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the case of Israel, the phenomenon of “contribution by inaction” seemed insufficiently common to merit in-cluding in and thereby complicating the theory. The coalition against the Taliban, however, also included a country that contributed by inaction. India offered assistance to the coalition, but it was clear that Indian participation would reduce Pakistan’s willingness to assist the United States, so the United States demurred at India’s offer. The Israeli and Indian cases also appear similar in that each country arguably used the U.S. desire to keep them on the sidelines as a source of leverage over U.S. policies of interest to each country (respectively, Middle East peace talks and the status of Kashmir). An opportunity may thus exist to add a new type to the theory by including a variable for “relations among potential contributors” and studying these and other cases of contribution by inaction.

Finally, these burden-sharing studies demonstrate how a complex typological theory can be presented as a causal diagram, albeit a complex one. The causal diagram in Figure 11.1, from the first study, corresponds with the typological theory in Table 11.1 (the correspondence is inexact because Table 11.1 collapses the two domestic variables into one for presentational simplicity). Figure 11.1 groups together similar outcomes while still allowing for alternative paths to these outcomes (i.e., equifinality). Moving from the left to the right of the figure, the five boxes on the left represent the independent variables, the three “outcome” boxes represent the dependent variable, and the boxes on the right represent path-dependent interpretations of cases that might have arrived at the same outcome through different routes. Each possible path through the five boxes on the left corresponds with one of the types in Table 11.1 (except that the table has one fewer variable and hence 16 fewer possible paths). For example, Iran and China both arrived at Outcome 1, but through very different processes. Iran greatly valued the goal of an Iraqi defeat, but did not depend on the United States; it “rode free” on the efforts of the U.S.-led coalition that fought Iraq. China did not greatly value the goal of an Iraqi defeat, so it kept its distance by making only the minimal political contribution of not exercising its veto on the UN Security Council.

More generally, the four path-dependent interpretations in the figure turn on whether a state’s contribution, or lack thereof, matched the value it placed on the public good of reversing the Iraqi invasion. First, a state “rides free” if, like Iran, it values the good but does not contribute. Second, a state “keeps its distance” if, like China, it does not value the good and does not contribute. Third, a state “reveals its preferences and pays up” if it values the good and contributes. This could arise through various contributions of perceived threat, alliance dependence, and domestic politics, as in the cases of Britain and Egypt. Fourth, a state is “entrapped” if it does not value the good but contributes anyway due to alliance dependence, as in the cases of Japan and Germany.

Figure 11.1. Decision-Making Model of Security Coalition Contributions based on Perceptions of Public Good.

Limitations of Typological Theory and Potential Remedies

Despite the strengths and flexibility of typological theories, the development of typological theory suffers from important limitations.507 Researchers are liable to miss some possible causal relationships and to face indeterminacy in assessing others. The main reason for this is that extant historical cases may represent only a few of the combinations of variables that are possible in the social world. In addition, left-out variables and probabilistic causal mechanisms can further weaken causal inferences from case studies and the development of typological theories.

In practice, the severity of these limitations may be reduced through rigorous case study methods. First, as noted above, not all cases are equally theoretically informing, and a single