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We examine the characteristics of schools preferred by parents in New Orleans, Louisiana, where a "portfolio" of school choices is available. This tests the conditions under which school choice induces healthy competition between public and private schools through the threat of student exit. Using unique data from parent applications to as many as eight different schools (including traditional public, charter, and private schools), we find that many parents include a mix of public and private schools among their preferences, often ranking public schools alongside or even above private schools on a unified application. Parents who list both public and private schools show a preference for the private sector, all else equal, and are willing to accept lower school performance scores for private schools than otherwise equivalent public options. These parents reveal a stronger preference for academic outcomes than other parents and place less value on other school characteristics such as sports, arts, or extended hours. Public schools are more likely to be ranked with private schools and to be ranked higher as their academic performance scores increase.