Sponsored Search, Market Equilibria, and the Hungarian Method

📝 Abstract

Matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market. Here, as in other markets of that kind, market equilibria correspond to feasible, envy free, and bidder optimal outcomes. For settings without budgets such an outcome always exists and can be computed in polynomial-time by the so-called Hungarian Method. Moreover, every mechanism that computes such an outcome is incentive compatible. We show that the Hungarian Method can be modified so that it finds a feasible, envy free, and bidder optimal outcome for settings with budgets. We also show that in settings with budgets no mechanism that computes such an outcome can be incentive compatible for all inputs. For inputs in general position, however, the presented mechanism—as any other mechanism that computes such an outcome for settings with budgets—is incentive compatible.

💡 Analysis

Matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market. Here, as in other markets of that kind, market equilibria correspond to feasible, envy free, and bidder optimal outcomes. For settings without budgets such an outcome always exists and can be computed in polynomial-time by the so-called Hungarian Method. Moreover, every mechanism that computes such an outcome is incentive compatible. We show that the Hungarian Method can be modified so that it finds a feasible, envy free, and bidder optimal outcome for settings with budgets. We also show that in settings with budgets no mechanism that computes such an outcome can be incentive compatible for all inputs. For inputs in general position, however, the presented mechanism—as any other mechanism that computes such an outcome for settings with budgets—is incentive compatible.

📄 Content

arXiv:0912.1934v6 [cs.GT] 20 Dec 2012 Sponsored Search, Market Equilibria, and the Hungarian Method∗ Paul D¨utting† Monika Henzinger‡ Ingmar Weber§ Abstract Matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market. Here, as in other markets of that kind, market equilibria correspond to feasible, envy free, and bidder optimal outcomes. For settings without budgets such an outcome always exists and can be computed in polynomial-time by the so-called Hungarian Method. Moreover, every mechanism that computes such an outcome is incentive compatible. We show that the Hungar- ian Method can be modified so that it finds a feasible, envy free, and bidder optimal outcome for settings with budgets. We also show that in settings with budgets no mechanism that computes such an outcome can be incentive compatible for all in- puts. For inputs in general position, however, the presented mechanism—as any other mechanism that computes such an outcome for settings with budgets—is incentive compatible. 1 Introduction In a matching market n bidders have to be matched to k items. A prime example of such a market is the sponsored search market, where bidders correspond to advertisers and items correspond to ad slots. In this market each bidder has a per-click valuation vi, each item j has a click-through rate αj, and bidder i’s valuation for item j is vi,j = αj · vi. More generally, each bidder i has a valuation vi,j for each item j. In addition, each item j has a reserve price rj. A mechanism is used to compute an outcome (µ, p) consisting of a matching µ and per-item prices pj. The bidders have quasi-linear utilities. That is, bidder i’s utility is ui = 0 if he is unmatched and it is ui = vi,j −pj if he is matched to item j ∗A preliminary version of this paper appeared in D¨utting et al. [5]. †Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Station 14, CH-1015 Lausanne, Switzerland, Email: paul.duetting@epfl.ch. ‡University of Vienna, Faculty of Computer Science, W¨ahringer Straße 29/6.32, A-1090 Vienna, Austria, Email: monika.henzinger@univie.ac.at. §Yahoo! Research Barcelona, Avinguda Diagonal 177 (8th Floor), E-08018 Barcelona, Spain, Email: ingmar@yahoo-inc.com. 1 at price pj. The valuations are private information and the bidders need not report their true valuations if it is not in their best interest to do so. Ideally, the market should be in equilibrium. In the context of matching markets this typically means that the outcome computed by the mechanism should be feasible, envy free, and bidder optimal. An outcome is feasible if all bidders have non-negative utilities and if the price of all matched items is at least the reserve price. It is envy free if it is feasible and if at the current prices no bidder would get a higher utility if he was assigned a different item. It is bidder optimal if it is envy free and if the utility of every bidder is at least as high as in every other envy free outcome. Another requirement is that the mechanism should be incentive compatible. A mechanism is incentive compatible if each bidder maximizes his utility by reporting truthfully no matter what the other bidders report. For matching markets of the above form a bidder optimal outcome always exists [10], can be computed in polynomial time by the so-called Hungarian Method [8], and every mechanism that computes such an outcome is incentive compatible [9]. The above model, however, ignores the fact that in practice bidders often have budgets. Concrete examples include Google’s and Yahoo’s ad auction. Budgets are also challenging theoretically as they lead to discontinuous utility functions and thus break with the quasi-linearity of the original model without budgets. In our model each bidder can specify a maximum price for each item. If bidder i specifies a maximum price of mi,j for item j, then he cannot pay any price pj ≥mi,j. Hence the utility of bidder i is ui = 0 if he is unmatched, it is ui = vi,j −pj if he is matched to item j at price pj < mi,j (strict inequality), and it is ui = −∞otherwise.1 As before an outcome is feasible if all bidders have non-negative utilities and if the price of all matched items is at least the reserve price. It is envy free if it is feasible and if at the current prices no bidder would get a higher utility if he was assigned a different item. It is bidder optimal if it is envy free and if the utility of every bidder is at least as high as in every other envy free outcome. For this model we show that the Hungarian Method can be modified so that it always finds a bidder optimal outcome in polynomial time. We also show that no mechanism that computes such an outcome is incentive compatible for all inputs. For inputs in general position, i.e., inputs with the property that in a certain weighted multi-graph defined on the basis of the input no two walks have exactly the same weight, our mechanism—as any other mechanism that computes a bidder opti

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