Symmetry restoration by pricing in a duopoly of perishable goods

Competition is a main tenet of economics, and the reason is that a perfectly competitive equilibrium is Pareto-efficient in the absence of externalities and public goods. Whether a product is selected in a market crucially relates to its competitiveness, but the selection in turn affects the landscape of competition. Such a feedback mechanism has been illustrated in a duopoly model by Lambert et al., in which a buyer's satisfaction is updated depending on the {\em freshness} of a purchased product. The probability for buyer $n$ to select seller $i$ is assumed to be $p_{n,i} \propto e^{ S_{n,i}/T}$, where $S_{n,i}$ is the buyer's satisfaction and $T$ is an effective temperature to introduce stochasticity. If $T$ decreases below a critical point $T_c$, the system undergoes a transition from a symmetric phase to an asymmetric one, in which only one of the two sellers is selected. In this work, we extend the model by incorporating a simple price system. By considering a greed factor $g$ to control how the satisfaction depends on the price, we argue the existence of an oscillatory phase in addition to the symmetric and asymmetric ones in the $(T,g)$ plane, and estimate the phase boundaries through mean-field approximations. The analytic results show that the market preserves the inherent symmetry between the sellers for lower $T$ in the presence of the price system, which is confirmed by our numerical simulations.