A core party is undefeated in pair-wise majority preference comparisons against all other political parties. We use CSES survey self-reported voter preferences data and moment inequalities testing methods to identify potential core and Condorcet winner parties. We carry out tests for 1,168 parties over 195 election surveys. We cannot reject the hypothesis of a core party in nearly half the tests, while we cannot reject the hypothesis that a party is a Condorcet winner in only about 7.36% of tests. We relate the incidence of apparent core parties to party-specific observables; and find that parties that pass the core test are over 25% more likely to participate in a post-election government, even when controlling for common predictors of inclusion in the cabinet.