价格战仍在继续：关于竞争性多产品定价
The Pricing War Continues: On Competitive Multi-Item Pricing
摘要：
我们研究了一个博弈与\emph{strategic}供应商谁拥有多个项目和一个单一的买家与一个子模块的估值功能。供应商的目标是通过物品的定价来最大化他们的收入，因为买方将购买使其净收益最大化的物品。我们表明，这个游戏可能并不总是有一个纯粹的纳什均衡，在以前的结果相比，每个供应商拥有一个单一的项目的特殊情况下。我们通过将我们的博弈与一个中间的、离散的博弈相关联来做到这一点，在这个博弈中，供应商只选择可用的物品，然后他们的价格是外生的。我们进一步利用中间的游戏，提供严格的界限上的无政府代价价格的子集游戏，有纯纳什均衡，我们发现，在以前的特殊情况下达到的最佳PoA不成立，但只有一个对数。最后，我们证明了对于次模函数的特殊情况，有效纯纳什均衡总是存在的。
英文摘要：
We study a game with \emph{strategic} vendors who own multiple items and a single buyer with a submodular valuation function. The goal of the vendors is to maximize their revenue via pricing of the items, given that the buyer will buy the set of items that maximizes his net payoff. We show this game may not always have a pure Nash equilibrium, in contrast to previous results for the special case where each vendor owns a single item. We do so by relating our game to an intermediate, discrete game in which the vendors only choose the available items, and their prices are set exogenously afterwards. We further make use of the intermediate game to provide tight bounds on the price of anarchy for the subset games that have pure Nash equilibria; we find that the optimal PoA reached in the previous special cases does not hold, but only a logarithmic one. Finally, we show that for a special case of submodular functions, efficient pure Nash equilibria always exist.
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