Budget Feasible Procurement Auctions

In online discussion forums, users are more motivated to take part in discussions when observing other users’ participation—the effect of social influence among forum users. In this paper, we study how to utilize social influence for increasing the overall forum participation. To this end, we propose a mechanism to maximize user influence and boost participation by displaying forum threads to users. We formally define the participation maximization problem, and show that it is a special instance of the social welfare maximization problem with submodular utility functions and it is NP-hard. However, generic approximation algorithms is impracticable for real-world forums due to time complexity. Thus we design a heuristic algorithm, named Thread Allocation Based on Influence (TABI), to tackle the problem. Through extensive experiments using a dataset from a real-world online forum, we demonstrate that TABI consistently outperforms all other algorithms in maximizing participation. The results of this work demonstrates that current recommender systems can be made more effective by considering future influence propagations. The problem of participation maximization based on influence also opens a new direction in the study of social influence.

Sales promotions are important in the fast-moving consumer goods (FMCG) industry due to the significant spending on promotions and the fact that a large proportion of FMCG products are sold on promotion. This paper considers the problem of planning sales promotions for an FMCG product in a grocery retail setting. The category manager has to solve the promotion optimization problem (POP) for each product, i.e., how to select a posted price for each period in a finite horizon so as to maximize the retailer’s profit. Through our collaboration with Oracle Retail, we developed an optimization formulation for the POP that can be used by category managers in a grocery environment. Our formulation incorporates business rules that are relevant, in practice. We propose general classes of demand functions (including multiplicative and additive), which incorporate the post-promotion dip effect, and can be estimated from sales data. In general, the POP formulation has a nonlinear objective and is NP-hard. We then propose a linear integer programming (IP) approximation of the POP. We show that the IP has an integral feasible region, and hence can be solved efficiently as a linear program (LP). We develop performance guarantees for the profit of the LP solution relative to the optimal profit. Using sales data from a grocery retailer, we first show that our demand models can be estimated with high accuracy, and then demonstrate that using the LP promotion schedule could potentially increase the profit by 3%, with a potential profit increase of 5% if some business constraints were to be relaxed. The online appendix is available at https://doi.org/10.1287/opre.2016.1573

Many information systems aim to present results that maximize the collective satisfaction of the user population. The product search of an online store, for example, needs to present an appropriately diverse set of products to best satisfy the different tastes and needs of its user population. To address this problem, we propose two algorithms that can exploit observable user actions (e.g. clicks) to learn how to compose diverse sets (and rankings) that optimize expected utility over a distribution of utility functions. A key challenge is that individual users evaluate and act according to their own utility function, but that the system aims to optimize collective satisfaction. We characterize the behavior of our algorithms by providing upper bounds on the social regret for a class of submodular utility functions in the coactive learning model. Furthermore, we empirically demonstrate the efficacy and robustness of the proposed algorithms for the problem of search result diversification.KeywordsOnline LearningCoactive LearningImplicit FeedbackDiversified Retrieval

Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (text {MSR}). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the text {MSR} problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.

4 - Recursive Enumerability

Sensitivity of a general class of shape functionals to topological changes

The multilinear framework for submodular maximization was developed to achieve a tight $$1-1/e$$ 1 - 1 / e approximation for maximizing a monotone submodular function subject to a matroid constraint, including as special case the submodular welfare problem. The framework has a continuous optimization step (solving the multilinear extension of a submodular function) and a rounding part (rounding a fractional solution to an integral one). We extend both parts to provide a framework for a wider array of applications. The continuous part works for a more general class of continuous functions parameterized by a new smoothness parameter $$\sigma $$ σ . A twice differential function F is called $$\sigma $$ σ -one-sided-smooth ( $$\sigma $$ σ -OSS) if its second derivatives are bounded as follows: $$\frac{1}{2}u^T\nabla ^2 F(x) u \le \sigma \cdot \frac{\Vert u\Vert _1}{\Vert x\Vert _1} u^T \nabla F(x)$$ 1 2 u T ∇ 2 F ( x ) u ≤ σ · ‖ u ‖ 1 ‖ x ‖ 1 u T ∇ F ( x ) for all $$u,x\ge 0$$ u , x ≥ 0 , $$x\ne 0$$ x ≠ 0 . For $$\sigma =0$$ σ = 0 this includes previously studied continuous DR-Submodular functions as well as quadratics defined by copositive matrices. We give a modification of the continuous greedy algorithm which finds a solution for maximizing a monotone $$\sigma $$ σ -OSS F over a polytope in the non-negative orthant; the solution approximates the optimum to within factors which are functions of $$\sigma $$ σ which depend on additional properties. Interestingly, $$\sigma $$ σ -OSS functions arise as the multilinear extensions of set functions associated with several well-studied diversity maximization problems: $$\max f(S) = \sum _{i,j \in S} A_{ij} : |S| \le k$$ max f ( S ) = ∑ i , j ∈ S A ij : | S | ≤ k . For instance, when $$A_{ij}$$ A ij defines a $$\sigma $$ σ -semi-metric, its extension is $$\sigma $$ σ -OSS. In these settings, we also develop rounding schemes to approximate the discrete problem.

In interference coupled wireless systems, where it is not possible to orthogonalize all the users in the system, we characterize the impact of interference coupling on the convexity of certain utility functions and problems. We introduce a general class of competitive user utility functions and natural competitive user utility functions. We further introduce the signal-to-interference based utility functions, which are based on physical layer parameters in wireless systems. We prove the conditions, which when satisfied - result in a competitive user utility function being a signal-to-interference ratio based utility function. We further show that there exists no natural competitive user utility function, which is convex or concave. Furthermore, we show that a sum of weighted combination of natural competitive user utility functions is not convex or concave. Such functions are commonly encountered in wireless communication systems, e.g. rate or MMSE as a function of signal-to-interference ratio. We show that such rate maximization or MMSE minimization problems are not convex programs under our specified conditions.

