreign country ∗i imports

M∗i(p∗i(T ∗i∗i , pw0 ), pw0 ) = M∗i. (5)

Next, the home country imports

M(p(τ̃ , pw0 ), pw0 ) = [1/pw0 ]E(p(τ̃ , pw0 ), pw0 ) (6)

= [1/pw0 ]∑i=1,2

M∗i(p∗i(T ∗i∗i , pw0 ), pw0 )

= [1/pw0 ][M∗1 +M∗2]

< Mh,

where the first equality follows from trade balance for the home country, thesecond equality follows from market clearing in good y, the third equality uses(5), and the inequality employs (2).Finally, we confirm that, given the implied import volume limits and world

price, the assigned tariff vector also maximizes the value of trade volume.To see this, we consider an arbitrary vector of tariffs, (τ , τ∗1, τ∗2), for whichp̃w(τ , τ∗1, τ∗2) = pw0 and such that no country imports a volume in excess of itsimplied import volume. We then define the associated value of trade volume as

TV (τ , τ∗1, τ∗2) ≡ pw0M(p(τ , pw0 ), pw0 ) +∑i=1,2

M∗i(p∗i(τ∗i, pw0 ), pw0 ). (7)

Using trade balance for the home country, market clearing for good y, and (5),we obtain

TV (τ , τ∗1, τ∗2) = E(p(τ , pw0 ), pw0 ) +∑i=1,2

M∗i(p∗i(τ∗i, pw0 ), pw0 ) (8)

= 2[M∗1(p∗1(τ∗1, pw0 ), pw0 ) +M∗2(p∗2(τ∗2, pw0 ), pw0 )]

≤ 2[M∗1 +M∗2]

= 2[M∗1(p∗1(T ∗1∗1 , pw0 ), pw0 ) +M∗2(p∗2(T ∗2∗2 , p

w0 ), pw0 )]

= TV (τ̃ , T ∗1∗1 , T∗2∗2 ),

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where the inequality follows from the restriction that imported volumes forforeign countries not exceed their respective implied import volume limits. Sinceby (5) the assigned tariff vector ensures that both foreign countries importvolumes that equal their respective implied import volume limits, the assignedvector of tariffs thus achieves the maximum value for the value of trade volume.In fact, given the proposals, any distinct tariffvector (τ , τ∗1, τ∗2) 6= (τ̃ , T ∗1∗1 , T

∗2∗2 )

for which p̃w(τ , τ∗1, τ∗2) = pw0 and such that no country imports a volume inexcess of its implied import volume must deliver strictly less trade volume:TV (τ , τ∗1, τ∗2) < TV (τ̃ , T ∗1∗1 , T

∗2∗2 ). This is because the proposal can be distinct

while maintaining the initial world price only if (τ∗1, τ∗2) 6= (T ∗1∗1 , T∗2∗2 ). It fol-

lows that (τ , τ∗1, τ∗2) implies a distinct trade volume for at least one foreigncountry; thus, since M∗i(p∗i(T ∗i∗i , p

w0 ), pw0 ) = M∗i and M∗i(p∗i(τ∗i, pw0 ), pw0 ) ≤

M∗i for all i = 1, 2, we may conclude that TV (τ , τ∗1, τ∗2) < TV (τ̃ , T ∗1∗1 , T∗2∗2 ) =

2[M∗1+M∗2]. Consequently, our baseline rules are suffi cient to ensure the uniquedetermination of the assigned tariff vector when the tariff proposals satisfy (2).In particular, for such tariff proposals, our constructed mechanism must assignthe tariff vector (τ̃ , T ∗1∗1 , T

∗2∗2 ) where τ̃ is defined by (4).

Home Match: We next consider the case in which the tariff proposals are suchthat the home country matches:

pw0Mh = M∗1 +M∗2. (9)

This case is analogous to the case in which the home country is long. Given thetariff proposals, we again maximize the value of trade volume while respectingimplied import volume limits and maintaining the initial world price by assigningthe tariffvector (3) where as before τ̃ = τ̃(T ∗1∗1 , T

∗2∗2 ) is defined to satisfy (4). The

assigned foreign tariffs again satisfy (5), so that each foreign country importsits implied import volume. Given (9), we may now repeat the steps in (6) withthe exception that the final inequality there now takes the form of an equality.Thus, we now have that

M(p(τ̃ , pw0 ), pw0 ) = Mh, (10)

which is to say that the home country also imports its implied import volume.Since the world price is fixed at the initial value, pw0 , we may conclude that theassigned tariff vector in (3) uses the proposals made by each country for its owntariff:

(τ̃ , T ∗1∗1 , T∗2∗2 ) = (Thh , T

∗1∗1 , T

∗2∗2 ) (11)

Of course, as noted previously, this conclusion does not imply that the propos-als agree, since, for example, the home country may propose a tariff for someforeign country ∗i that differs from the tariff that foreign country ∗i proposesfor itself. Finally, given the proposals, we may argue as in the case in which thehome country is long that the assigned tariff vector now given in (11) uniquelymaximizes the value of trade volume among tariff vectors that deliver the initialworld price and that do not require any country to import a volume in excess

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of its implied import volume. Thus, our baseline rules are suffi cient to ensurethe unique determination of the assigned tariff vector when the tariff proposalssatisfy (9). In particular, for such tariff proposals, our constructed mechanismmust assign the tariff vector (τ̃ , T ∗1∗1 , T

∗2∗2 ) = (Thh , T

∗1∗1 , T

∗2∗2 ).

Home Short: The final case of disagreement occurs when the tariff proposalsare such that the home country is short:

pw0Mh < M∗1 +M∗2. (12)

An initial point is that under our baseline rules we cannot now assign tariffsthat achieve the implied import volumes for the foreign countries, since to doso would violate the implied import volume limit for the home country. To seethis point, let us assume to the contrary that we assign an applied tariff vector,(τ , τ∗1, τ∗2), for which p̃w(τ , τ∗1, τ∗2) = pw0 and M

∗i(p∗i(τ∗i, pw0 ), pw0 ) = M∗i foreach i. Using trade balance for the home country, market clearing for good y,M∗i(p∗i(τ∗i, pw0 ), pw0 ) = M∗i for each i, and (12), we then obtain

pw0M(p(τ , pw0 ), pw0 ) = E(p(τ , pw0 ), pw0 )

=∑i=1,2

M∗i(p∗i(τ∗i, pw0 ), pw0 )

