Competitive Equilibria in Matching Markets with Budgets

arXiv:1004.2565v2 [cs.GT] 16 Apr 2010 Competitive Equilibria in Matching Markets with Budgets Ning Chen∗ Xiaotie Deng† Arpita Ghosh‡ November 21, 2018 Abstract We study competitive equilibria in the classic Shapley-Shubik assignment model with indivisible goods and unit-demand buyers, with budget constraints: buyers can specify a maximum price they are willing to pay for each item, beyond which they cannot afford the item. This single discontinuity intro- duced by the budget constraint fundamentally changes the properties of equilibria: in the assignment model without budget constraints, a competitive equilibrium always exists, and corresponds exactly to a stable matching. With budgets, a competitive equilibrium need not always exist. In addition, there are now two distinct notions of stability, depending on whether both or only one of the buyer and seller can strictly benefit in a blocking pair, that no longer coincide due to the budget-induced discontinuity. We define weak and strong stability for the assignment model with transferable utilities, and show that competitive equilibria correspond exactly to strongly stable matchings. We consider the algorithmic question of efficiently computing competitive equilibria in an extension of the assignment model with budgets, where each buyer specifies his preferences over items using utility functions uij, where uij(pj) is the utility of buyer i for item j when its price is pj. Our main result is a strongly polynomial time algorithm that decides whether or not a competitive equilibrium exists and if yes, computes a minimum one, for a general class of utility functions uij. This class of utility functions includes the standard quasi-linear utility model with a budget constraint, and in addition, allows modeling marketplaces where, for example, buyers only have a preference ranking amongst items subject to a maximum payment limit for each item, or where buyers want to optimize return on investment (ROI) instead of a quasi-linear utility and only know items’ relative values. ∗Division of Mathematical Sciences, Nanyang Technological University, Singapore. Email: ningc@ntu.edu.sg. †Department of Computer Science, City University of Hong Kong, Hong Kong. Email: csdeng@cityu.edu.hk. ‡Yahoo! Research, Santa Clara, CA, USA. Email: arpita@yahoo-inc.com. 1 Introduction Consider a market with n unit demand buyers and m sellers, each selling one unit of an indivisible good. The buyers specify their preferences over items via utility functions uij(pj), which is the utility of buyer i for item j when its price is pj. So far, this is the classic Shapley-Shubik assignment model [36] which captures a variety of matching markets including housing markets and ad auctions [23, 39], except for the extension to general utility functions instead of the quasi-linear utilities in the original model. Shapley and Shubik show that a competitive equilibrium always exists in their model, and later work [14, 33, 24] shows that a competitive equilibrium must also exist for the model with general utility functions uij(·), provided these uij(·) are strictly decreasing and continuous everywhere. Now suppose we extend the assignment model with an extra budget constraint: a buyer i can specify a maximum price bij that he is able to pay for item j, above which he cannot afford the item. That is, the utility function uij(·) can now (possibly) have a discontinuity at pj = bij. Budgets are a very real constraint in many marketplaces such as advertising markets, and have led to a spate of recent work on auction design [10, 4, 6, 31, 21, 5, 9], where the addition of the budget constraint, while seemingly innocuous, introduces fundamental new challenges to the problem. As we will see, the same happens in the Shapley-Shubik assignment model: the discontinuity introduced by the budget constraint is not merely technical, but fundamentally changes the properties of competitive equilibria. First, a competitive equilibrium no longer always exists (Example A.1 in Appendix A). Second, and related to the first, while competitive equilibria in the original model correspond precisely to stable matchings [36], this is not quite true with budgets since the different notions of stability no longer coincide as in the original model, i.e., there is no longer a single unique notion of stability. A weakly stable matching is one where there is no unmatched buyer-seller pair where both the buyer and seller can strictly benefit by trading with each other, where a seller’s payoffis the payment he receives for his item. A strongly stable matching is one where there is no unmatched buyer-seller pair where one party strictly benefits and the other weakly benefits from the deviation (an example of the seller only weakly benefiting is when the buyer i strictly prefers to buy seller j’s item at its current price, but not at any higher price). Without the budget-induced discontinuity, these two notions can be shown to be identical given the contin

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