Consumer bounded rationality and rigidity/flexibility retail price patterns

I revisit the model of market competition with boundedly rational consumers due to Spiegler (2006), in which firms compete in price distributions and consumers use a naive sampling procedure to evaluate them. I assume that firms can assign weight to arbitrarily low prices, and consumers have a non-trivial ex ante outside option. In symmetric Nash equilibrium, firms charge a high "regular price"with positive probability, and in addition randomize continuously over an interval of "sale"prices that are bounded away from the regular price. Sales become less frequent but more drastic as the number of competitors increases and as the consumer's outside option becomes more attractive.