CD8 T cells which recognize neoepitopes are central to immune control of cancers. T cells, through T cell receptors, recognize neoepitope peptide-MHC I on cancer cells. The strength (or avidity) of the T cell receptor- peptide-MHCI interaction is a critical variable in immune control of cancers in vivo. Here, analyzing neoepitope-specific CD8 cells of distinct avidities to validated neoepitopes of a mouse tumor, low avidity T cells are observed to be the sole mediators of cancer control in vivo. Low avidity cells are also observed to be solely responsive to checkpoint blockade in mice and in humans. High avidity T cells are not only ineffective but suppress immune response. The mechanistic basis of the differences in the anti-tumor activity of cells of low and high avidities, lies in the higher exhaustion status of high avidity cells. High avidity T cells have a distinct transcriptomic profile which creates an Avidity Score, used here to identify in silico, the low and high avidity T cell populations in mice and humans. Surprisingly, CD8 T cells with identical TCRs exhibit a wide variation in avidities, suggesting a novel level of regulation of T cell activity. These observations, and the mechanisms that underlie them, shed new light on the endogenous T cell response to cancer and suggest new avenues in cancer immunotherapy.