Machine learning can provide powerful tools for the design and analysis of multi-agent systems, as well as a novel lens on important questions in the area. In this talk I will focus on the use of machine learning for understanding submodular functions, an important class of discrete functions that model laws of diminishing returns and enjoy many important applications in multi-agent settings. For example, submodular functions are commonly used to model valuation functions for bidders in auctions, the influence of various subsets of agents in social networks, and the benefits of performing different actions in a variety of situations. Traditionally it is assumed that these functions are known to the decision maker; however, for large scale systems, in the age of big data, it is often the case they must be learned from observations. In this talk, I will discuss a recent line of work on studying the learnability of submodular functions, and highlight its applications to the analysis of multi-agent systems. I will discuss both general algorithms for learning such functions, as well as even better guarantees that can be achieved for important classes appearing in multi-agent scenarios that exhibit additional structure. These classes include: probabilistic coverage functions that can be used to model the influence function in classic models of information diffusion in networks; functions with bounded complexity used in modeling bidder valuation functions in auctions, including XOS and gross-substitutes; and classes of functions appearing in cooperative game theory for expressing the values of various types of coalitions. I will additionally discuss a large scale application of our algorithms for learning the influence functions in social networks, that significantly outperforms existing approaches empirically in both synthetic and real world data.

CHAPTER 5 - Household Behavior with Endogenous Fertility

Delay aversion under a general class of preferences

Auctions are widely used on the Web. Applications range from sponsored search to platforms such as eBay. In these and in many other applications the auctions in use are single-/multi-item auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is the limited expressiveness that they offer to the bidders. The General Auction Mechanism (GAM) of Aggarwal et al. [2009] takes a first step toward addressing the problem of limited expressiveness by computing a bidder optimal, envy-free outcome for linear utility functions with identical slopes and a single discontinuity per bidder-item pair. We show that in many practical situations this does not suffice to adequately model the preferences of the bidders, and we overcome this problem by presenting the first mechanism for piecewise linear utility functions with nonidentical slopes and multiple discontinuities . Our mechanism runs in polynomial time. Like GAM it is incentive compatible for inputs that fulfill a certain nondegeneracy assumption, but our requirement is more general than the requirement of GAM. For discontinuous utility functions that are nondegenerate as well as for continuous utility functions the outcome of our mechanism is a competitive equilibrium . We also show how our mechanism can be used to compute approximately bidder optimal, envy-free outcomes for a general class of continuous utility functions via piecewise linear approximation. Finally, we prove hardness results for even more expressive settings.

Journal Article The Ces Utility Function, Non-Linear Budget Constraints and Labour Supply. Results on Female Participation and Hours Get access A. Zabalza A. Zabalza London School of Economics Search for other works by this author on: Oxford Academic Google Scholar The Economic Journal, Volume 93, Issue 370, 1 June 1983, Pages 312–330, https://doi.org/10.2307/2232795 Published: 01 June 1983

The paper presents a model of the market for early child care services, where the preferences of households and local government are influenced by relative prices, disposable income and cost conditions, as well as «social norms» such as the attitude to working women and the value assigned to the educational (as well as supervisory) role of institutional child care. A simple partial equilibrium model describes alternative outcomes in the allocation of private and public provision. In the case of the latter the allocation scheme could be influenced by the welfare model adopted by local governments: this could explain their paternalistic policies and may lead to conflict with the motivations of households, which may be affected by the «consolidation of needs» induced by a long tradition of public provision at prices lower than average costs. Moreover public budget constraints interfere with the diverging attitudes of the various players by introducing rationing schemes and redistributive goals that reduce «market» transparency. The empirical importance of the «non economic» motivations referred to above therefore seems to make them a major issue to be considered when evaluating the allocation process. Using a survey on economic and social conditions of the households in the province of Modena in 2006, which directly surveys the willingness to pay for child care services, the second part of the paper explores the demand for early child care through an econometric estimation of the reservation price with particular attention to «economic» and «non economic» determinants. There are complex relationships between the motivations of local government and households: the paternalistic behaviour of public providers is clear from their supply and price strategies; on the other hand, households seem to be affected by a form of needs consolidation which is a barrier to the correct evaluation of the costs and benefits of services, and by a greater focus on mere «child-minding». The tensions between the different attitudes of households and local governments reflects the different welfare models which implicitly underlie their behaviour. This points to the need for a more explicit dialogue between local governments and citizens on the costs and benefits of the services involved, to improve the transparency of this «market».

This paper investigates a decomposition approach for binary optimization problems with nonlinear objectives and linear constraints. Our methodology relies on the partition of the objective function into separate low-dimensional dynamic programming (DP) models, each of which can be equivalently represented as a shortest-path problem in an underlying state-transition graph. We show that the associated transition graphs can be related by a mixed-integer linear program (MILP) so as to produce exact solutions to the original nonlinear problem. To address DPs with large state spaces, we present a general relaxation mechanism that dynamically aggregates states during the construction of the transition graphs. The resulting MILP provides both lower and upper bounds to the nonlinear function, and it may be embedded in branch-and-bound procedures to find provably optimal solutions. We describe how to specialize our technique for structured objectives (e.g., submodular functions) and consider three problems arising in revenue management, portfolio optimization, and healthcare. Numerical studies indicate that the proposed technique often outperforms state-of-the-art approaches by orders of magnitude in these applications. Data and the online appendix are available at https://doi.org/10.1287/mnsc.2017.2849 . This paper was accepted by Yinyu Ye, optimization.