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174 CHAPTER 6 Fully Distributed Learning Algorithms ·························································································· · · the hybrid mean field limit game dynamics generated by these learning schemes satisfies · · · also the weighted equilibrium stationarity property. · Proof. If x is an equilibrium then x is a rest point of all the dynamics in the support ~ of . We will now prove that any rest point of the combined dynamics in the support ~ of is an equilibrium. Suppose that this is not the case. Then there exists at least one j such that x is not a rest point of the dynamics generated by the learning scheme j which satisfies (WES). This means that x is not an equilibrium. We conclude any rest point of combined dynamics is an equilibrium. This completes the proof. Remark (How to eliminate the rest points that are not equilibria?) Consider the , family of learning schemes generated by a ,a = max(0, u (x(t)) - u (x(t))) , 1. a a It is easy to see that this family satisfies the property (WES). We deduce that if the population is constituted of 99% of players using a learning scheme via , and 1% of the population use a replicator-based learning scheme, then the resulting com- bined dynamics will satisfy the property (WES). We conclude that every rest point of the replicator dynamics which is a non-Nash equilibrium will be eliminated using