= M∗1 +M∗2

> pw0Mh,

which means that the import volume for home under this tariff vector mustexceed the home country’s implied import volume limit: M(p(τ , pw0 ), pw0 ) > Mh.Hence, the tariff vector that we assign must be such that at least one foreigncountry imports a volume that is strictly lower than its implied import volume.Our discussion to this point suggests that our assignment for this case will

be such that the home country imports a volume that equals that implied by itsproposal: M(p(τ , pw0 ), pw0 ) = Mh. At least one foreign country will then importa volume that falls strictly below its implied import volume. In fact, there area continuum of possible ways to allocate a fixed value of trade volume, pw0Mh,across the two foreign countries, even while maintaining the initial world priceand ensuring that no foreign country imports a volume that exceeds its impliedimport volume. We require additional rules, therefore, if we seek a basis for aunique assigned tariff vector when the tariff proposals are such that the homecountry is short.One approach might be to construct a mechanism that assigns tariffs so

that foreign countries split the difference, with both foreign countries importingless than the volumes implied by their respective proposals. Looking aheadtoward our dominant-strategy arguments, however, a potential danger with thisapproach is that a foreign country might overstate its desired import volume inorder to diminish the extent to which its assigned import volume falls short ofthe import volume that it actually prefers.4 We thus pursue a different approach

4For example, if pw0Mh < M∗1 +M∗2 with M∗1 = M∗2, then one approach might assign

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here. The approach that we use is in the spirit of “serial dictator”assignmentschemes. The general idea is to pick a foreign country at random, and specify forthat country a tariff that delivers its implied import volume provided that thevalue of that volume is no greater than the value of the home country’s impliedimport volume. Otherwise, the selected foreign country imports a volume equalin value to the home country’s implied import volume. The tariff of the otherforeign country is set so as to import the remaining value, if any, of the homecountry’s implied import volume. With these assignments in place, the homecountry’s tariff is then set so as to deliver the initial world price as the market-clearing world price. As we establish below, the home country’s proposed tarifffor itself is then the home country tariff that delivers the initial world price.Suppose then that the proposals are such that (12) holds and that foreign

country ∗i is randomly selected as the “first” country. In the assigned tariffvector, foreign country ∗i then sets the tariff τ∗i such that

M∗i(p∗i(τ∗i, pw0 ), pw0 ) = min{M∗i, pw0Mh}. (13)

There are thus two cases, which we consider in turn.The first case arises if

M∗i ≥ pw0Mh. (14)

For this case, τ̂∗i is set so as to satisfy

M∗i(p∗i(τ∗i, pw0 ), pw0 ) = pw0Mh. (15)

Next, τ̂∗j is set at a prohibitive level, so that

M∗j(p∗j(τ∗j , pw0 ), pw0 ) = 0.5 (16)

Finally, we define τ̂ so that the world price is maintained, given these foreigntariffs:

p̃w(τ̂ , τ̂∗1, τ̂∗2) = pw0 . (17)

This assigned tariff vector, (τ̂ , τ̂∗1, τ̂∗2), delivers the initial world price andis also such that no country imports a volume that exceeds its implied importvolume. It is immediately clear that neither foreign country imports a volumethat exceeds its implied import volume. The home country imports a volumethat equals its implied import volume. To see this, we respectively use trade

tariffs such that foreign country ∗i imports pw0Mh/2 < M∗i. If the mechanism further specifiesthat foreign country ∗i achieves its implied import volume limit when it instead makes aproposal that implies the import volume limit of M

′∗i = M∗i + ε, then foreign country ∗i

would have incentive to propose for itself a lower tariff that implies the import volume M′∗i,

even if its preferred volume is M∗i, since M′∗i = M∗i + ε is closer to M∗i than is pw0Mh/2.

Hence, when the home country is short, dominant strategy implementation may fail undersome natural (Bertrand-like) assignment rules.

5Notice that we assume here that, given the initial world price, there exists a finite tarifffor foreign country ∗j at and above which import volume into foreign country ∗j is zero.

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balance for the home country, market clearing for good y, (16) and (15) toobtain

pw0M(p(τ̂ , pw0 ), pw0 ) = E(p(τ̂ , pw0 ), pw0 ) (18)

= M∗i(p∗i(τ̂∗i, pw0 ), pw0 ) +M∗j(p∗j(τ̂∗j , pw0 ), pw0 )

= M∗i(p∗i(τ̂∗i, pw0 ), pw0 )

= pw0Mh,

and thusM(p(τ̂ , pw0 ), pw0 ) = Mh. Given this equality, it now follows that τ̂ = Thh ;thus, in the first case, the tariff that is assigned to the home country is the homecountry’s proposed tariff for itself.Finally, we confirm that it is not possible to find another tariff vector that

generates a greater value for trade volume while also delivering the initial worldprice and ensuring that no country imports a volume in excess of its implied im-port volume. To see this, we consider an arbitrary vector of tariffs, (τ , τ∗1, τ∗2),for which p̃w(τ , τ∗1, τ∗2) = pw0 and such that no country imports a volume in ex-cess of its implied import volume. We then define the associated value of tradevolume TV (τ , τ∗1, τ∗2) as in (7). Using trade balance for the foreign countries,market clearing for good x, the requirement that the import volume of the homecountry not exceed its implied import volume, and (18), we obtain

TV (τ , τ∗1, τ∗2) = pw0M(p(τ , pw0 ), pw0 ) + pw0∑i=1,2

E∗i(p∗i(τ∗i, pw0 ), pw0 )

= 2pw0M(p(τ , pw0 ), pw0 )

≤ 2pw0Mh

= 2pw0M(p(τ̂ , pw0 ), pw0 )

= TV (τ̂ , τ̂∗1, τ̂∗2).

For the first case, the assigned tariff vector thus achieves the maximum valuefor the value of trade volume, given the initial world price and the restrictionthat no country imports a volume in excess of its implied import volume.We note in this first case that our baseline rules do not uniquely identify an

assigned tariffvector; for example, we could achieve the same trade volume whilesatisfying the other constraints by slightly increasing (lowering) the implementedtariff for foreign country ∗i ( foreign country ∗j) in a fashion that maintains theaggregate implied import volume for the two foreign countries.6 Thus, when theproposals induce this first case, we impose some additional rules in constructingour mechanism so as to arrive at the assigned tariff vector, (τ̂ , τ̂∗1, τ̂∗2).The second case arises if

M∗i < pw0Mh. (19)

For this case, τ∗i is set so as to satisfy

M∗i(p∗i(τ∗i, pw0 ), pw0 ) = M∗i, (20)6We could also achieve the same outcome by simply raising the assigned tariff for foreign

country ∗j, since foreign country ∗j’s trade volume would also be prohibited at a higher tarifflevel.

