Decentralized wireless networks require efficient channel access protocols to enable wireless nodes (WNs) to access dedicated frequency channels without any coordination. In this paper, we develop a distributed spectrum access protocol for the case where the WNs are equipped with multiple radio transceivers. We consider the case where the channels are identical and duly separated so that each of the users' antenna can access only one of the available channels. To model the competition amongst WNs, we formulate a particular multi-agent multi-carrier spectrum access game, where each WN has to decide at each iteration how many antennas and which frequency channels it has to access. To study the resulting equilibrium, we solve a multi-objective optimization problem and design a bi-level learning algorithm which is proven to converge towards a socially efficient and max-min fair equilibrium state.