Budget-feasible mechanisms for proportionally selecting agents from groups

In many social domains involving collective decision-making (e.g., committee selection and survey sampling), it is often desirable to select individuals from different population groups to achieve proportional representation (e.g., to represent the opinions of each group). For instance, in the selection of a committee (e.g., to form a working group within a company), the planner would like to select agents from different groups to represent their respective groups proportionally. Typically, there are intrinsic private costs for agents to represent their groups, and the planner would like to compensate the selected agents via some form of payments, which is constrained by the planner's available budget. As the costs are unknown to the planner, the planner is required to design incentive mechanisms to elicit agents' real costs and provide payments (or monetary incentives) to the selected agents to ensure proportional representation and the total payments do not exceed the budget. Such a mechanism design setting falls into the budget-feasible mechanism design paradigm. However, existing budget-feasible mechanisms only consider all agents to be in the same group with non-proportional objectives. To study the above-mentioned setting, we consider the problem of designing budget-feasible mechanisms for selecting agents with private costs from various groups to ensure proportional representation, where the minimum proportion of the overall value of the selected agents from each group is maximized. We study this problem by first considering the setting with homogeneous agents who have identical values to the planner. Depending on agents' membership in the groups, we consider two models: a single group model where each agent belongs to only one group, and a multiple group model where each agent may belong to multiple groups. We propose novel budget-feasible proportion-representative mechanisms for these models that require different selection methods, i.e., a novel greedy mechanism that considers all possible proportion ratios for the single group model and a novel mechanism that leverages the Max-Flow algorithm to evaluate the proportional representation for the multiple group model, to choose representative agents from each group. The proposed mechanisms guarantee theoretical properties of individual rationality, budget-feasibility, truthfulness, and approximation performance on maximizing the minimum proportional representation of each group. We also provide a matching lower bound for budget-feasible proportion-representative mechanisms. Finally, we non-trivially extend these mechanisms to the settings of heterogeneous agents who can have different values to the planner under the two models.