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from which it follows that τ∗i = T ∗i∗i . Next, τ∗j is set so that

M∗j(p∗j(τ∗j , pw0 ), pw0 ) = min{M∗j , pw0Mh −M∗i} = pw0Mh −M∗i, (21)

where the second equality follows from (12). Finally, we define τ so that theinitial world price is maintained, given these foreign tariffs:

p̃w(τ , τ∗1, τ∗2) = pw0 . (22)

This assigned tariff vector, (τ , τ∗1, τ∗2), delivers the initial world price andis also such that no country imports a volume that exceeds its implied importvolume. It is immediately clear that neither foreign country imports a volumethat exceeds its implied import volume. The home country imports a volumethat equals its implied import volume. To see this, we respectively use tradebalance for the home country, market clearing for good y, (20) and (21) toobtain

pw0M(p(τ , pw0 ), pw0 ) = E(p(τ , pw0 ), pw0 ) (23)

= M∗i(p∗i(τ∗i, pw0 ), pw0 ) +M∗j(p∗j(τ∗j , pw0 ), pw0 )

= M∗i +M∗j(p∗j(τ∗j , pw0 ), pw0 )

= M∗i + pw0Mh −M∗i

= pw0Mh,

and thusM(p(τ , pw0 ), pw0 ) = Mh. Given this equality, it now follows that τ = Thh ;thus, in the second case as well, the home country’s assigned tariff is the tariffthat it proposed for itself.Finally, we confirm that it is not possible to find another tariff vector that

generates a greater value for trade volume while also delivering the initial worldprice and ensuring that no country imports a volume in excess of its impliedimport volume. To see this, we employ a similar argument to that above for thefirst case. Specifically, we consider an arbitrary vector of tariffs, (τ , τ∗1, τ∗2),for which p̃w(τ , τ∗1, τ∗2) = pw0 and such that no country imports a volume inexcess of its implied import volume. We then define the associated value of tradevolume TV (τ , τ∗1, τ∗2) as in (7). Using trade balance for the foreign countries,market clearing for good x, the requirement that the import volume of the homecountry not exceed its implied import volume, and (23), we obtain

TV (τ , τ∗1, τ∗2) = pw0M(p(τ , pw0 ), pw0 ) + pw0∑i=1,2

E∗i(p∗i(τ∗i, pw0 ), pw0 )

= 2pw0M(p(τ , pw0 ), pw0 )

≤ 2pw0Mh

= 2pw0M(p(τ , pw0 ), pw0 )

= TV (τ , τ∗1, τ∗2)

Thus, for the second case, the assigned tariff vector achieves the maximum valuefor the value of trade volume, given the initial world price and the restrictionthat no country imports a volume in excess of its implied import volume.

11

We note in this second case that our baseline rules do not uniquely identifyan assigned tariff vector; for example, we could achieve the same trade volumewhile satisfying the other constraints by slightly increasing (lowering) the im-plemented tariff for foreign country ∗i ( foreign country ∗j) in a fashion thatmaintains the aggregate implied import volume for the two foreign countries.Thus, when the proposals induce this second case, we impose some additionalrules in constructing our mechanism so as to arrive at the assigned tariff vector,(τ , τ∗1, τ∗2).

The Constructed Mechanism: Our constructed mechanism is defined bythe strategy space S and an outcome function g that takes tariff proposalsand assigns tariffs. We may now summarize the tariffs that our constructedmechanism assigns as a function of the tariff proposals:

A. If the tariff proposals agree, then the home tariff proposal is assigned. Thetariffproposals can agree only if the home country matches: pw0Mh = M∗1+M∗2.

B. If the tariff proposals do not agree and the home country is long, so thatpw0Mh > M∗1 + M∗2, then the assigned tariff vector is (τ̃ , T ∗1∗1 , T

∗2∗2 ) where τ̃

satisfies p̃w(τ , T ∗1∗1 , T∗2∗2 ) = pw0 .

C. If the tariff proposals do not agree and the home country matches, so thatpw0Mh = M∗1+M∗2, then the assigned tariffvector is (τ̃ , T ∗1∗1 , T

∗2∗2 ) where τ̃ = Thh

satisfies p̃w(τ , T ∗1∗1 , T∗2∗2 ) = pw0 .

D. If the tariff proposals do not agree and the home country is short, so thatpw0Mh < M∗1 +M∗2, then there are two cases:

1. In the first case, the randomly selected first country, foreign country ∗i, makesa proposal such thatM∗i ≥ pw0Mh. The assigned tariff vector is then (τ̂ , τ̂∗1, τ̂∗2)where τ̂∗i satisfiesM∗i(p∗i(τ∗i, pw0 ), pw0 ) = pw0Mh, τ̂

∗j satisfiesM∗j(p∗j(τ∗j , pw0 ), pw0 ) =0 and τ̂ = Thh satisfies p̃

w(τ , τ̂∗1, τ̂∗2) = pw0 .

2. In the second case, the randomly selected first country, foreign country∗i, makes a proposal such that M∗i < pw0Mh. The assigned tariff vector isthen (τ , τ∗1, τ∗2) where τ∗i satisfies M∗i(p∗i(τ∗i, pw0 ), pw0 ) = M∗i, τ

∗j satisfiesM∗j(p∗j(τ∗j , pw0 ), pw0 ) = pw0Mh−M∗i, and τ = Thh satisfies p̃

w(τ , τ∗1, τ∗2) = pw0 .

2 Dominant Strategies

We are now prepared to consider the endogenous determination of tariffs in theconstructed mechanism when governments use only dominant strategies. In thissection, we take the first step in this process and characterize the respective setsof dominant strategies for foreign country ∗i and the home country when theconstructed mechanism is used.

