Switching costs in infinitely repeated games

We show that small switching costs can have surprisingly dramatic effects in infinitely repeated games if these costs are large relative to payoffs in a single period. This shows that the results in Lipman and Wang do have analogs in the case of infinitely repeated games [Lipman, B., Wang, R., 2000. Switching costs in frequently repeated games. J. Econ. Theory 93, August 2000, 149-190]. We also discuss whether the results here or those in Lipman-Wang imply a discontinuity in the equilibrium outcome correspondence with respect to small switching costs. We conclude that there is not a discontinuity with respect to switching costs but that the switching costs do create a discontinuity with respect to the length of a period.