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Minimal winning coalitions are alignments of parties or politicians just large enough to defeat rivals and no larger. In such coalitions, defection by a single member is suﬃcient to render the coalition no longer large enough to ‘win.’ Riker (1962) introduced the idea of minimal winning coalitions in the study of electoral and legislative politics as an alternative to the view expressed in Downs (1957). Downs argued that politicians are primarily oﬃce seekers rather than policy makers or allocators of resources. As such, they maximize electoral support and, therefore, forge coalitions as large as possible. Riker’s decision makers make authoritative allocation decisions and so seek to minimize the number of claimants on the distribution of resources. A vast literature on coalition formation and government stability has grown out of the debate between Riker and Downs.
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The Downsian model indicates that on unidimensional issues and in winner-take-all elections, politicians adopt (usually centrist) policy positions in order to maximize their vote share. Downs’s politicians care only about winning oﬃce. They do not concern themselves with the policy or private goods concessions they must make to others in order to win. Riker, in contrast, argued that maximizing votes is costly. Voters are attracted to a candidate by promises about personal beneﬁts. Candidates have preferences of their own about the distribution of scarce resources in the form of private goods to their backers and leftover resources for their own use. To attract the maximum number of votes, politicians must pay a cost by sacriﬁcing some personal interests or granting private side-payments to prospective supporters in an eﬀort to avoid alienating potential voters. Riker argued that rational politicians, motivated primarily by a desire to control resources, seek to attract just enough votes to win and no more, subject to variation above minimal winning size only because of uncertainty about the preferences of voters or their loyalty. By forming minimal winning coalitions, politicians make as few concessions as possible while still controlling suﬃcient support to maintain governmental authority and pass legislation. Larger than minimal winning coalitions reduce the vulnerability of leaders to defections by individual legislators or small, highly issue-focused parties, but at a price.
2. Policy Preferences and Coalition Formation
In neither Downs’s nor Riker’s theories are decision makers motivated primarily by policy preferences. Subsequent developments in the coalitions literature reﬁne and challenge elements in the Riker–Downs debate, in part by introducing policy preference as another factor, besides victory or resource maximization, that motivates coalition formation. Axelrod (1970) and De Swaan (1973) argue that policy preferences restrict electoral coalitions to political parties with similar policy agendas. Both maintain that ideologically diverse minimal winning coalitions are less likely to form or survive than are ideologically compact coalitions. Axelrod calls such coalitions minimal-connected coalitions and contends that these arrangements are privileged when prospective coalition partners negotiate with one another. The contention that ideological aﬃnities play a part in selecting minimal winning coalitions is generally supported by the experiences of political parties in, for example, Israel, but is contradicted by the experiences with coalition formation in India.
3. Issue Trading and Multidimensional Issues
The idea that parties or politicians are election oriented, resource oriented, and policy oriented has been developed further. Several scholars extend these ideas by examining coalition formation in multidimensional spaces, that is, in circumstances when policy preferences cannot be portrayed on a single line, but rather reﬂect greater complexity or linkages across issues. Stokman and Van den Bos (1992) constructed an exchange model that identiﬁes likely vote-trading partners and the trades that can bring them into coalitional alignment. This model and variations on it have been subjected to extensive empirical testing, with impressive results. Others, notably Laver and Schoﬁeld (1990) and Laver and Shepsle (1996) have used game theory and spatial models to investigate the theoretical and empirical basis for arguing that coalition formation is compact in terms of policy diﬀerences in parliamentary settings with multidimensional issue spaces. This body of research, also with signiﬁcant empirical testing, moves the study of minimal winning coalitions beyond the single-issue or unidimensionality envisioned in the original Downs–Riker debate and places the issues in the context of issue trading and logrolling. Furthermore, it moves beyond the cooperative game theory framework proposed by Riker, making use of a mix of insights from noncooperative game theory in which promises are not binding, but are enforced only by self-interest, and from advances in social choice theory. This expanded literature examines how political systems can use institutions or agenda control to circumvent the problems identiﬁed in Arrow’s Impossibility Theorem. That theorem shows that under speciﬁc, generally reasonable conditions of democratic choice, no rule for aggregating preferences exists that can guarantee a positive direct translation of individual preferences into policy outcomes, with the exceptions of unanimity or dictatorship. In analyses of multiple, connected issues, relatively closely shared policy preferences among contending coalition partners is a necessary condition for coalition formation. Especially in Laver and Shepsle, multidimensional issue preferences are used to determine coalition membership and membership is rewarded through policy inﬂuence. Policy inﬂuence is disseminated, according to them, by using cabinet portfolios as the principal payoﬀ to parties or individuals in parliamentary coalitions.