12

An initial observation is that, for any foreign country ∗i, the proposal strat-egy is completely described by T ∗i∗i . To see the point, consider foreign country ∗1.Since foreign country ∗1 is restricted to set T ∗2∗1 = τ∗20 and to set Th∗1 = Th∗1(T

∗1∗1 )

at the unique value given T ∗1∗1 that delivers pw0 ≡ p̃w(Th∗1, T

∗1∗1 , τ

∗20 ), its proposal

(Th∗1, T∗1∗1 , T

∗2∗1 ) = (Th∗1(T

∗1∗1 ), T ∗1∗1 , τ

∗20 ) is completely determined by its selection

of T ∗1∗1 . By contrast, as noted previously, the home country’s proposal is notfully determined by its proposal for its own tariff, Thh , since the home country’sproposal includes levels for both foreign tariffs and these tariffs can be combinedin different ways to generate a world price that equals the initial world price.To characterize dominant strategies, we must specify payoff functions. Fol-

lowing the main paper, and for a given initial world price pw0 , we assume thatthe home country payoff is defined by the function W (p(τ , pw0 ), pw0 ) while thepayoff for foreign country ∗i is defined by the function W ∗i(p∗i(τ∗i, pw0 ), pw0 ).It is convenient now to recall from the main paper that the politically optimalreaction tariff for home satisfies

Wp(p(τ , pw0 ), pw0 ) = 0 (24)

while the politically optimal reaction curve tariff for foreign country ∗i satisfies

W ∗ip∗i(p

∗i(τ∗i, pw0 ), pw0 ) = 0. (25)

Let τPO and τ∗iPO represent the respective politically optimal reaction tariffs forthe home country and foreign country ∗i.Taking the initial world price pw0 as fixed, we assume that each country

has single-peaked preferences with respect to its own tariff (or equivalently,with respect to its local price).7 Formally, for the given pw0 , we assume thatWp(p(τ , p

w0 ), pw0 ) > 0 for p(τ , pw0 ) < p(τPO, p

w0 ) and Wp(p(τ , p

w0 ), pw0 ) < 0 for

p(τ , pw0 ) > p(τPO, pw0 ), where we recall from the main paper that p(τ , pw0 ) = τpw0 .

Similarly, for the given pw0 , we assume that W∗ip∗i(p

∗i(τ∗i, pw0 ), pw0 ) > 0 forp∗i(τ∗i, pw0 ) < p∗i(τ∗iPO, p

w0 ) and W ∗i

p∗i(p∗i(τ∗i, pw0 ), pw0 ) < 0 for p∗i(τ∗iPO, p

w0 ) >

p(τPO, pw0 ), where we recall from the main paper that p∗i(τ∗i, pw0 ) = (1/τ∗i)pw0 .

These assumptions are all understood to hold for tariffs such that the corre-sponding country has positive trade volume.

Dominant Strategies for Foreign Country ∗i: Consider foreign country∗i. We wish to argue that foreign country ∗i’s dominant strategy is to proposeT∗i,PO defined by T ∗i∗i,PO = τ∗iPO with T ∗j∗i,PO = τ∗j0 and Th∗i,PO = Th∗i(τ

∗iPO)

then set to deliver the initial world price, pw0 , as the market-clearing worldprice under the proposal. We compare this proposal strategy to an alternativestrategy T∗i associated with a different own tariff for foreign country ∗i, T ∗i∗i ≡τ∗i 6= T ∗i∗i,PO = τ∗iPO, where as described T

∗j∗i = τ∗j0 and Th∗i = Th∗i(τ

∗i) then

7 In the main paper, we capture this assumption with our assumption that global second-order conditions hold for all maximization problems. The global second-order condition asso-cated with the maximization problem leading to home’s politically optimal reaction tariff, forexample, is that Wpp(p, pw0 ) < 0.

13

follow. Let M∗i,PO ≡ M∗i(p∗i(τ∗iPO, pw0 ), pw0 ) and M∗i = M∗i(p∗i(τ∗i, pw0 ), pw0 )

denote the corresponding implied import volumes.If the proposals of the home country and foreign country ∗j are such that

the tariff proposals agree when foreign country ∗i proposes T∗i,PO, then thealternative proposal T∗i cannot possibly represent an improvement for foreigncountry ∗i. This follows since under proposal T∗i,PO foreign country ∗i enjoysits favorite local price for the given initial world price. By similar reasoning,if the proposals of the home country and foreign country ∗j are such that thehome country is long or matches when foreign country ∗i proposes T∗i,PO, thenthe proposal T∗i,PO again results in foreign country ∗i’s favorite local price forthe given initial world price, and so the alternative proposal T∗i cannot possiblyrepresent an improvement for foreign country ∗i.8 The remaining possibilityis that the proposal T∗i,PO and the proposals of other countries are such thatthe home country is short. To address this remaining possibility, we distinguishbetween two cases for the alternative proposal, T∗i.The first case is that the alternative proposal entails a lower tariff for for-

eign country ∗i: τ∗i < τ∗iPO and thus M∗i > M∗i,PO. Given that the pro-posals of other countries are such that the home country is short when for-eign country ∗i proposes T∗i,PO, the home country will again be short in thisfirst case under the alternative proposal, T∗i. If foreign country ∗i is not ran-domly selected to go first, then its assigned tariff does not depend on its pro-posal (conditional on the home country being short) and so it is then indif-ferent between the two proposals. If foreign country ∗i is randomly selectedto go first, then it enjoys its favorite local price under the proposal T∗i,PO ifM∗i,PO ≤ pw0Mh. The alternative proposal then cannot represent an improve-ment for foreign country ∗i. If foreign country ∗i is randomly selected to go firstandM∗i,PO > pw0Mh, then the assigned tariff for foreign country ∗i is τ̂∗i whereτ̂∗i satisfies M∗i(p∗i(τ∗i, pw0 ), pw0 ) = pw0Mh. The alternative proposal entails aneven higher implied import volume, M∗i > M∗i,PO, and thus leads to the sameassigned tariff for foreign country ∗i. Hence, when the home country is short,and in the first case where τ∗i < τ∗iPO, we conclude that the politically optimalproposal T∗i,PO is always at least weakly preferred to the alternative proposal,T∗i.

The second case is that the alternative proposal entails a higher tariff for for-eign country ∗i: τ∗i > τ∗iPO and thusM∗i < M∗i,PO. Suppose first that the otherproposals are such that the home country remains short under the alternativeproposal (despite the fact that the alternative proposal implies a lower tradevolume for foreign country ∗i). If foreign country ∗i is not randomly selectedto go first, then its assigned tariff does not depend on its proposal (conditionalon the home country being short) and so it is then indifferent between the twoproposals. If foreign country ∗i is randomly selected to go first, then it enjoysits favorite local price under the proposal T∗i,PO if M∗i,PO ≤ pw0Mh. The al-

8Recall that, under the constructed mechanism defined above, foreign country ∗i’s proposedtariff for itself is assigned under agreement and also under disagreement when the homecountry is long or matches.

