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原文摘要

We study a pricing game in markets with budget-constrained buyers, where each seller offers a single good and sets its price. We consider both non-rivalrous goods, such as datasets, and rivalrous goods, such as physical goods. For non-rivalrous goods, each buyer chooses a utility-maximizing bundle within her budget. For rivalrous goods, an allocation rule selects a Pareto-optimal allocation subject to the buyers' budgets and the goods' supply constraints. Pure Nash equilibria often fail to exist in this game. A key reason is that Nash equilibrium assumes that the other sellers keep their prices fixed when one seller deviates. In practice, however, sellers harmed by a price change may respond, for example, by matching the new price. We propose a new equilibrium notion called Response-Aware Rational Equilibrium (RARE). At a RARE, every deviation by a seller can be deterred by a response from the other sellers, in which no responder earns less than she did right after the deviation. Every Nash equilibrium is a RARE. Assuming that the allocation rule is continuous in prices, we show that a RARE always exists for rivalrous goods, and for non-rivalrous goods when buyers have equal budgets. Moreover, the RARE that we construct extracts the buyers' entire budget. For non-rivalrous goods with unequal budgets, we show that an approximate RARE exists, where a deviating seller can increase her revenue by a factor equal to the ratio of the average budget to the minimum budget. We also study a natural dynamic in which sellers repeatedly raise their prices to earn the revenue that they could guarantee themselves by deviating. We show that, in the settings where we prove exact existence, this dynamic reaches a RARE with additive error $δ$ within $\lfloor B/δ\rfloor$ rounds, where $B$ is the buyers' total budget. This paper's main results have been formally verified in Lean 4.