  With the exponential growth of data and its crucial impact on our lives and
decision-making, the integrity of data has become a significant concern.
Malicious data poisoning attacks, where false values are injected into the
data, can disrupt machine learning processes and lead to severe consequences.
To mitigate these attacks, distance-based defenses, such as trimming, have been
proposed, but they can be easily evaded by white-box attackers. The evasiveness
and effectiveness of poisoning attack strategies are two sides of the same
coin, making game theory a promising approach. However, existing
game-theoretical models often overlook the complexities of online data
poisoning attacks, where strategies must adapt to the dynamic process of data
collection.
  In this paper, we present an interactive game-theoretical model to defend
online data manipulation attacks using the trimming strategy. Our model
accommodates a complete strategy space, making it applicable to strong evasive
and colluding adversaries. Leveraging the principle of least action and the
Euler-Lagrange equation from theoretical physics, we derive an analytical model
for the game-theoretic process. To demonstrate its practical usage, we present
a case study in a privacy-preserving data collection system under local
differential privacy where a non-deterministic utility function is adopted. Two
strategies are devised from this analytical model, namely, Tit-for-tat and
Elastic. We conduct extensive experiments on real-world datasets, which
showcase the effectiveness and accuracy of these two strategies.
