Appendices for No-regret Learning in Price Competitions under Consumer Reference Effects A Expanded Literature Review

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.