Faculty & Research -How Do Coalitions Break Down? An Alternative View

How Do Coalitions Break Down? An Alternative View

Motivated by recent coalition splitting events through unilateral countries’ withdrawals, we propose an alternative theory based on two key assumptions that: i) the payoff sharing rule within coalitions is not necessarily set according to any optimality criterion, and, ii) players initially behave as if the coalition will last forever. Using dynamic game theory applied to international environmental agreements, we find that splitting countries are precisely those which use to benefit the most from the coalition. Suitable sharing rules should be used to prevent coalition splitting.

The recent years have noticed numerous withdrawals of countries from international organizations:

  1. the most recent happens on July 7, 2020, the Trump administration formally notified the United Nations that it is pulling out of the World Health Organization, which effective as of July 6th, 2021;
  2. during the same Trump presidency period, on June 1, 2017, President Trump announced that the U.S. would cease all participation in the 2015 Paris
  3. Agreement on climate change mitigation until some fair conditions to the U.S.A can be negotiated;
  4. the United Kingdom withdrew from the European Union on January 31, 2020;
  5. Canada withdrew from Kyoto Protocol on December 13, 2011…etc

Beside these institutional withdrawals, companies’ splitting happens all the time. Some of the splitting also has quite influential social and market effects, such as when, through October 2014, technology giant Hewlett-Packard, known as HP, split into two companies as a result of separating its computer and printer businesses from its faster-growing corporate hardware and services operations. This led to the loss of 5,000 jobs as part of its turnaround plan. On July 2015, PayPal spun off from eBay and this split benefitted both eBay’s marketplace business by letting it accept different forms of electronic payment and also gives PayPal more autonomy to work with other potential partners, such as Amazon or Alibaba.

Obviously, this kind of phenomena has always drawn the attention of academic economists, theorists and empirists. A few economic papers have already explored the diverse impacts of Brexit (e.g. Sampson, 2017), and many more papers have been devoted to the withdrawals from International  Environmental Agreements (IEAs) starting from the U.S. withdrawal from the Kyoto Protocol (e.g. Bucher et al., 2002). Nonetheless, none of the articles belonging to these literature streams has already explored the (optimal) lifetime of coalitions, being the related economic literature essentially static, relying on inherent static stability concepts (see Bréchet et al., 2011). Even the much more dynamic literature on IEAs has not addressed explicitly so far the issue of unilateral splitting of a given coalition member and related optimal timings. Our paper proposes a direct and explicit formulation of the unilateral splitting problem, characterizes the associated optimality conditions, and draws the corresponding policy recommendations.

Conceptual framework

We shall present here the main distinctive features of our conceptual framework. We abstract away from technicalities (which can be found in the original article).

Our paper departs from the above mentioned dynamic games literature in two major ways:

  1. First of all, the sharing rule within coalitions is not necessarily set according to any optimality and/or stability criterion.
  2. Second, players initially behave as if the coalition will last for ever.

At first glance, the two departures seem rather realistic. The coalitions are typically based on a number of, say, constitutional rules specifying the duties and benefits corresponding to each member of the coalition. Usually, the coalition may obviously entail large heterogeneities across members, in particular in multi-country coalitions, technological, demographic or geographic notably. It’s unlikely that the constitutional rules at the dawn of the coalition can cope with all these discrepancies, and meet any kind of optimality as it’s understood in economics. It’s also quite reasonable that when engaging in essential (and somehow existential) collective initiatives like a strategic political alliance or a long-term environmental agreement, no member will start playing against it, they will rather act optimally in accordance with the constitutional rules as if the coalition will last till the end time agreed upon. If the constitutional rules are non-negotiable or if renegotiation is very costly, compliance to these rules may become unbearable as time passes, either because the rules, being too rigid, would make exit preferable (endogenous exit) or because an exogenous (symmetric or asymmetric) shock occurs undermining the political, economic or historical rational behind the coalition creation. We shall consider the endogenous exit problem in this paper, specializing in the case of time-invariant and non-negotiable constitutional rules.

While realistic and fitting a variety of situations in very different contexts, our assumptions entail two theoretically unpleasant features: predetermined (non-optimal) shares under the coalition setting and time inconsistency. As we will see clearly when solving the dynamic games posed, our alternative frame involves the absence of forward-looking behavior in the coalition stage, which enables a forward induction solution method. We study this case till its ultimate consequences, including policy implications (which does make perfect sense as this case is based on reasonably realistic assumptions). Nonetheless, we also provide with the solution to the standard forward-looking counterpart where players do anticipate the coalition breakdown from $t=0$ and behave accordingly. Despite that the induced solution scheme is opposite to the counterpart in our alternative case and consists in backward induction, we show that the ultimate optimal splitting problems are analogous. Moreover the main policy implication (related to the sharing rule) is qualitatively the same. That’s to say the equilibrium outcomes generated by our alternative frame is quite far from irrationality (if rationality corresponds to the pure forward-looking case).

In our theory, since we focus on coalition splitting, we specialize in the simple case where one single country can potentially break down its tie to the coalition. After the split, if any, a two-players non-cooperative game sets in: the splitting country and the remaining coalition treated as a single player. This mimics many of the recent coalition withdrawals occurrences mentioned above, and we believe the theoretical approach taken is deep enough to highlight some of the key determinants of coalition breakdowns. In our illustrative example, we use a two-stage pollution game where in the first stage players (countries) cooperatively decide about their production (and thus, about their pollution emissions) to minimize the cost of the (common) global pollution. A second stage of the game would arise if and only if one of the countries of the coalition decides to split, in such a case a non-cooperative game would start between the splitting country and the remaining coalition. We characterize the resulting equilibria (Markov Perfect Equilibria). Needless to say, for this conceptual frame to make sense, players need to be somehow heterogenous. They are so at least in their technological level (or productivity), in the extent of pollution damage (capturing the ecological cost in welfare terms) and in their relative institutional power in the initial coalition (as captured by their share of the joint coalition payoff), the three magnitudes play a central role in the splitting decision.