14

ternative proposal then cannot represent an improvement for foreign country∗i. If foreign country ∗i is randomly selected to go first and M∗i,PO > pw0Mh,then the assigned tariff for foreign country ∗i under the proposal T∗i,PO isτ̂∗i where τ̂∗i satisfies M∗i(p∗i(τ∗i, pw0 ), pw0 ) = pw0Mh. If the alternative pro-posal satisfies M∗i ≥ pw0Mh, then the same tariff is assigned for foreign country∗i. If the alternative proposal satisfies M∗i < pw0Mh, then the assigned tarifffor foreign country ∗i under the alternative proposal T∗i is τ∗i which satisfiesM∗i(p∗i(τ∗i, pw0 ), pw0 ) = M∗i. Given M∗i < pw0Mh < M∗i,PO, we then see thatτ∗i > τ̂∗i > τ∗iPO. We conclude that the politically optimal proposal T∗i,POis then preferred to the alternative proposal T∗i, since the assigned tariff forforeign country ∗i is closer to its politically optimal reaction tariff under theproposal T∗i,PO.

Continuing with the second case where M∗i < M∗i,PO, we suppose secondthat the other proposals are such that the home country is short under theproposal T∗i,PO but is not short under the alternative proposal T∗i. Thus, wenow focus on the scenario where M∗i + M∗j ≤ pw0Mh < M∗i,PO + M∗j . If theother proposals are such that M∗j > pw0Mh, then the home country is sureto be short under any proposal by foreign country ∗i, since foreign country ∗icannot induce a negative value for M∗i. Thus, the scenario under considerationis possible only if M∗j ≤ pw0Mh. If the other proposals are such that M∗j =pw0Mh, then the scenario under consideration is possible only if the alternativeproposal for foreign country ∗i specifies a prohibitive tariff so that M∗i = 0.But foreign country ∗i then does better under the proposal T∗i,PO, since itwould then also receive zero import volume (if foreign country ∗j were randomlyselected) or receive positive import volume corresponding to either M∗i,PO (ifM∗i,PO ≤ pw0Mh) or pw0Mh (if M∗i,PO > pw0Mh). Foreign country ∗i clearlydoes better when enjoying a positive import volume, since its assigned tariff isthen either τ∗iPO or at least closer to τ

∗iPO than is the prohibitive tariff.

9 Supposethen that the other proposals are such that M∗j < pw0Mh. Under the proposalT∗i,PO, if foreign country ∗i is randomly selected, then it enjoys a trade volumeof either M∗i,PO (if M∗i,PO ≤ pw0Mh) or pw0Mh (if M∗i,PO > pw0Mh). Underthe proposal T∗i,PO, if foreign country ∗i is not randomly selected, then itenjoys a trade volume of pw0Mh −M∗j > 0. Under the alternative proposal, thehome country is not short and so foreign country ∗i enjoys a trade volume ofM∗i ≤ pw0Mh −M∗j . Thus, foreign country ∗i enjoys a trade volume closer toM∗i,PO under the proposal T∗i,PO than under the proposal T∗i.Having now considered all possible trade-volume scenarios when foreign

country ∗i proposes T∗i,PO, and the corresponding possibilities were foreigncountry ∗i instead to propose an alternative proposal T∗i, we now concludethat alternative proposals T∗i which entail τ∗i 6= τ∗iPO are dominated by theproposal T∗i,PO which entails τ∗i = τ∗iPO. Furthermore, it is straightforward toargue that any tariff proposal for foreign country ∗i such that T ∗i∗i 6= T∗i,PO is

9We are assuming here and throughout that politically optimal reaction tariffs generatestrictly positive import volume at the initial world price.

15

not a dominant strategy.10 As a consequence, we may now draw the followingconclusion:

Lemma 6 Given the constructed mechanism, the set of dominant strategy pro-posals for foreign country ∗i is a singleton defined by T∗i,PO, where T ∗i∗i,PO =

τ∗iPO, T∗j∗i,PO = τ∗j0 and Th∗i,PO = Th∗i(τ

∗iPO) is then uniquely set to deliver pw0 as

the market-clearing world price.

Dominant Strategies for the Home Country: We consider next the homecountry’s tariff proposal. We wish to argue that the set of dominant strategiesfor the home country is defined by a set of proposals under which the homecountry proposes the tariff τPO for itself and tariffs for the foreign countries thatin combination with τPO deliver the initial world price as the market-clearingworld price. To fix ideas, let us select any proposal, Th,PO, from this set, wherethe proposal Th,PO is thus defined by Thh,PO = τPO with (T ∗1h,PO, T

∗2h,PO) then

satisfying pw0 ≡ p̃w(τPO, T∗1h,PO, T

∗2h,PO). We compare this proposal strategy to

an alternative strategy Th for which Thh ≡ τ 6= Thh,PO = τPO with (T ∗1h , T ∗2h )

then satisfying pw0 ≡ p̃w(τ , T ∗1h , T ∗2h ). Recall that, given the initial world price,the home-country’s implied import volume is determined by its proposed tarifffor itself. Let Mh,PO ≡ M(p(τPO, p

w0 ), pw0 ) and Mh = M(p(τ , pw0 ), pw0 ) denote

the corresponding implied import volumes.If the proposals of the foreign countries are such that the tariff proposals

agree when the home country proposes Th,PO, then the alternative proposal Thcannot possibly represent an improvement for the home country. This followssince under proposal Th,PO the home country enjoys its favorite local price forthe given initial world price. By similar reasoning, if the proposals of the foreigncountries are such that the home country matches or is short when the homecountry proposes Th,PO, then we recall from above that the home country alsoapplies the tariff τPO and thus enjoys its favorite local price for the given initialworld price; hence, as before, the alternative proposal Th again cannot possiblyrepresent an improvement for the home country.11

The remaining possibility is that the proposal Th,PO and the proposals ofthe foreign countries are such that the home country is long so that pw0Mh,PO >M∗1 + M∗2. Under the proposal Th,PO, the home country is then assigned

10Let T∗i be any tariff proposal for foreign country ∗i for which T ∗i∗i 6= τ∗iPO. Consider nowtariff proposals for foreign country ∗j and the home country such that the home country islong, matches or agrees, whether foreign country ∗i proposes T∗i or T∗i,PO . Given such tariffproposals for foreign country ∗j and the home country, foreign country ∗i imports a volumethat equals its implied import volume, which is M∗i under the tariff proposal T∗i and M∗i,POunder the tariff proposal T∗i,PO , respectively, withM∗i 6=M∗i,PO . The tariff proposal T∗i,POis then strictly better for foreign country ∗i than is the tariff proposal T∗i. In making thisargument, we assume that, for any tariff proposal for foreign country ∗i, there exist tariffproposals for the home country and foreign countries ∗j such that the home country is long,matches or agrees.11As previously noted, when the home country matches or is short, (10), (18) and (23)

ensure that the home country imports a volume that equals its implied import volume, fromwhich it follows that the home country applies the tariff that it proposed for itself.

