This paper investigates the strategic behavior of validators in blockchain systems utilizing the Proof-of-Stake (PoS) consensus mechanism through the application of game theory. A mathematical model of a non-cooperative game with complete information is proposed, where validators act as rational agents aiming to maximize their expected payoff by choosing between honest validation and malicious actions, specifically a double-spending attack. The model incorporates key economic parameters of the system: block and attestation rewards, transaction fees, operational costs, slashing penalties, and the probability of detecting protocol violations. Utility functions for two primary strategies – honest and attacking – are formalized, and conditions for the existence of Nash equilibrium, the central solution concept in game theory, are analyzed. The analysis demonstrates that under effective punishment mechanisms, the "all-honest" equilibrium is stable: an individual validator has no incentive to deviate from protocol-compliant behavior, as potential losses from penalties significantly outweigh any gains from a failed attack. Conversely, the "all-attackers" equilibrium, while theoretically possible, is practically unattainable due to the prohibitively high cost of acquiring a majority stake, rendering such a strategy economically infeasible. A quantitative example based on a hypothetical network of 1000 validators confirms these findings and highlights the critical importance of balancing incentives for honest behavior with strong disincentives for malicious actions. The study emphasizes the crucial role of economic security in PoS systems, where stability is ensured not only by technical safeguards but also by carefully designed economic mechanisms. The developed model can be used by blockchain protocol designers to calibrate consensus parameters, thereby promoting decentralization, resilience, and long-term network reliability. Future research can extend the model by incorporating heterogeneous validators, repeated games, and the analysis of other attack vectors. Keywords: Proof-of-Stake, validators, game theory, Nash equilibrium, economic security, slashing, double-spending attack, game model, blockchain, consensus.
Viktor Godliuk (Fri,) studied this question.