This phase III trials studies whether maintenance immunotherapy (nivolumab) following definitive treatment with radiation and chemotherapy (cisplatin) result in significant improvement in overall survival (time being alive) and progression-free survival (time being alive without cancer) for patients with intermediate risk human papillomavirus (HPV) positive oropharynx cancer (throat cancer) that has spread to nearby tissue or lymph nodes. Drugs used in chemotherapy such as cisplatin work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Radiation therapy uses high energy rays to kill tumor cells and shrink tumors. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. It is not yet known whether chemotherapy and radiation therapy followed by maintenance nivolumab therapy works better than chemotherapy and radiation therapy alone in treating patients with HPV positive oropharyngeal cancer.

PRIMARY OBJECTIVE: I. To assess the efficacy of concurrent definitive therapy followed by nivolumab compared with concurrent definitive therapy followed by observation in terms of overall survival (OS). SECONDARY OBJECTIVES: I. To further assess the efficacy of nivolumab compared with observation in terms of: Ia. To evaluate treatment effect within the subset of patients tested as PD-L1+ . Ib. To evaluate the prognostic effect of baseline saliva and/or plasma HPV status . Ic. To evaluate the prognostic effect of mutation burden among patients on the nivolumab arm. Id. To evaluate the association of 12-week post therapy fludeoxyglucose F-18 (FDG) positron emission tomography(PET)/computed tomography (CT) OS and progression free survival (PFS). Ie. To establish the prognostic value of standardized uptake value (SUV) max of primary tumor or neck nodal metastasis of baseline FDG PET/CT for OS (and/or PFS). If. To correlate SUV max of primary tumor or nodal metastasis of baseline FDG PET/CT with PD-L1 expression (positive versus \[vs.\] negative). Ig. To compare the PET based therapy response assessment (Hopkins criteria) to the Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 assessment at 12 week post chemoradiation therapy, for patients who have a PET/CT scan at 12 weeks. II. To assess the efficacy of concurrent definitive therapy followed by nivolumab in terms of progression free survival (PFS). OUTLINE: Patients are randomized to Arm A or Arm B. Patients in Arm B may cross-over to Arm C with clearly documented disease progression. ARM A: Patients receive cisplatin intravenously (IV) over 60 minutes weekly and intensity modulated radiation therapy (IMRT) 5 days a week for 7 weeks for a total of 35 fractions. Within 4 weeks after completion of concurrent therapy, patients receive nivolumab IV once weekly over 30 minutes every 4 weeks for 12 months in the absence of disease progression or unacceptable toxicity. ARM B: Patients receive cisplatin IV over 60 minutes weekly and IMRT 5 days a week for 7 weeks for a total of 35 fractions, and then go on observation. Patients will be offered the option to cross-over to Arm C if they have clearly documented progression within 12 months from the end of cisplatin/radiation therapy. ARM C: Patients receive nivolumab IV over 30 minutes every 4 weeks for 12 months in the absence of disease progression or unacceptable toxicity. All patients undergo computed tomography (CT) or Fludeoxyglucose F-18 (FDG) positron emission tomography (PET)/CT scans throughout the trial. Patients may undergo echocardiography (ECHO) as clinically indicated. Additionally, patients undergo blood sample collection during screening. After completion of study treatment, patients are followed up every 3-6 months for 3 years and then annually for a total of 10 years.