A Spatial Model of Multilateral Negotiations
Kevin L. Cope and James D. Morrow
University of Michigan
Multilateral Negotiations
• Much of international law is created through
multilateral treaties that aspire to universality:
– Human rights
– International humanitarian law
– Trade
• Much progress has been made in studying the effects
of these treaties, but we know little systematically
about the negotiations that create these treaties.
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• Characteristics of multilateral bargaining
– Consensus required.
• States select into international law through treaty ratification.
– Multiple issues to bargain over
– Could drop/ignore some parties with following
consequences...
• Excluded party could add value to the agreement.
– The broader/deeper tradeoff
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Spatial Model of Negotiations
• Spatial models allow us to analyze how actors with
different positions reconcile their differences to
produce a common outcome.
– Elections
– Legislatures
– Bilateral negotiations
• Core concepts:
– Issues modeled spatially
– Ideal points for each actor
– Declining utility moving away from ideal point
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The Game
• Actors: M States, D-Drafting committee
• N-dimensional issue space
• Preferences:
– D: Ideal point at the origin
– States: ideal points xi
– Value for the treaty depends on how many states ratify.
Each state has weight wi with
𝑀
𝑊+
𝑤𝑖 = 1
𝑖=1
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– For D, payoff is
𝑢𝐷 𝑥 = 𝑊 +
𝑤𝑖 − 𝑥
2
𝑖 𝑟𝑎𝑡𝑖𝑓𝑖𝑒𝑠
– For i ϵ M, value ViIN if i ratifies, value ViOUT if i does not, with
Δi = ViIN – ViOUT.
• Values vary with nature of good produced by treaty and expressive value for
ratification.
– i’s payoff is
𝑢𝑖 𝑥 = ∆𝑖 𝑊 +
𝑤𝑖 − 𝑥 − 𝑥𝑖
2
𝑖 𝑟𝑎𝑡𝑖𝑓𝑖𝑒𝑠
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Time Line of Game:
1. D proposes treaty x.
2. All i simultaneously ratify x or not. Payoffs are
received.
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Characterizing the Equilibrium
• For K M, define i ϵ M to be favorable under K if
∆𝑖 𝑊 +
𝑗∈𝐾 𝑤𝑗
− 𝑥𝑖
2
≥0
and unfavorable under K otherwise.
• Coalition K is feasible if all i ϵ K are favorable under
K and all i ∉ K are unfavorable.
• Coalition K is the base coalition if it is feasible
and 𝑤 𝐾 = 𝑖∈𝐾 𝑤𝑖 is a maximum over all feasible K.
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• Let i be unfavorable under the base coalition. Define
Ω𝑖 = 1 −
∆𝑖 𝑊+ 𝑗∈𝐵+𝑖 𝑤𝑗
𝑥𝑖
𝑥𝑖
• Choose treaty t ϵ {Ωi,0} to maximize uD(t).
• Equilibrium: D proposes t. i ratifies if ui(t) ≥ 0.
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• Here are three actors:
• Each has ideal point and
indifference curve for
what treaties it will
accept.
• B is favorable and
forms the base coalition.
• A and C are unfavorable
under base coalition.
C
A
B
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• Treaty will be whichever
of {0,ΩA,ΩC,ΩAC}
maximizes uD(t).
C
A
C
0
A
B
 AC
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What is to Be Done?
• Finalize proofs.
• Use the model to drive the estimation procedure.
– Rome Conference is test data.
– Much of the data is missing compared to votes and decisions.
• Apply procedure to analyze other treaties, particularly
human rights treaties.
• Substantive questions:
– Do conference rules matter?
– Variation across treaties
– The logic of exclusion and its effect on resulting treaties
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Extra Slides
The Rome Statute
• The Rome Statute of 1998 constituted the
International Criminal Court and gave it power to
prosecute war crimes and crimes against humanity.
• Final conference held in Rome during June-July 1998
with 160 states represented.
• Plenary sessions collected comments on draft treaty
leading to revisions by the Drafting Committee.
• Final vote followed by ratification afterwards.
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• ICC created after 60 states ratified the Rome Statute.
• 124 states are parties to the ICC now.
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Estimating Ideal Points
• We use the negotiating record published by the UN as
data course to estimate state ideal points.
• Each comment by a state was coded as
1. Critical of the proposed language
2. Supportive of the proposed language
3. No opinion
• 218 issues were coded; some articles present multiple
issues.
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• We adapt techniques used for legislatures and courts
to estimate ideal points from votes and decisions
(Poole and Rosenthal, Martin and Quinn, Voeten).
• Bayesian spatial modeling based on item-response
theory.
• 135 states expressed at least one view during
negotiations.
– Of 29,430 (= 135*218) potential state views , 4364
expressed, most critical.
• We recover one dimension from the procedure.
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• Procedure gives
distributions for the
estimate of each state’s
position.
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• Estimated ideal point
predicts ratification of
the treaty afterwards.
– Ratified in blue
– Not ratified in red
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