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By analysing an infinitely repeated game where unit costs alternate stochastically between low and high states and where firms follow a price-matching punishment strategy, we demonstrate that the best collusive prices are rigid over time when the two cost levels are sufficiently close. This provides game theoretic support for the results of the kinked demand curve. In contrast to the kinked demand curve, it also generates predictions regarding the level and the determinants of the best collusive price, which in turn has implications for the corresponding collusive profits. The relationships between such price rigidity and the expected duration of a high-cost phase, the degree of product differentiation, and the number of firms in the market are also investigated.