Abstract: Average selection operators in random environments are introduced in the first place, and it is proved that the sequence of probability distributions in agents' strategies set, when repeatedly operated by average selection operator, converges to the probability distribution in the subset of strategies that takes the largest value of average fitness. Then, the concept of average selection operator is extended to the multi-agents game problems, and it is
proved that the fixed points of such an operator are just the Nash equilibria of games.

Key words: Average selection operators, Evolution of probability distribution, Nash equilibrium