Overweighting nations: optimal voting rules under participation constraints

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From theory to application
Antonin Macé and Rafael Treibich
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From Theory to Application
The recent vote on the Brexit, and the withdrawal of the US from the Paris climate agreement have reminded us of the fragility and the difficulty of international cooperation. We investigate the effect of voluntary participation on the design of multilateral institutions, in which several countries agree to delegate their sovereignty on one or several policy areas. In principle, these institutions should reconcile two objectives: be representative of all citizens, and be acceptable to all participating countries. As the following historical examples illustrate, this tension between representation and acceptability has often played a critical role in shaping the design of multilateral institutions.

In 1787, the founding fathers of the US Constitution faced a contentious challenge: how to accommodate a fair representation of states at the federal level, while preserving a say to small states in the new institutions? The issue was resolved by the so-called Connecticut Compromise, under which states received a weight proportional to their populations in the House, but an equal weight in the Senate. This distinctive feature of the US Constitution was decisive in its adoption and has remained unchanged until now.

Another prominent example is the Council of the European Union (EU), one of its main decision-making bodies, whose decision rule has often been changed after fierce debates. The Treaty of Lisbon established the latest rule, known as « qualified majority »: a reform is adopted if it is approved by at least 55% of the Member States representing at least 65% of the EU population.[1] This rule exhibits again a compromise between proportionality and equality, which ensures its acceptability by the smallest Member States.




“Scene at the Signing of the Constitution of the United States”, by Howard Chandler Christy (1940).

A final example is the UN Security Council, in which the five permanent members can veto any resolution, and consequently benefit from a disproportionate power. The Charter of the UN was ratified at the San Francisco Conference in 1945. As Francis O. Wilcox, an adviser to US delegation, put it: « the issue was made crystal clear by the leaders of the Big Five [China, France, Russia, UK, US]: it was either the Charter with the veto or no Charter at all » [3]. In that case, the acceptability of the rule entailed not only overweighting some countries, but also ensuring them a veto power.

The goal of our research is to shed light on the tension between the representativity of a voting rule and its acceptability, in the context of international organisations. Should smaller countries be overweighted? Are weighted voting rules, wherein each country is given a fixed voting weight, normatively compelling? Can some countries legitimately claim a veto power? To address these questions, we follow a « second-best » approach to institutional design: we look for normatively appealing rules among those that are politically feasible, assuming that countries’ participation is voluntary.




The San Francisco Conference, 25 April - 26 June 1945: The United States sign the United Nations Charter.[2]

Our model features a group of countries choosing whether to delegate some of their competences to a supranational entity, a «union»[3] . In a first step, countries choose to delegate at the unanimity: the union is formed if all countries are willing to participate. If cooperation is agreed upon, future decisions are made collectively according to a fixed voting rule. If cooperation is rejected, countries remain sovereign and take independent national decisions. Crucially, cooperation must be agreed ex-ante, before countries know the details of the reforms proposed in the union, with some uncertainty on whether they will agree or not with other cooperating countries.

What can motivate countries to join the union? On the one hand, cooperation generates substantial efficiency gains, such as coordination externalities, economies of scale, increased bargaining power, etc. On the other hand, cooperating implies a loss of decision power since countries sometimes end up disagreeing with the collective decision. The voting rule, which determines how much influence each country exerts on the collective decision, thus plays a critical role in inducing cooperation.


Enforcable Decisions

We start from the assumption that collective decisions are perfectly enforced: once a reform is approved through the voting rule, all countries implement it. As first shown by Barberà and Jackson [1], maximizing social welfare, conceived as the sum of all citizens’ utilities in the union, requires using a weighted majority rule. Each country is assigned a voting weight equal to its « stake » in the collective decision. Intuitively, countries whose preferences correlate more with overall social welfare (because they are larger, more democratic or more affected by the policy areas) get a larger say. However, this « efficient » voting rule may not necessarily lead to cooperation if some of the countries in the union do not benefit enough from cooperation or disagree too often with the efficient collective decision.

In order to capture the effect of the voting rule on cooperation, we introduce a constitutional stage where countries decide whether they want to cooperate or stay sovereign, before actually learning about future decisions. A voting rule induces cooperation, it satisfies the « participation constraints », if all countries expect to be better off cooperating, by taking decisions under this rule, than remaining sovereign.

