For decades, market researchers have used conjoint analysis to understand how consumers make decisions when faced with multi-dimensional choices. In such analyses, respondents are asked to score or rank a set of alternatives, where each alternative is defined by multiple attributes which are varied randomly or intentionally. Political scientists are frequently interested in parallel questions about decision-making, yet to date conjoint analysis has seen little use within the field. In this manuscript, we demonstrate the potential value of conjoint analysis in political science, using examples about vote choice and immigrant admission to the United States. In doing so, we develop a set of statistical tools for drawing causal conclusions from stated preference data based on the potential outcomes framework of causal inference. We discuss the causal estimands of interest and provide a formal analysis of the assumptions required for identifying those quantities. Prior conjoint analyses have typically used designs which limit the number of unique conjoint profiles. We employ a survey experiment to compare this approach to a fully randomized approach. Both our formal analysis of the causal estimands and our empirical results highlight the potential biases of common approaches to conjoint analysis which restrict the number of profiles.