There are also very recent works that address the dynamic pricing problem with consumer reference effects under uncertain demand. Nevertheless, these two lines of works are oblivious to consumer reference effects. In contrast to these two papers, our work studies price competitions over an infinite time horizon where reference prices adjust over time, and provides theoretical guarantees for the convergence of pricing strategies under the partial information setting. In their setting, the subgradient for each bidder's objective is a function of all bidders' decisions as well as its budget rate (i.e. total fixed budget divided by a given time horizon), which can be B.1 Proof of Theorem 3.1 (i) By first order conditions, we know that arg max We now follow a similar proof to that of Tarski's fixed point theorem: consider the set Note that convergence is monotonic because U () is nondecreasing. This implies that under Assumption 1, the interior SNE is unique.

Summary and Contributions: This paper studies a multi-period model: in each period, each of two firms posts a price for each product. The consumers demand for each of the products is linear in both prices and in addition linear in the reference price which captures past prices. Specifically, it is a weighted average of the previous-period reference price and the two previous-period posted prices. The firms do not know the specific demand function and have access only to the derivative of their revenue (a function that maps a price to revenue which is equal to demand times price) which is denoted by g_i(p_i). Note that g_i() depends on the other parameters but the firm views them as constants and is able to feed g_i with a possible choice of p_i and then get the revenue derivative for that price choice.