How can we design electoral institutions to achieve racially and ethnically fair representation in modern democracies? While a body of research examines the relationships between different electoral systems and the level of minority representation, remarkably less is known about how "changing" electoral systems from one form to another would affect minority representation. To overcome this limitation, this dissertation develops three new methodologies for studying the effects of electoral engineering on minority representation. In the first chapter, I offer a parsimonious mathematical model to explain and predict when racial minority candidates run for office and win in a particular district under first-past-the-post. Using novel datasets from Louisiana mayoral elections and state legislative general elections, I show that the mathematical model can accurately predict both minority candidate emergence and electoral victory while demonstrating that the model can answer relevant questions in redistricting and voting rights cases. In the second chapter, I propose a potential outcomes framework to study the causal effects of policy interventions on ranked outcome data. To illustrate the advantages of the framework, I reanalyze a survey experiment gauging the effect of different information on people's attitudes toward police violence and study ballot order effects in ranked-choice voting. In the third chapter, I introduce a spatial model for ethnic party competition in ethnically divided societies to study whether and under what conditions switching from first-past-the-post to ranked-choice voting yields moderation in ethnic party competition and reduces the level of racial and ethnic polarization. To simulate ethnic party competition under various conditions, I develop an algorithm based on agent-based modeling that is readily accessible to researchers and practitioners. Combining clustering and ecological inference with ranked ballot data from Bay Area Mayoral elections, I also show that switching from first-past-the-post to ranked-choice voting does not mitigate the level of racial polarization in the particular context. By integrating substantive knowledge with methodological innovation, this dissertation provides new opportunities for future research on electoral engineering and minority representation.