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... bedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract We consider the classical mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single-parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [19] . Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [24]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multi-dimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time. This work can be viewed as an extension and improvement of the single-agent algorithmic pricing work of [10] to the setting of multiple agents where the designer has combinatorial feasibility constraints on which agents can simultaneously obtain each service. Consider the following hotel rooms example with one room, two attendees, and attendee values independently and identically distributed uniformly between $100 and $200. The optimal mechanism is the Vickrey auction and its expected revenue is $133. The optimal sequential posted pricing is for the organizers to offer the room to attendee 1 at a price of $150. If the attendee accepts, then the room is sold, otherwise it is offered to attendee 2 for $100. The expected revenue of this SPM is $125. We are interested in comparing the optimal mechanism to the optimal posted pricing in general settings. A special class of SPMs is one where mechanisms have provable performance guarantees for any sequence of the agents. These order-oblivious posted pricings (OPM) are mechanisms defined by a price for each agent and the semantics that each agent is offered their corresponding price in some arbitrary sequence as a take-it-or-leave-it while-supplies-last offer. In single-dimensional settings, the advantages of sequential posted pricings speak to the many reasons optimal auctions are rarely seen in practice [1], and explain why posted pricings are ubiquitous [14] . First, take-it-or-leaveit offers result in trivial game dynamics: truthful responding is a dominant strategy. Second, SPMs satisfy strong notions of collusion resistance, e.g., group strategyproofness (see [11] ): the only way in which an agent can "help" another agent is to decline an offer that he could have accepted, thereby hurting his own utility. Third, agents do not need to precisely know or report their value, they must only be able to evaluate their offer; therefore, they risk minimal exposure of their private information. Fourth, agents learn immediately whether they will be served or not. In conclusion, the robustness of SPMs in single-dimensional settings makes their approximation of optimal mechanisms independently worthy of study. The final robustness property of SPMs, which is of paramount importance to our study of the multi-dimensional setting, is that they minimize the role of agent competition which implies that single-dimensional SPMs can be used "as-is" in multi-dimensional settings with only a constant factor loss in performance. In our translation from the multidimensional setting to the single-dimensional setting, each multi-dimensional agent has many single-dimensional representatives. A good OPM for the single-dimensional setting can be viewed as an OPM for the multi-dimensional setting by grouping all representatives of an agent together and making their offers simultaneously to the agent. The agent will of course accept the offer that maximizes their utility. The resulting mechanism is truthful and achieves the same performance guarantee as the single-parameter OPM. For SPMs where we are not free to group each multidimensional agent's single-dimensional representative together, an agent possibly faces a strategic dilemma of whether to accept an offer (e.g., for one hotel room) early on or wait for a later offer (e.g., another hotel room) which may or may not still be available. Our guarantee is that regardless of the actions of any agent with such a strategic option (i.e., implementation in undominated strategies, see, e.g., [4]) our performance is a constant fraction of the original SPM's performance. Given the advantages of SPMs over truthful mechanisms, such a non-truthful SPM may be more practically relevant than a truthful implementation. Finally, we note that most of our results for posted pricings are constructive and give efficient algorithms for them. A posted price mechanism has two components where computation is necessary: an offline computation of the prices to post (and for SPMs, the sequence of agents) and an online while-supplies-last offering of said prices. 1 The agents are only present for the online part where the mechanism is trivial. All of the computational burden for an SPM is in the offline part. The offline computation of our posted price mechanisms is based on a subroutine that repeatedly samples the distribution of agent values and simulates Myerson's mechanism on the sample. This clearly requires more computation than just running Myerson's mechanism on the real agents in the first place; however, we benefit from the robustness that comes from the trivial online implementation of posted pricings. Related work. See [24] and references therein for work in economics on optimal multi-dimensional mechanism design. See [10] and references therein for work in computer science on multi-dimensional pricing for a single agent. We extend the setting from [10] to multiple agents and improve their approximation for a single agent from 3 to 2. Sequential posted price mechanisms have been considered previously in single-dimensional settings. Sandholm and Gilpin [23] show experimentally that these mechanisms compare favorably to Myerson's optimal mechanisms. Blumrosen and Holenstein [7] show how to compute the optimal posted prices in the special case where agents' values are distributed identically, and also show that in this case the revenue of these mechanisms approaches the optimal revenue asymptotically. Several papers study revenue maximization through online posted pricings in the context of adversarial values, albeit in the simpler context of multi-unit auctions [6, 15, 5] .