Value loss in allocation systems with provider guarantees

Many operational settings share the following three features: (i) a centralized planning system allocates tasks to workers or service providers, (ii) the providers generate value by completing the tasks, and (iii) the completion of tasks influences the providers' welfare. In such cases, the planning system's allocations often entail trade-offs between the service providers' welfare and the total value that is generated (or that accrues to the system itself), and concern arises that allocations that are good under one metric may perform poorly under the other. We propose a broad framework for quantifying the magnitude of value losses when allocations are restricted to satisfy certain desirable guarantees to the service providers.} We consider a general class of guarantees that includes many considerations of practical interest arising, e.g., in the design of sustainable two-sided markets, in workforce welfare and compensation, or in sourcing and payments in supply chains, among other application domains. We derive tight bounds on the relative value loss, and show that this loss is limited for any restriction included in our general class. Our analysis shows that when many providers are present, the largest losses are driven by fairness considerations, whereas when few providers are present, they are driven by the heterogeneity in the providers' effectiveness to generate value; when providers are perfectly homogenous, the losses never exceed 50\%. We study additional loss drivers and find that less variability in the value of jobs and a more balanced supply-demand ratio may lead to larger losses. Lastly, we demonstrate numerically using both real-world and synthetic data that the loss can be small in several cases of practical interest