A Study of the Pharmacokinetics and Safety of MK-8808 (MK-8808-002)

Inclusion Criteria: * Female participants of reproductive potential must demonstrate a serum β-human chorionic gonadotropin (hCG) level consistent with the nongravid state at the pre-study (screening) visit, and a negative urine pregnancy test within 24 hours prior to all doses and agree to use (and/or have their partner use) two acceptable methods of birth control beginning at least 2 weeks prior to administration of the first dose of study drug, throughout the study (including washout intervals between treatment periods/panels) and until at least 12 months after administration of the last dose of study drug in the last treatment period * The participant has a Body Mass Index (BMI) ≤35 kg/m\^2 at the prestudy (screening) visit * For Part A Only: The participant has a body surface are (BSA) ≤2.0 m\^2 at the prestudy (screening) visit. * Has satisfied at least 4 of 7 American Rheumatology Association (ARA) 1987 revised criteria for the diagnosis of RA * Is American College of Rheumatology (ACR) Functional Class I, II, or III * Had a diagnosis of RA made at least 6 months prior to the prestudy (screening) visit, was ≥ 16 years of age when diagnosed, and has active disease * Is on a stable oral, IM, or SC dose of methotrexate and is continuing to take methotrexate * Has an inadequate response or intolerance to at least one disease-modifying antirheumatic drug (DMARD) * For Part A: Participant is either naïve to biological therapy for RA or has had an inadequate response to previous or current treatment with an anti-tumor necrosis factor (TNF) treatment (patient could have failed up to three anti-TNF agents treatments) or participant has had intolerance up to three anti-TNF treatments. * For Part B: Participant has had an inadequate response to previous or current treatment with an anti-TNF treatment (patient could have failed up to three anti-TNF agents treatments) or participant has had intolerance up to three anti-TNF treatments * Participant has no clinically significant abnormality on electrocardiogram performed at the prestudy (screening) visit and/or prior to administration of the initial dose of study drug * For Part B Only: Participant is positive for rheumatoid factor (RF) or, if negative for RF, is positive for anti-CCP at screening visit * For Part C Only: Participant must have completed the first 52 weeks of treatment in the base study * For Part C Only: Participant achieved a minimum 20% response from baseline on the American College of Rheumatology (ACR) Responder Index (ACR20) at Visit 19 (last visit for the base study) Exclusion Criteria: * Mentally or legally incapacitated, has significant emotional problems at the time of the prestudy (screening) visit or during the conduct of the study or has a history of a clinically significant psychiatric disorder over the last 5 years * Creatinine clearance of ≤ 80 mL/min * History of stroke, chronic seizures or major neurological disorder * History of neoplastic disease, except treated basal cell carcinoma or carcinoma in situ of the cervix or other malignancies which have been successfully treated ≥ 5 years * History of leukemia, lymphoma, malignant melanoma, or myeloproliferative disease regardless of the time since treatment * History of coronary artery disease, congestive heart failure (New York Heart Association Class I-IV), or a history of clinically significant arrhythmia (including any history of atrial fibrillation, atrial flutter, or any sustained ventricular arrhythmia) * Hypersensitivity or allergy to rituximab or any of the excipients of MK-8808 or rituximab (MabThera® or Rituxan® ) * History of a rheumatic autoimmune disease other than RA (e.g. systemic lupus erythematosus (SLE), polymyositis, etc.) * Severe active infection of any type or history of a medically serious infection as defined by a history of treatment requiring hospitalization, long term IV outpatient treatment for systemic bacterial, viral or fungal infection, use of IV antibiotics within 30-days of screening, or use of antibiotic therapy three or more times in the last six months prior to screening * History of opportunistic infection * Active-virus vaccination within 4 weeks * Active tuberculosis with or without adequate treatment, history of latent tuberculosis without written confirmation from health care provider of adequate prophylaxis or any evidence of tuberculosis on a chest X-ray performed within 3 months of dosing * Chronic hepatitis B or hepatitis C infection or has human immunodeficiency virus (HIV) infection * Previously treated with rituximab (MabThera® or Rituxan®) or any investigational anti-CD20 antibody * Active use or planned use of a prohibited DMARD during the course of study participation, and/or insufficient washout from a prohibited DMARD at the time of the planned first dose of MK-8808/rituximab (MabThera® or Rituxan®) * Had major surgery, donated or lost 1 unit of blood (approximately 500 mL) within 4 weeks * Participated in another investigational study with length of time within at least 5 half-lives of the previous investigational study drug * Pregnant or breastfeeding or expecting to conceive * Allergy to murine proteins * Allergy or sensitivity to components of the drug vial or any of the materials used for infusion