For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) buyers' types must be distributed independently (not necessarily identically), (ii) objective function must be linearly separable over the buyers, and (iii) except for the supply constraints, there should be no other inter-buyer constraints. Our framework is general in the sense that it makes no explicit assumption about buyers' valuations, type distributions, and single buyer constraints (e.g., budget, incentive compatibility, etc). We present two generic multi buyer mechanisms which use single buyer mechanisms as black boxes; if an α-approximate single buyer mechanism can be constructed for each buyer, and if no buyer requires more than 1 k of all units of each item, then our generic multi buyer mechanisms are γ k α-approximation of the optimal multi buyer mechanism, where γ k is a constant which is at least 1 − 1 √ k+3. Observe that γ k is at least 1 2 (for k = 1) and approaches 1 as k → ∞. As a byproduct of our construction, we present a generalization of prophet inequalities. Furthermore , as applications of our framework, we present multi buyer mechanisms with improved approximation factor for several settings from the literature.

We study an abstract optimal auction problem for selecting a subset of self-interested agents to whom to provide a service. A feasibility constraint governs which subsets can be simultaneously served; however, the mechanism may additionally choose to bundle unconstrained attributes such as payments or add-ons with the service. An agent's preference over service and attributes is given by her private type and may be multi-dimensional and non-linear. A single-agent problem is to optimizes a menu to offer an agent subject to constraints on the probabilities with which each of the agent's types is served. We give computationally tractable reductions from multi-agent auction problems to these single-agent problems. Our discussion focuses on maximizing revenue, but our results can be applied to other objectives (e.g., welfare).From each agent's perspective, any multi-agent mechanism and distribution on other agent types induces an interim allocation rule, i.e., a probability that the agent will be served as a function of the type she reports. The resulting profile of interim allocation rules (one for each agent) is feasible in the sense that there is a mechanism that induces it. An optimal mechanism can be solved for by inverting this process.(1) Solve the single-agent problem of finding the optimal way to serve an agent subject to an interim allocation rule as a constraint (taking into account the agent's incentives).(2) Optimize over interim feasible allocation profiles, i.e., ones that are induced by the type distribution and some mechanism, the cumulative revenue of the single-agent problems for the interim allocation profile. (3) Find the mechanism that induces the optimal interim feasible allocation profile in the previous step.For a large class of auction problems and multi-dimensional and non-linear preferences each of the above steps is computationally tractable. We observe that the single-agent problems for (multi-dimensional) unitdemand and budgeted preferences can be solved in polynomial time in the size of the agent's type space via a simple linear program. For feasibility constraints induced by single-item auctions interim feasibility was characterized by Border (1991); we show that interim feasible allocation rules can be optimized over and implemented via a linear program that has quadratic complexity in the sum of the sizes of the individual agents' type spaces. We generalize Border's characterization to auctions where feasible outcomes are the independent sets of a matroid; here the characterization implies that the polytope of interim feasible allocation rules is a polymatroid. This connection implies that a concave objective, such as the revenue from the single-agent problems, can be maximized over the interim feasibility constraint in polynomial time. The resulting optimal mechanism can be viewed as a randomization over the vertices of the polymatroid which correspond to simple greedy mechanisms: given an ordering on a subset all agent types, serve the agents in this order subject to feasibility.Re...

We study the problem of online prophet-inequality matching in bipartite graphs. There is a static set of bidders and an online stream of items. We represent the interest of bidders in items by a weighted bipartite graph. Each bidder has a capacity, i.e., an upper bound on the number of items that can be allocated to her. The weight of a matching is the total weight of edges matched to the bidders. Upon the arrival of an item, the online algorithm should either allocate it to a bidder or discard it. The objective is to maximize the weight of the resulting matching. We consider this model in a stochastic setting where we know the distribution of the incoming items in advance. Furthermore, we allow the items to be drawn from different distributions, i.e., we may assume that the t th item is drawn from distribution Dt. In contrast to i.i.d. model, this allows us to model the change in the distribution of items throughout the time. We call this setting the Prophet-Inequality Matching because of the possibility of having a different distribution for each time. We generalize the classic prophet inequality by presenting an algorithm with the approximation ratio of 1 − 1 √ k+3 where k is the minimum capacity. In case of k = 2, the algorithm gives a tight ratio of 1 2 which is a different proof of the prophet inequality. We also consider a model in which the bidders do not have a capacity, instead each bidder has a budget. The weight of a matching is the minimum of the budget of each vertex and the total weight of edges matched to it, when summed over all bidders. We show that if the bid to the budget ratio of every bidder is at most 1 k then a natural randomized online algorithm has an approximation ratio of 1 − k k e k k! ≈ 1 − 1 √ 2πk compared to the optimal offline (in which the ratio goes to 1 as k becomes large).We also present the applications of our model in Adword Allocation, Display Ad Allocation, and AdCell Model.

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