Decentralized data-injection attack construction with minimum mean-square-error state estimation is studied in a game-theoretic setting. Within this framework, the interaction between the network operator and the set of attackers, as well as the interactions among the attackers, are modeled by a game in normal form. A novel utility function that captures the trade-off between the maximum distortion that an attack can introduce and the probability of the attack being detected by the network operator is proposed. Under the assumption that the state variables can be modeled as a multivariate Gaussian random process, it is shown that the resulting game is a potential game. The cardinality of the corresponding set of Nash Equilibria (NEs) of the game is analyzed. It is shown that attackers can agree on a data-injection vector construction that achieves the best trade-off between distortion and detection probability by sharing only a limited number of bits offline. Interestingly, this vector construction is also shown to be an NE of the resulting game.