Replication Robust Payoff Allocation in Submodular Cooperative Games

Submodular functions have been a powerful mathematical model for a wide range of real-world applications. Recently, submodular functions are becoming increasingly important in machine learning (ML) for modelling notions such as information and redundancy among entities such as data and features. Among these applications, a key question is payoff allocation, i.e., how to evaluate the importance of each entity towards the collective objective? To this end, classic solution concepts from cooperative game theory offer principled approaches to payoff allocation. However, despite the extensive body of game-theoretic literature, payoff allocation in submodular games are relatively under-researched. In particular, an important notion that arises in the emerging submodular applications is redundancy, which may occur from various sources such as abundant data or malicious manipulations where a player replicates its resource and act under multiple identities. Though many game-theoretic solution concepts can be directly used in submodular games, naively applying them for payoff allocation in these settings may incur robustness issues against replication. In this paper, we systematically study the replication manipulation in submodular games and investigate replication robustness, a metric that quantitatively measures the robustness of solution concepts against replication. Using this metric, we present conditions which theoretically characterise the robustness of semivalues, a wide family of solution concepts including the Shapley and Banzhaf value. Moreover, we empirically validate our theoretical results on an emerging submodular ML application, i.e., the ML data market.