16

the tariff τ̃ defined given the foreign proposals (T ∗1∗1 , T∗2∗2 ) to deliver pw0 ≡

p̃w(τ̃ , T ∗1∗1 , T∗2∗2 ). Given the foreign proposals, if the home country remains long

under the altenative proposal Th, then the same tariff vector is assigned andso the alternative proposal fails to offer an improvement for the home country.Suppose then the home country achieves agreement or is short or matches underthe alternative proposal Th. For the given foreign proposals, under any of thesecases for the alternative proposal Th, the tariff that the home country proposesfor itself, τ , is assigned, and so we have that pw0M(p(τ , pw0 ), pw0 ) = pw0Mh ≤M∗1 + M∗2 = pw0M(p(τ̃ , pw0 ), pw0 ) < pw0Mh,PO where the final equality followsfrom (6). It follows that τPO < τ̃ ≤ τ , and so the alternative proposal Th resultsin an assigned home-country tariff τ that is (weakly) further from τPO than isthe assigned home-country tariff τ̃ that results from the proposal Th,PO.Having now considered all possible trade-volume scenarios when the home

country proposes Th,PO, and the corresponding possibilities were the home coun-try instead to propose an alternative proposal Th, we now conclude that alter-native proposals Th which entail τ 6= τPO are dominated by the proposal Th,POwhich specify a home-country tariff of τPO. Since Th,PO is an arbitrary selectionfrom the set of home proposals for which Thh,PO = τPO, we conclude that thealternative proposal Th is dominated by all home-country proposal strategiesthat specify a home-country tariff of τPO.

Finally, we compare distinct home-country proposal strategies that bothspecify a home-country tariff of τPO, and we argue that for any given for-eign proposals such home-country proposal strategies must result in the sameassigned tariff vector. A corollary is that one home-country strategy in thisclass cannot dominate another.12 To develop this argument, we use Th,PO =(τPO, T

∗1h,PO, T

∗2h,PO) to denote one such proposal, and we denote an alterna-

tive such proposal as T′

h,PO = (τPO, T∗1′h,PO, T

∗2′h,PO), where (T ∗1h,PO, T

∗2h,PO) 6=

(T ∗1′

h,PO, T∗2′h,PO). For given foreign proposals, if Th,PO achieves agreement, then

T′

h,PO does not achieve agreement. In this case, however, the home country

would match under the alternative proposal T′

h,PO, and so the same tariff vec-tor would be assigned under either home-country proposal. Given the foreignproposals, if the home country matches or is short under Th,PO, then the homecountry similarly matches or is short under T

′

h,PO. For these cases, we mayargue as above and conclude that, whether the home country proposes Th,POor T

′

h,PO, it imports a volume that equals its implied import volume, whichunder either proposal is Mh,PO. The same tariff vector thus would be assignedunder either home-country proposal.13 Finally, given the foreign proposals, ifthe home country is long under Th,PO, then the home country is long as wellunder T

′

h,PO. The assigned tariff vector is then (τ̃ , T ∗1∗1 , T∗2∗2 ) whether the home

12We develop our arguments here for the case in which the two home-country proposalstrategies call for a common home-country tariff of τPO . Our arguments apply more generally,though, for any home-country tariff τ .13This statement is understood to refer to expected values when the home country is short

and a “first”foreign firm is selected at random. The key point is that the selection probabilityis random and thus independent of the specific home-country proposal.

17

country proposes Th,PO or T′

h,PO.Having now considered all possible trade-volume scenarios when the home

country proposes Th,PO, and the corresponding possibilities were the home coun-try instead to propose alternative proposal Th, we now conclude that alternativeproposals Th which entail τ 6= τPO are dominated by the proposal Th,PO whichentails τ = τPO. We have also argued that the home-country proposal Th,POoffers the same assigned tariff vector for any given foreign proposals as does thehome-country proposal T

′

h,PO, where Th,PO and T′

h,PO both entail τ = τPO butpropose different foreign tariffs. Furthermore, it is straightforward to argue thatany tariff proposal for the home country such that Thh 6= τPO is not a dominantstrategy.14 As a consequence, we may now draw the following conclusion:

Lemma 7 Given the constructed mechanism, the set of dominant strategy pro-posals for the home country is a set whose members are the home-country pro-posal strategies for which Thh = τPO, with T ∗1h and T ∗2h then specified in anyfashion ensuring that p̃w(τPO, T

∗1h , T ∗2h ) = pw0 .

3 Dominant Strategy Implementation

Having characterized the sets of dominant strategies for foreign country ∗i andthe home country, we now characterize the tariffvectors that can be implementedin the constructed mechanism when governments use only dominant strategies.Our discussion shows how the final negotiation outcome actually emerges fromdominant strategy proposals, and we also assess the effi ciency of the negotiatedoutcome. We distinguish between two cases: pw0 6= pwPO and p

w0 = pwPO.

Case 1: pw0 6= pwPO. In this case, a first point is that, when dominant strategyproposals are used, agreement does not occur. Assume to the contrary thatagreement occurs. The home proposal is then assigned, and so the resultingtariffvector is (Thh , T

∗1h , T ∗2h ), where we also know from our discussion above that

T ∗ih = T ∗i∗i under agreement. For dominant strategy proposals, we further havethat (Thh , T

∗1h , T ∗2h ) = (τPO, τ

∗1PO, τ

∗2PO). But such an outcome is not feasible,

since p̃w(τPO, τ∗1PO, τ

∗2PO) = pwPO 6= pw0 . Second, we observe similarly that, when

dominant proposals are used, it is also infeasible that the home country matches.If the home country were to match, then we would have that pw0Mh = M∗1+M∗2,with the own-tariff proposal of each country obtaining in the assigned tariffvector. Under dominant strategy proposals, this leads again to the contradictionthat p̃w(τPO, τ

∗1PO, τ

∗2PO) = pwPO 6= pw0 .

