If mixed strategies are permitted, then even without utility transfers, the players can improve on the (d,d) outcome in the asymmetric case. For instance, with the first asymmetric PD given above, if Alice cooperates 45% of the time, and Bob cooperates 70% of the time, Alice will have an average payoff of 5.91, while Bob will have a payoff of 7.95
If mixed strategies are permitted, then even without utility transfers, the players can improve on the (d,d) outcome in the asymmetric case. For instance, with the first asymmetric PD given above, if Alice cooperates 45% of the time, and Bob cooperates 70% of the time, Alice will have an average payoff of 5.91, while Bob will have a payoff of 7.95