We consider a problem of prediction based on opinions elicited from heterogeneous rational agents with private information. Making an accurate prediction with a minimal cost requires a joint design of the incentive mechanism and the prediction algorithm. Such a problem lies at the nexus of statistical learning theory and game theory, and arises in many domains such as consumer surveys and mobile crowdsourcing. In order to elicit heterogeneous agents' private information and incentivize agents with different capabilities to act in the principal's best interest, we design an optimal joint incentive mechanism and prediction algorithm called COPE (COst and Prediction Elicitation), the analysis of which offers several valuable engineering insights. First, when the costs incurred by the agents are linear in the exerted effort, COPE corresponds to a "crowd contending" mechanism, where the principal only employs the agent with the highest capability. Second, when the costs are quadratic, COPE corresponds to a "crowd-sourcing" mechanism that employs multiple agents with different capabilities at the same time. Numerical simulations show that COPE improves the principal's profit and the network profit significantly (larger than 30% in our simulations), comparing to those mechanisms that assume all agents have equal capabilities.
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