Optimal Budget Allocation in Social Networks: Quality or Seeding

In this paper, we study a strategic model of marketing and product consumption in social networks. We consider two competing firms in a market providing two substitutable products with preset qualities. Agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for the consumptions. At some point in time, firms receive a limited budget which they can use to trigger a larger consumption of their products in the network. Firms have to decide between marginally improving the quality of their products and giving free offers to a chosen set of agents in the network in order to better facilitate spreading their products. We derive a simple threshold rule for the optimal allocation of the budget and describe the resulting Nash equilibrium. It is shown that the optimal allocation of the budget depends on the entire distribution of centralities in the network, quality of products and the model parameters. In particular, we show that in a graph with a higher number of agents with centralities above a certain threshold, firms spend more budget on seeding in the optimal allocation. Furthermore, if seeding budget is nonzero for a balanced graph, it will also be nonzero for any other graph, and if seeding budget is zero for a star graph, it will be zero for any other graph too. We also show that firms allocate more budget to quality improvement when their qualities are close, in order to distance themselves from the rival firm. However, as the gap between qualities widens, competition in qualities becomes less effective and firms spend more budget on seeding.Comment: 7 page

Slopey quantizers are locally optimal for Witsenhausen's counterexample

We study the perfect Bayesian equilibria of a leader-follower game of incomplete information. The follower makes a noisy observation of the leader's action (who moves first) and chooses an action minimizing her expected deviation from the leader's action. Knowing this, leader who observes the realization of the state, chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. We show the existence of what we call “near piecewise-linear equilibria” when there is strong complementarity between the leader and the follower and the precision of the prior is poor. As a major consequence of this result, we prove local optimality of a class of slopey quantization strategies which had been suspected of being the optimal solution in the past, based on numerical evidence for Witsenhausen's counterexample

An Analytics Approach To Designing Patient Centered Medical Home

Recently the patient centered medical home (PCMH) model has become a popular team based approach focused on delivering more streamlined care to patients. In current practices of medical homes, a clinical based prediction frame is recommended because it can help match the portfolio capacity of PCMH teams with the actual load generated by a set of patients. Without such balances in clinical supply and demand, issues such as excessive under and over utilization of physicians, long waiting time for receiving the appropriate treatment, and non continuity of care will eliminate many advantages of the medical home strategy. In this research, we formulate the problem into two phases. At the first phase we proposed a multivariate version of multilevel structured additive regression (STAR) models which involves a set of health care responses defined at the lowest level of the hierarchy, a set of patient factors to account for individual heterogeneity, and a set of higher level effects to capture heterogeneity and dependence between patients within the same medical home team and facility. We show how a special class of such models can equivalently be represented and estimated in a structural equation-modeling framework. A Bayesian variable selection with spike and slab prior structure is then developed that allows including or dropping single effects as well as grouped coefficients representing particular model terms. We use a simple parameter expansion to improve mixing and convergence properties of Markov chain Monte Carlo simulation. A detailed analysis of the VHA medical home data is presented to demonstrate the performance and applicability of our method. In addition, by extending the hierarchical generalized linear model to include multivariate responses, we develop a clinical workload prediction model for care portfolio demands in a Bayesian framework. The model allows for heterogeneous variances and unstructured covariance matrices for nested random effects that arise through complex hierarchical care systems. We show that using a multivariate approach substantially enhances the precision of workload predictions at both primary and non primary care levels. We also demonstrate that care demands depend not only on patient demographics but also on other utilization factors, such as length of stay. Our analyses of a recent data from Veteran Health Administration further indicate that risk adjustment for patient health conditions can considerably improve the prediction power of the model. For the second phase, with the help of the model developed in first phase, we are able to estimate the annual workload demand portfolio for each patient with given attributes. Together with the healthcare service supply data, and based on the principles of balancing supply and demand, we developed stochastic optimization models to allocate patients to all PCMH teams in order to make balance between supply and demand in healthcare system. We proposed different stochastic models and two solution approaches such as Progressive Hedging and L shaped Benders Decomposition. We described the application of the two mentioned algorithms and finally we compared the performance of the two methods