Budgetary effects on pricing equilibrium in online markets

Following the work of Babaioff et al [4], we consider the pricing game with strategic vendors and a single buyer, modeling a scenario in which multiple competing vendors have very good knowledge of a buyer, as is common in online markets. We add to this model the realistic assumption that the buyer has a fixed budget and does not have unlimited funds. When the buyer's valuation function is additive, we are able to completely characterize the different possible pure Nash Equilibria (PNE) and in particular obtain a necessary and sufficient condition for uniqueness. Furthermore, we characterize the market clearing (or Walresian) equilibria for all submodular valuations. Surprisingly, for certain monotone submodular function valuations, we show that the pure NE can exhibit some counterintuitive phenomena; namely, there is a valuation such that the pricing will be market clearing and within budget if the buyer does not reveal the budget but will result in a smaller set of allocated items (and higher prices for items) if the buyer does reveal the budget. It is also the case that the conditions that guarantee market clearing in Babaioff et al [4] for submodular functions are not necessarily market clearing when there is a budget. Furthermore, with respect to social welfare, while without budgets all equilibria are optimal (i.e. POA = POS = 1), we show that with budgets the worst equilibrium may only achieve 1/n-2 of the best equilibrium.