We show that the optimal voting rule, i.e. the rule that maximizes social welfare under participation constraints, is also a weighted majority rule. However, in contrast with the efficient rule, the optimal voting rule involves overweighting certain countries relative to their stakes. Only countries for which the participation constraint is binding may be overweighted.

We first observe that, even when considering participation constraints, weighted voting is optimal. This rules out the « qualified majority » currently used in the Council of the EU, which does not to belong to the class of weighted voting rules. Second, we note that the prediction that overweighted countries have a binding constraint resonates with the example of the UK, which has successfully negotiated a special status in the EU for years,[4] before taking the decision to leave the union, with a very close margin.


Non-Enforcable Decisions

The assumption of enforceable decisions may be too strong in the context of international organizations, in particular when cooperation is weak. Departing from this assumption, we posit that a country may choose not to implement the reform even if it has been approved collectively. Even in this scenario, countries may still choose to conform if the long-term benefit of sustaining cooperation outweighs the short-term loss of implementing an unfavorable reform.

Following Maggi and Morelli [2], we consider a repeated interaction between countries, an « infinitely repeated game », that allows to consider forward-looking behavior. At each stage, countries first decide whether to cooperate, and then either make a decision collectively (if cooperation has been agreed upon) or independently (if cooperation has been rejected). Finally, in case of cooperation, each country decides whether to implement the collective decision. A voting rule is said to be self-enforcing if all countries always choose to cooperate and to respect the collective decision.

We first show that the requirement of self-enforceability is more stringent than the participation constraints: inducing a sustainable cooperation is more difficult when decisions are not enforceable. Then, we characterise the optimal self-enforcing rule: it takes the form of a weighted majority rule, but with an additional veto power for some countries.

This provides a potential (cautious) justification for the type of rule used at the UN Security Council: while giving a veto to some countries clearly violates the democratic principle of equal representation, it may constitute a lesser evil, as it enables cooperation on important and sensible decisions, even in the absence of an external enforcement mechanism.



In order to obtain sharper recommendations we also consider a simpler model where countries differ in their population size but are otherwise identical. In such a model of « apportionment », we show that countries must receive weights proportional to their populations, except for the smallest ones, which must all receive an equal weight larger than their population. The result thus offers a rationale for a minimum representation of smaller countries, as well as the principle of « degressive proportionality »,[5] two explicit requirements for the composition of the European Parliament stated in the Treaty of Lisbon. It also echoes the distribution of weights at the US Electoral College, where each state is allocated a baseline of 2 electors plus a number of electors proportional to its population.


[1] This rule prevails for most decisions. To be precise, the rule also precludes a group of less than four countries to block a reform.

[3] For instance, the EU has exclusive competence over custom unions, competition policy, monetary policy (for countries in the Eurozone), common fisheries policy, and common commercial policy. The EU also holds shared competence over various other domains such as the internal market, agricultural policy, environmental policy, and consumer protection.

[4] The UK rebate on its contribution to the EU budget dates from 1984.

[5] The allocation of seats in a parliament is « degressively proportional » when more populous districts have more seats than smaller ones, but less seats per capita. 



[1] Barbera, S. and M. O. Jackson (2006): On the weights of nations: assigning voting weights in a heterogeneous union," Journal of Political Economy, 114, 317-339.

[2] Maggi, G. and M. Morelli (2006): “Self-enforcing voting in international organizations," The American Economic Review, 96, 1137-1158

[3] Wilcox, F. O. (1945): “The Yalta Voting Formula." American Political Science Review, 39, 943-956.


About the Authors



Antonin Macé is a researcher in economics at the CNRS, working at the Aix-Marseille School of Economics. He obtained a Ph.D. in economics from Ecole Polytechnique in 2014. His research focuses on collective decisions, analysed with the tools from social choice, game theory and experimental economics. 






Rafael Treibich is an assistant professor in the department of Business and Economics at the University of Southern Denmark. He holds a Ph.D. in economics from Ecole Polytechnique. His research focuses on political economy and microeconomic theory. 



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