In this chapter I explore three extensions to the mathematical model presented in Chapter 4. First, I show that the results do not depend upon the balance of minority to majority assumed in Chapter 4. Instead, at any ratio of majority to minority size, candidates might side with an intense minority over a less-intense majority. Candidates do not, however, side with any intense minority, siding only with those of a sufficient combination of size and intensity. Second, I consider the free-rider problem in large electorates. I discuss a modification to the mathematical model where increasing the size of the electorate does not prevent some voters from choosing to incur costly political action. Finally, I consider the social welfare implications of intensity theory showing that utilitarian welfare can sometimes be higher with frustrated majorities and costly political action than under simple majority rule.

In this chapter I present results from a mathematical formalization of intensity theory. I apply tools from game theory to analyze the dynamics of electoral competition when voters vary in how much they care about policy and when candidates do not know which voters care more and less intensely. I show that candidates for office choose policy platforms as a function of the size and intensity of opposing policy coalitions. Candidates sometimes set policy with an intense minority even though they know that a less-intense majority wants the opposite policy. But they also sometimes choose not to frustrate majorities even with an intense minority.

In this chapter I walk through the how and why of a theory of issue intensity and electoral competition and build the basics of the mathematical model used to explore intensity theory. I lay out six foundational assumptions of the model drawn from existing theories of elections in political science or political economy. The assumptions rule out current explanations for frustrated majorities so that I can show that the combination of issue intensity and electoral competition alone can cause frustrated majorities. I show how costly political action becomes an important part of the story when candidates cannot perfectly observe the issue intensity of voters. I then present a simple mathematical model with numerical examples to provide intuition for analysis in subsequent chapters.