Collaborative Mean Estimation Among Heterogeneous Strategic Agents: Individual Rationality, Fairness, and Truthful Contribution

We study a collaborative learning problem where m agents aim to estimate a vector μ =(μ1,ldots,μd)∈ ℝd by sampling from associated univariate normal distributions N(μk, σ2)k∈[d]. Agent i incurs a cost ci,k to sample from N(μk, σ2). Instead of working independently, agents can exchange data, collecting cheaper samples and sharing them in return for costly data, thereby reducing both costs and estimation error. We design a mechanism to facilitate such collaboration, while addressing two key challenges: ensuring individually rational (IR) and fair outcomes so all agents benefit, and preventing strategic behavior (e.g. non-collection, data fabrication) to avoid socially undesirable outcomes. We design a mechanism and an associated Nash equilibrium (NE) which minimizes the social penalty-sum of agents' estimation errors and collection costs-while being IR for all agents. We achieve a O(√m)-approximation to the minimum social penalty in the worst case and an O(1)-approximation under favorable conditions. Additionally, we establish three hardness results: no nontrivial mechanism guarantees (i) a dominant strategy equilibrium where agents report truthfully, (ii) is IR for every strategy profile of other agents, (iii) or avoids a worst-case Omega(√m) price of stability in any NE. Finally, by integrating concepts from axiomatic bargaining, we demonstrate that our mechanism supports fairer outcomes than one which minimizes social penalty.