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This paper explores the relation between equilibrium coarsenings and equilibrium refinements via Bertrand competition example and similar situations, it shows that the typical equilibrium coarsening --- a unique correlated equilibrium --- is equivalent to the unique Nash equilibrium itself, is also equivalent to the equilibrium refinement, for the standard n-firms Bertrand competition model with linear demand and symmetric, linear costs in the most special and simplest case, and compares some wonderful and remarkable differences of the existence, uniqueness, stability, connectivity, and strategic property of Nash equilibrium and correlated equilibrium between Cournot and Bertrand model. We also propose some open questions.