Perplexity, a metric often used to evaluate the performance of machine learning models, particularly in natural language processing, has limitations that may not be immediately obvious. The authors argue that while perplexity measures a model's surprise at its outputs, it doesn't reliably indicate which model is better in terms of accuracy. They present a theoretical framework using recent findings on Transformer models to demonstrate that a model could exhibit low perplexity while still failing to predict certain outputs correctly. Essentially, achieving low perplexity doesn’t guarantee strong generalization or accuracy. The authors prove that if a compact decoder-only Transformer model predicts a sequence confidently, there must be another sequence with low perplexity that the model does not predict well. This relationship complicates the use of perplexity as a sole criterion for model selection. Their analysis of iso-perplexity plots reveals that a model's confidence must be matched by an increase in accuracy for it to be deemed superior based on perplexity. The findings challenge the assumption that lower perplexity always correlates with better performance, suggesting that relying solely on this metric may misguide researchers and practitioners in model evaluation and selection.