14Let Th be any tariff proposal for the home country for which Thh 6= τPO. Consider nowtariff proposals for foreign countries ∗1 and ∗2 such that the home country is short, matches oragrees, whether the home country proposes Th or Th,PO . Given such foreign tariff proposals,the home country imports a volume that equals its implied import volume, which is Mh

under the tariff proposal Th and Mh,PO under the tariff proposal Th,PO , respectively, withMh 6=Mh,PO . The tariff proposal Th,PO is then strictly better for the home country than isthe tariff proposal Th. In making this argument, we assume that, for any tariff proposal forthe home country, there exist tariff proposals for foreign countries ∗1 and ∗2 such that thehome country is short, matches or agrees.

18

Thus, when pw0 6= pwPO, there are two possible outcomes under dominantstrategy implementation. One possibility is that under dominant strategy pro-posals the home country is long. This possibility occurs if and only if

pw0Mh,PO > M∗1,PO +M∗2,PO, (26)

whereMh,PO ≡M(p(τPO, pw0 ), pw0 ) andM∗i,PO ≡M∗i(p∗i(τ∗iPO, p

w0 ), pw0 ). When

home is long so that (26) obtains, the assigned tariff vector is (τ̃ , T ∗1∗1 , T∗2∗2 ) where

p̃w(τ̃ , T ∗1∗1 , T∗2∗2 ) = pw0 defines τ̃ . Under dominant strategy proposals, T

∗i∗i = τ∗iPO,

and so τ̃ is defined to satisfy p̃w(τ̃ , τ∗1PO, τ∗2PO) = pw0 . Since p

wPO 6= pw0 , it follows

that τ̃ 6= τPO.15 An implication is that the implemented tariff vector is ineffi -cient when countries use only dominant strategies and when the home countryis long and pw0 6= pwPO.

16

When pw0 6= pwPO, it is interesting to compare the implemented tariff vectorwith the proposed tariffs when countries use dominant strategy proposals andthe home country is long. The home-country proposal entails Thh = τPO alongwith proposals for foreign tariff that preserve the initial world price, and thisproposal is clearly not employed since τ̃ 6= τPO when pwPO 6= pw0 . Provided thatτ∗j0 6= τ∗jPO, foreign country ∗i’s proposal is also not exactly implemented. Butthe implemented tariff vector is in foreign country ∗i’s equivalence class givenits proposal. Thus, for each foreign country ∗i, we can think in this case of itsproposal being accepted, once world-price-preserving modifications are made toits proposed tariffs for the home country and for foreign country ∗j.

The other possibility when pw0 6= pwPO is that the home country is short. Thispossibility occurs if and only if

pw0Mh,PO < M∗1,PO +M∗2,PO, (27)

whereMh,PO ≡M(p(τPO, pw0 ), pw0 ) andM∗i,PO ≡M∗i(p∗i(τ∗iPO, p

w0 ), pw0 ). When

home is short so that (27) obtains, the implemented home tariff is Thh = τPO.The implemented foreign tariffs depend on whether for the randomly selectedforeign country ∗i we have M∗i,PO ≥ pw0Mh,PO or M∗i,PO < pw0Mh,PO. Underthe first inequality, the implemented tariff, τ̂∗i, for the randomly selected for-eign country ∗i satisfies τ̂∗i ≥ τ∗iPO while foreign country ∗j sets a prohibitivetariff τ̂∗j which thus satisfies τ̂∗j > τ∗jPO.

17 Under the second inequality, theimplemented tariff, τ∗i, for the randomly selected foreign country ∗i satisfiesτ∗i = τ∗iPO while foreign country ∗j sets a tariff τ∗j under which by (21) wehave M∗j(p∗j(τ∗j , pw0 ), pw0 ) < M∗j,PO and so τ∗j > τ∗jPO. Given that the homecountry applies the tariff τPO while at least one foreign country applies a tariffthat differs from its politically optimal reaction tariff, we conclude that the im-plemented tariff vector is ineffi cient when countries use only dominant strategiesand when the home country is short and pw0 6= pwPO.

15 In particular, τ̃ > τPO if pw0 < pwPO, and τ̃ < τPO if pw0 > pwPO .16A policy vector fails to be effi cient if some but not all countries respectively apply tariffs

equal to their politically optimal reaction tariffs. See Bagwell and Staiger (1999, 2002) forfurther discussion.17Recall that we assume that politically optimal reaction tariffs generate strictly positive

import volume at the initial world price.

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When pw0 6= pwPO, it is also interesting to compare the implemented tariff vec-tor with the proposed tariffs when countries use dominant strategy proposalsand the home country is short. The home-country proposal entails Thh = τPOalong with proposals for foreign tariffs that preserve the initial world price. Sincethe implemented tariff vector specifies that the home country apply the tariffτPO, any difference between the home-country proposal and the implementedtariff vector must involve the associated foreign-country tariffs. Provided thatτ∗j0 6= τ∗jPO, the randomly selected foreign country ∗i’s proposal is also notexactly implemented, even when (as under the second inequality above) its im-plemented tariff is τ∗iPO. The dominant strategy proposal for foreign country∗j (i.e., the foreign country which is not randomly selected) sets T ∗j∗j = τ∗jPO;thus, its proposal is clearly not implemented, since the implemented tariff vectorspecifies either τ̂∗j > τ∗jPO or τ

∗j > τ∗jPO for foreign country ∗j. It thus follows aswell that the implemented tariff vector is not in foreign country ∗j’s equivalenceclass given its proposal. We can think in this case of the home-country proposalbeing accepted as is (if foreign country tariffs are specified appropriately), or af-ter world-price-preserving modifications are made to the home-country proposalfor foreign-country tariffs.18

Case 2: pw0 = pwPO. In this case, under dominant strategy proposals, we havethat

pw0Mh,PO = M∗1,PO +M∗2,PO, (28)

where Mh,PO ≡ M(p(τPO, pw0 ), pw0 ) and M∗i,PO ≡ M∗i(p∗i(τ∗iPO, p

w0 ), pw0 ). The

proposals agree or are such that the home-country proposal matches.It is certainly possible that the proposals agree. The set of dominant strategy

proposals for the home country includes the following proposal: (Thh , T∗1h , T ∗2h ) =

(τPO, τ∗1PO, τ

∗2PO). This proposal is now feasible, since p̃w(τPO, τ

∗1PO, τ

∗2PO) ≡

pwPO = pw0 . Recall that the dominant strategy set for foreign country ∗i is asingleton defined by T∗i,PO, where T ∗i∗i,PO = τ∗iPO, T

∗j∗i,PO = τ∗j0 and Th∗i,PO =

Th∗i(τ∗iPO) is then uniquely set to deliver pw0 as the market-clearing world price.

