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We consider a mixed duopoly service system with two service providers (SPs): one private and the other public. The public SP's objective is welfare maximization, whereas the private SP is profit driven. Customers are heterogeneous in terms of their quality tastes and choose from three options, namely, joining the private queue, joining the public queue, and balking (or taking an outside option). We first consider the scenario where the private SP provides a premium service and the public SP provides a regular service. Paradoxically, we find that although the public SP's objective is welfare maximization, welfare in the system can be less than that obtained with only profit-seeking private SPs. We further demonstrate that the maximum social welfare is achieved by partially privatizing the public SP, that is, by including both welfare and profit maximization as arguments in its objective function. In extreme cases in which the public SP's capacity is very small, fully privatizing the public SP can be socially desirable. We then consider the alternate setting where the public SP provides a premium service and the private SP provides a regular service and obtain similar conclusions. These findings, however, rely on the assumption that an outside option exists for balking customers, and one should be cautious in generalizing to no-balking situations. This paper was accepted by Jayashankar Swaminathan, operations management. Funding: W. Zhou was supported by the National Natural Science Foundation of China [Grants 71925002 and 71731006]. W. Huang acknowledges the financial support from the National Natural Science Foundation of China [Grants 71801096 and 72101090] and the Guangdong Basic and Applied Basic Research Foundation [Grant 2022A1515011983]. V. N. Hsu was supported by the General Research Fund of the Hong Kong Research Grants Council [Grant CUHK14526216]. P. Guo received financial support from the Hong Kong Research Grants Council [Grant 15502820]. Supplemental Material: The e-companion and data are available at https://doi.org/10.1287/mnsc.2022.4424. [ABSTRACT FROM AUTHOR]