For the recursive formulation of the model, markets are sequentially incomplete and we show that there is room for Pareto improvements in terms of intergenerational risk sharing. Finally, we examine whether the introduction of capital income taxation improves the allocation of risk.

Our goal is to understand whether the dependence on n is required in the multi-row case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is (1 − ε)-competitive as long as the right-hand sides are Ω(m/ϵ² log n/ϵ) . Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP’s are handled by perturbing the input columns, which can be seen as making the learning problem more robust.

In this paper, we present the first randomized online algorithms for this problem with poly-logarithmic competitive ratio. To do this, we employ the LP formulation of a scaled reward version of the problem. Then we extend a weighted-majority type update rule along with uncrossing properties of tight sets in the matroid polytope to find an approximately optimal fractional LP solution. We use the fractional solution values as probabilities for a online randomized rounding algorithm. To show that our rounding produces a sufficiently large reward independent set, we prove and use new covering properties for randomly rounded fractional solutions in the matroid polytope that may be of independent interest.

The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles having speeds {λ _i } _i = 1 ^k , the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated.

The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for minimum makespan scheduling in unrelated parallel machines of Lenstra et al. [19]. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees [18]. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the lengths of the parts and their distances to the depot

Specifically, we show it is optimal to segment customers into service grades which are differentiated by their mean service rate (or equivalently mean service time). For such a system we derive the closed-form optimal strategies of service rate differentiation, showing that optimal service rates and grade utilizations both form geometric distributions. We also compute the asymptotic system performance (as the number of grades increases to infinity) and illustrate the sensitivity of the benefits of differentiation with respect to jobs’ characteristics: marginal service value, marginal waiting cost and variation of processing time. We find that providing differentiated service can improve system performance by 5% without any additional capacity investment.

Using the peer voting score, we are able to infer the true potential of ideas, whereas the cost to the firm for implementing the idea is indirectly imputed from the idea implementation data. We find that individuals tend to significantly underestimate the costs to the firm for implementing their ideas but overestimate the potential of their ideas in the initial stages of the crowdsourcing process. Therefore, the “idea market” is initially overcrowded with ideas that are less likely to be implemented. However, individuals learn about both their abilities to come up with high potential ideas as well as the cost structure of the firm from peer voting on their ideas and the firm response to contributed ideas. We find that the individuals learn rather quickly about their abilities to come up with high potential ideas, but the learning regarding the firm's cost structure is quite slow. We also find that an individual’s discontent adversely affects the individual’s continuous participation in idea contributions. As a result of the learning process, the crowdsourcing market becomes more efficient. Contributors of low potential ideas eventually drop out, while the high potential idea contributors remain active. Over time, the average potential of generated ideas increases, while the number of ideas contributed decreases. Hence, the firm can reduce the cost of screening ideas without losing high potential ideas. In our policy simulation, we show that providing more precise cost signals to individuals can accelerate the filtering process. Increasing the total number of ideas to respond to and improving the response speed will lead to more idea contributions. However, failure to distinguish between high and low potential ideas and between high and low ability idea generators lead to the overall potential of the ideas generated to drop significantly.

Our proposed model is applied to a wholesale food distributor and we find widespread heterogeneity in purchase patterns. The empirical results shed light on the competitive elements of customer demand that cannot be studied with traditional reduced form response models. For example, we found that some costumers satisfy most of their requirement from one of their distributors, while others consistently split their demands across suppliers. Also, we found that price sensitivity of customers making most of their purchases with the focal supplier are less affected by the volume of purchases in previous periods. We expect this result to provide valuable information for vendors to negotiate prices with the customers. It allows the distributor to make efficient inferences about competitors in those occasions when competitor price is not directly observed.