Game-Theoretic Learning and Distributed Optimization in Memoryless Multi-Agent Systems by Tatiana Tatarenko

Author:Tatiana Tatarenko
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

To prove this theorem we need the following lemma:

Lemma 3.6.1

Let be the distribution with the density π β(t) defined in (3.29). Then, if ϕ is a continuous function defined on the compact set ,

(3.37)

where .

Proof

According to the formula for total variation distance (Theorem A.​1.​1 in Appendix A.1) and (3.29),

Since ϕ is continuous on and is compact, I(β) can be differentiated under the integral sign and . Hence,

Thus, we get

□

Remark 3.6.1

Note that the lemma above does not use any information about the measure of the set and, thus, proposes another way to prove the statement in Theorem 3.6.4 in the case of the following assumption on the utility and potential functions: for any i ∈ [N],  for any , and . Indeed, because of (3.37),

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