This phase II/III trial studies how well a reduced dose of radiation therapy works with nivolumab compared to cisplatin in treating patients with human papillomavirus (HPV)-positive oropharyngeal cancer that is early in its growth and may not have spread to other parts of the body (early-stage), and is not associated with smoking. Radiation therapy uses high-energy x-rays to kill tumor cells and shrink tumors. Chemotherapy drugs, 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. 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. This trial is being done to see if a reduced dose of radiation therapy and nivolumab works as well as standard dose radiation therapy and cisplatin in treating patients with oropharyngeal cancer.