Since pwPO = pw0 , the home-country proposal is in the equivalence class for for-eign country ∗i that is defined by its dominant strategy proposal. The proposalsthen agree and effi ciency is obtained.19 In this case, we may understand thatthe home-country proposal is accepted as is; furthermore, for each foreign coun-try ∗i, we can think of its proposal being accepted, once world-price-preservingmodifications are made to its proposed tariffs for the home country and forforeign country ∗j.It is also possble that the proposals match but do not agree. In particular,

the home country may propose (Thh , T∗1h , T ∗2h ) where Thh = τPO, (T ∗1h , T ∗2h ) 6=

(τ∗1PO, τ∗2PO) and the market-clearing world price under the home-country’s pro-

posed tariffs equals pw0 = pwPO. This situation arises when the home-country18When making its proposal, the home country does not know which foreign country will

be randomly selected to go first; thus, the home-country proposal cannot be accepted as iswith probability one.19As noted in the main text, the politically optimal tariff vector, (τPO, τ∗1PO, τ

∗2PO), is effi -

cient. See also Bagwell and Staiger (1999, 2002).

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proposal balances T ∗ih > τ∗iPO against T∗jh < τ∗jPO, for i 6= j, so as to maintain the

initial world price as the market-clearing world price. Since the dominant strat-egy proposal for each foreign country ∗i satisfies T ∗i∗i,PO = τ∗iPO, T

∗j∗i,PO = τ∗j0

and Th∗i,PO = Th∗i(τ∗iPO), we see that agreement is not achieved in this case.

The implemented tariff vector in this case is (τPO, τ∗1PO, τ

∗2PO). Thus, an effi -

cient outcome is implemented, even though agreement does not occur. For eachforeign country ∗i, we can think in this case of its proposal being accepted,once world-price-preserving modifications are made to its proposed tariffs forthe home country and for foreign country ∗j.

Summary: We have established the following proposition:

Proposition 8 Given the constructed mechanism, under dominant strategy pro-posals, the implemented tariff vector is effi cient if and only if pw0 = pwPO.

As illustrated above, we can also interpret the process through which outcomesare achieved. In some cases, the outcome entails accepting the home-countryproposal as is, or after world-price-preserving modifications are made to thehome-country proposal for foreign-country tariffs. In other cases, we can think ofeach foreign country ∗i’s proposal as being accepted, once world-price-preservingmodifications are made to its proposed tariffs for the home country and forforeign country ∗j. When pw0 = pwPO, it is also possible that the proposalsagree, so that the home-country proposal is accepted as is and also the proposalof each foreign country ∗i is accepted once world-price-preserving modificationsare made to its proposed tariffs for the home country and for foreign country∗j.

4 Extensions

The analysis can be extended in multiple directions. We briefly discuss herethree extensions.

Multiple countries As a first example, we may extend the analysis to con-sider multiple foreign countries. Suppose for example that there are three foreigncountries. When the home country is short, one of the foreign countries, sayforeign country ∗i, is selected as the “first” foreign country. As above, in theconstructed mechanism, foreign country ∗i imports a volume that is the mini-mum of M∗i and pw0Mh. If pw0Mh > M∗i, then the value of the home country’simplied import volume is not exhausted by foreign country ∗i’s implied importvolume. We can then treat this residual value, pw0Mh −M∗i, as the value ofhome trade volume that is allocated across the remaining two foreign countries(i.e., pw0Mh −M∗i in the model with three foreign countries then plays the roleof Mh in the constructed mechanism defined above for the model with two for-eign countries). In this general way, we can proceed inductively to define theconstructed mechanism for any number of foreign countries.

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Private information A second extension allows for private information. Sup-pose payofffunctions are given asW (p(τ , pw0 ), pw0 ; θ) andW ∗i(p∗i(τ∗i, pw0 ), pw0 ; θ∗i),where θ ∈ Θ is privately observed by the home country and θ∗i ∈ Θ∗i is privatelyobserved by foreign country ∗i, i 6= j. We may think of Θ and Θ∗i as intervalson the real line, for example. The variables θ and θ∗i correspond to preferenceshocks and do not directly impact the determination of economic variables (i.e.,τ0, τ

∗10 , τ

∗20 , p

w0 and the function p̃

w are all independent of θ and θ∗i).For this extended model, the constructed mechanism remains the same as

above. The politically optimal tariff reactions are also defined as above, ex-cept that now they are defined in an ex post sense. Thus, τPO(θ) satisfiesWp(p(τ , p

w0 ), pw0 ; θ) = 0 and τ∗iPO(θ∗i) satisfies W ∗i

p∗i(p∗i(τ∗i, pw0 ), pw0 ; θ∗i) = 0.

Lemmas 6 and 7 hold as stated above, once τ∗iPO is replaced by τ∗iPO(θ∗i) inLemma 6 and once τPO is replaced by τPO(θ) in Lemma 7. In the extendedmodel, Lemmas 6 and 7 thus characterize ex post dominant strategies. Finally,Proposition 8 carries over as well, once pwPO is replaced by pwPO(θ, θ∗1, θ∗2)where pwPO(θ, θ∗1, θ∗2) ≡ p̃w(τPO(θ), τ∗1PO(θ∗1), τ∗2PO(θ∗2)). With this adjust-ment, Proposition 8 characterizes when an ex post effi cient outcome can beimplemented using dominant strategies in the private information model, whenthe constructed mechanism is used.

Principal supplier In the constructed mechanism, when the home countryis short, we specified that a randomly selected foreign country be given firstpriority in the allocation of trade volume. As we noted, alternative prioritizationschemes could be used, with the main requirement being that a foreign country’sproposal cannot affect its priority position (conditional on home being short).Suppose, for example, that foreign country ∗i is, for exogenous reasons, a moresignificant trading partner for the home country. The home country might thenregard foreign country ∗i as a principal supplier. An alternative prioritizationscheme could then be specified in which, when the home country is short, foreigncountry ∗i always assumes the first-priority position (i.e., conditional on homebeing short, foreign country ∗i is always treated as if it were randomly selectedto go “first” in the scheme above). The results above would continue to holdunder this alternative prioritization scheme as well.

5 References

Bagwell, Kyle and Robert W. Staiger (1999), “An Economic Theory of GATT,”American Economic Review 89(1), 215-48.

Bagwell, Kyle and Robert W. Staiger (2002). The Economics of the WorldTrading System. The MIT Press: Cambridge, MA.

Bagwell, Kyle, Robert W. Staiger and Ali Yurukoglu (2015), “Multilateral TradeBargaining: A First Look at the GATT Bargaining Records,”August.

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