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There is an inconsistency between theory and observations with regards to the contributions of strong and weak players to the public goods. Theory suggests that contributors are either strong players in asymmetric games or cooperative players in symmetric games, but experiments indicate that the weak players in asymmetric systems also contribute to public goods. To reconciling these conflicts, we here study an evolutionary volunteer's dilemma game by assuming different roles can be interchangeable. In this model, the evolutionary dynamics shows the dynamics of multiple equilibria that depend on initial conditions, which can be interpreted as the production modes of public goods under different circumstances. Precisely, we find that the survival of strong individuals with mixed strategies is associated with two different outcomes. One result is equal to Selten's (1980) model, and public goods are produced by strong players if the defectors are weak players, where strong defectors are scarce in the initial condition. In another result, the weak individuals with mixed strategies produce public goods if the defectors are strong individuals, where the strong cooperators are absent in the initial condition. Concretely, the game degenerates to a mixed population of strong individuals with the weak players going extinct, and the weak defectors are scarce in the initial condition. The studied evolutionary game may help to explain the emergence of diverse forms of cooperation in asymmetric evolutionary games.