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Since many random-access protocols are designed based on the assumption that all terminals follow the same medium-access-control (MAC) policy, they can be vulnerable to the selfish behavior of each terminal. This paper proposes robust random-access protocols for wireless networks in fading environments, where each terminal operates in a Nash equilibrium (NE) of a random-access game. Since any deviation of a terminal from an NE penalizes the terminal, the network with this protocol becomes more robust by preventing a selfish terminal from violating the access protocol. We model slotted ALOHA and carrier-sense multiple access (CSMA) as games and show that the strategies in symmetric equilibria of these games have the properties so that a terminal with a better channel state is more likely to access the channel. Thus, these protocols ensure multiuser diversity as well as robustness, a conclusion confirmed by numerical results.