Generative adversarial networks (GANs), modeled as a zero-sum game between a generator (G) and a discriminator (D), allow generating synthetic data with formal guarantees. Noting that D is a classifier, we begin by reformulating the GAN value function using class probability estimation (CPE) losses. We prove a two-way correspondence between CPE loss GANs and $f$-GANs which minimize $f$-divergences. We also show that all symmetric $f$-divergences are equivalent in convergence. In the finite sample and model capacity setting, we define and obtain bounds on estimation and generalization errors. We specialize these results to $α$-GANs, defined using $α$-loss, a tunable CPE loss family parametrized by $α\in(0,\infty]$. We next introduce a class of dual-objective GANs to address training instabilities of GANs by modeling each player's objective using $α$-loss to obtain $(α_D,α_G)$-GANs. We show that the resulting non-zero sum game simplifies to minimizing an $f$-divergence under appropriate conditions on $(α_D,α_G)$. Generalizing this dual-objective formulation using CPE losses, we define and obtain upper bounds on an appropriately defined estimation error. Finally, we highlight the value of tuning $(α_D,α_G)$ in alleviating training instabilities for the synthetic 2D Gaussian mixture ring as well as the large publicly available Celeb-A and LSUN Classroom image datasets.