We tackle fair multi-agent optimization problems and use a generalized Gini index to determine a fair and efficient solution. We claim that considering mixed solutions (i.e., lotteries over solutions) enables to enhance the fairness of an optimal solution. Interpreting a fair multi-agent optimization problem as a zero-sum two-player game between an optimization player choosing a solution and an adversary which has some control over the payoffs of the game, we propose two methods (a cutting-plane method and a double oracle method) to compute an optimal mixed solution. Numerical tests are provided to compare their efficiency.