We present a systematic framework of indices designed to characterize Large Language Model (LLM) responses when challenged with rebuttals during a chat. Assessing how LLMs respond to user dissent is crucial for understanding their reliability and behavior patterns, yet the complexity of human-LLM interactions makes systematic evaluation challenging. Our approach employs a fictitious-response rebuttal method that quantifies LLM behavior when presented with multiple-choice questions followed by deliberate challenges to their fictitious previous response. The indices are specifically designed to detect and measure what could be characterized as sycophantic behavior (excessive agreement with user challenges) or stubborn responses (rigid adherence to the fictitious response in the chat history) from LLMs. These metrics allow investigation of the relationships between sycophancy, stubbornness, and the model's actual mastery of the subject matter. We demonstrate the utility of these indices using two physics problems as test scenarios with various OpenAI models. The framework is intentionally generalizable to any multiple-choice format question, including on topics without universally accepted correct answers. Our results reveal measurable differences across OpenAI model generations, with trends indicating that newer models and those employing greater "Reasoning Effort" exhibit reduced sycophantic behavior. The FR pairing method combined with our proposed indices provides a practical, adaptable toolkit for systematically comparing LLM dialogue behaviors across different models and contexts.