This paper considers convex games involving multiple agents that aim to minimize their own cost functions using locally available information. A common assumption in the study of such games is that the agents are symmetric, meaning that they have access to the same type of information. Here we lift this assumption, which is often violated in practice, and instead consider asymmetric agents; specifically, we assume some agents have access to first-order gradient information and others have access to the zeroth-order oracles (cost function evaluations). We propose an asymmetric learning algorithm that combines the agent information mechanisms. We analyze the regret and Nash equilibrium convergence of this algorithm for convex and strongly monotone games, respectively. Specifically, we show that our algorithm always performs between pure first- and zeroth-order methods, and can match the performance of these two extremes by adjusting the number of agents with access to zeroth-order oracles. Therefore, our algorithm incorporates the pure first- and zeroth-order methods as special cases. We provide numerical experiments on a market problem for both deterministic and risk-averse games to demonstrate the performance of the proposed algorithm.