Findings and policy recommendations

The key result of the paper concerns the identification of the technological, ecological and institutional conditions leading to unilateral splitting, and the computation of the optimal splitting (if so) in closed form.  These conditions are summarized in a compact single one involving the three latter ingredients.

Basically, this compact splitting condition reads as follows:  for given technological gap (that’s the ratio between the average productivity of the remaining coalition to the productivity of the splitting country) and the pollution damage magnitude, the payoff share of the splitting country, say a, under coalition is required to be large enough for this player to engage in a coalition and to stay in for a finite time. Indeed, the constitutional parameter a is key in the optimal institutional dynamics: it’s key for the existence of an optimal finite time splitting, and it’s also key for the duration of the coalition.


Note: T is the splitting time (or equivalently, the duration of the coalition), player i is the splitting country, player J denotes the remaining coalition, ai and aJ are their respective productivity levels.  a is the share of the coalition payoff obtained by player i.

To dig deeper into the main formal result described above, one has to account for the ecological and technological heterogeneity aspects. A technologically lagged country may remain in the coalition for any value of the pollution ecological cost, provided the reward, as captured by a is large enough (in the case of two players, bigger than one-half). Suppose now that country is more advanced than average, what would the outcome be? Suppose that the pollution damage cost is small, then the benefits for this country to remain in the coalition are rather thin. In this case one may expect that the more advanced player’s technology, the larger the payoff share requested by this player to remain in the coalition. The contrary also holds true, if the pollution damage is large enough: in such a case, the more advanced the country, the lower the payoff share requested to remain in the coalition.

In the figure above, one can see how the splitting time evolves depending on the productivities of the players for two values of the constitutional parameter, a,  0.75 and 0.85, respectively. Two interesting findings emerge. First of all, the optimal splitting time is a decreasing function of the productivity of the splitting country for each α value. For fixed α, and a small enough unit cost of pollution, the incentives to remain in a coalition fade away as this country gets more productive relative to the remaining coalition , thus the declining patterns of the optimal switching time. Second, the optimal switching time may increase or decrease with a depending on the size of the technological gap: The switching time curve for α = 0.85 is above the counterpart for α = 0.75 when country i is relatively advanced, the reverse occurs when the latter country is relatively lagged. As country i is technologically lagged with respect to the remaining coalition, J, the smaller the share α, the longer this country will remain in coalition to take advantage of the technological “locomotive” J . When this is no longer the case, and country i becomes the technological leader, the technological gap magnitude does matter, and the previous finding may be reversed.

Several policy recommendations can be extracted from our theoretical analysis.  Let’s stress one of them:  only “big” enough countries may under certain conditions quit the coalition. More precisely, optimal finite time splitting requires the payoff share a to be big enough. Recall that the latter is determined by the Constitution of the coalition, reflecting in particular the relative historical, geographic, demographic and economic weight of the countries. If the Constitution is also meant to guarantee no-splitting, two avenues can be taken within our framework. One is to counterbalance the impact of too large payoff share by adding penalties to the constitution, making sure that penalties are increasing enough in the payoff share to discourage splitting. The second (non-exclusive) solution is to limit by Constitution the payoff share of all individual players, which guarantees that everlasting coalitions are the unique optimal institutional arrangement.


The paper is fully theoretical, the main results are established mathematically, they are completed with a few numerical and graphical illustrations. Technically, our analytical approach combines multistage optimal control tools with the typical techniques used to solve differential games. There exist an increasing number of papers using multistage optimal control to characterize optimal/ equilibrium regime transitions and the inherent optimal regime shift timings.  In contrast, much fewer papers merging multi-stage optimal control and dynamic games have come out, an exception is for example Boucekkine et al. (2011).

Applications and beneficiaries

This paper contributes to an increasingly active and impactful literature on the design of viable International Environmental Agreements (IEAs). Beside the academic impact, the context of the theory developed (the wave of unilateral withdrawals from multi-country political treaties and IEAs) and the political recommendations deduced (identifying the characteristics of the potential splitters) can be of interest to the international bodies coordinating and monitoring these coalitions.

Reference to the research:

Raouf Boucekkine, Carmen Camacho, Weihua Ruan, and Benteng Zou (2024), “How do coalitions break down? An alternative view,” Dynamic Games and Applications, 14, 157-194.

Consult the research paper

Related useful references (cited above)

Sampson T. (2017), “Brexit: The Economics of International Disintegration,” Journal of Economic Perspectives, 31 (4), 163-184.

Bucher B. C. Carraro, and Cersosimo, I. (2002), “Economic consequences of the US withdrawal from the Kyoto/Bonn Protocol,” Climate Policy, 2 (4), 273-292.

Bréchet, T., F. Gerard, and Tulkens, H.  (2011), “Efficiency vs. stability in climate coalitions: A conceptual and computational appraisal,” The Energy Journal, 32, 49-75.

Boucekkine R., J. Krawczyk, and Vallée, T. (2011), “Environmental quality versus economic performance: a dynamic game approach,” Optimal Control Applications and Methods, 32, 29-46.