4. Coalition Payoﬀs
The idea of linking coalition membership to cabinet posts goes back to the earliest formal and empirical examinations of coalition formation. The size of coalitions carries implications regarding the distribution of payoﬀs or beneﬁts to the members, especially regarding cabinet posts. Research on minimal winning coalitions, however, generally assumes that the beneﬁts distributed to members do not inﬂuence their subsequent legislative or electoral competitiveness, that is, the payoﬀs from joining a successful coalition—including cabinet posts, greater inﬂuence over the legislative agenda, control over patronage, etc.— frequently are treated as if they do not translate into resources that subsequently aﬀect the competitiveness of political parties. Browne and Franklin (1973), for instance, argue that such payoﬀs are distributed in proportion to the party’s size in the coalition, and give no consideration to whether payoﬀ allocations inﬂuence the future size of parties. Bueno de Mesquita (1978), in contrast, oﬀers evidence that the most valuable cabinet portfolios inﬂuence electoral prospects and, therefore, alter the future size of political parties. His analysis shows that party size is endogenous to decisions about coalition payoﬀs. This suggests that valuable payoﬀs are highly contested and so are not routinely distributed according to a simple proportionality formula.
5. Challenges to the Idea of Minimal Winning Coalitions
Groseclose and Snyder (1996) challenge the foundations of the debate over minimal winning coalitions. They note that virtually all formal models of coalition formation, as well as the related literatures on logrolling and vote-trading, either predict or assume minimal winning coalitions. They, however, suggest a model in which super-majority coalitions are preferred to minimal winning coalitions. In doing so, they endorse Riker’s idea that politicians seek to form coalitions that are as cheap as possible. They show, however, that if coalition builders—they call them vote buyers—move sequentially rather than simultaneously, and if the losing vote buyer is always given a last chance to seek defectors from the winning vote buyer’s coalition, then minimal winning coalitions will not generally be cheapest for the buyer. Rather, larger than minimal winning coalitions will generally be expected to form in equilibrium. Bueno de Mesquita et al. (2000) build on this work to show the conditions under which super-majority or minimal winning coalitions will form when politicians are motivated by reselection and, secondarily, by a desire to maximize their personal welfare. They show that political systems governed by small coalitions emphasize rentseeking at the expense of eﬃcient production of public goods, while leaders of systems dependent on larger coalitions shift resources into the provision of public goods and away from private goods allocations.
Strom (1990) also presents a challenge to the minimal winning coalition perspective. He examines minority coalitions, primarily in Europe, and shows that, in seeming contradiction to the equilibrium expectations formed in the literature, they are a common feature of the political landscape and in many cases they endure as the basis of on-going governance. Usually they consist of a single, minority party government that seeks legislative support on an ad hoc basis as it moves from one issue to another. Strom’s argument relies on political institutions to induce equilibria, in this case equilibria that include minority coalitions. He further supports minority coalition formation by showing a strategic dependence between party behavior in government and subsequent electoral accountability.
The size of winning coalitions proves to be highly variable empirically. Though there is a central tendency to favor nearly minimal winning coalitions, variations in the motivations of leaders seem to tilt them toward or away from minimal winning coalitions. Those who are more concerned with power and control, rather than policy, tend to pursue larger coalitions, as suggested by Downs. Those with a stronger concern for promoting a policy agenda tend to form coalitions just large enough to win and no larger. Because of this diﬀerence, the theory of minimal winning coalitions not only makes predictions about the size of legislative or electoral groupings, but also provides clues for discerning the motives of leaders from their strategies for gaining support to win and hold oﬃce.
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