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Following the work of Babaioff et al, we consider the pricing game withstrategic vendors and a single buyer, modeling a scenario in which multiplecompeting vendors have very good knowledge of a buyer, as is common in onlinemarkets. We add to this model the realistic assumption that the buyer has afixed budget and does not have unlimited funds. When the buyer's valuationfunction is additive, we are able to completely characterize the differentpossible pure Nash Equilibria (PNE) and in particular obtain a necessary andsufficient condition for uniqueness. Furthermore, we characterize the marketclearing (or Walresian) equilibria for all submodular valuations.  Surprisingly, for certain monotone submodular function valuations, we showthat the pure NE can exhibit some counterintuitive phenomena; namely, there isa valuation such that the pricing will be market clearing and within budget ifthe buyer does not reveal the budget but will result in a smaller set ofallocated items (and higher prices for items) if the buyer does reveal thebudget. It is also the case that the conditions that guarantee market clearingin Babaioff et al for submodular functions are not necessarily market clearingwhen there is a budget. Furthermore, with respect to social welfare, whilewithout budgets all equilibria are optimal (i.e. POA = POS = 1), we show thatwith budgets the worst equilibrium may only achieve 1/(n-2) of the bestequilibrium.

Budgetary Effects on Pricing Equilibrium in Online Markets