Abstract
Verifying whether rational participants in a BAR system (a distributed system including Byzantine, Altruistic and Rational participants) would deviate from the specified behaviour is important but challenging. Existing works consider this as Nash-equilibrium verification in a multi-player game. If the game is probabilistic and non-terminating, verifying whether a coalition of rational players would deviate becomes even more challenging. There is no automatic verification algorithm to address it. In this article, we propose a formalization to capture that coalitions of rational players do not deviate, following the concept of Strong Nash-equilibrium (SNE) in game-theory, and propose a model checking algorithm to automatically verify SNE of non-terminating probabilistic BAR systems. We implemented a prototype and evaluated the algorithm in three case studies.
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Notes
- 1.
Here, game refers to the atomic concept of Game Theory, defined as the study of mathematical models of conflict and cooperation between intelligent and rational decision-maker agents.
- 2.
In BAR model, the agents are divided in three categories, altruistic, rational or Byzantine. Only altruistic agents follow the system specification.
- 3.
We do not need to consider the case of \(\emptyset \) as there is no probability in this case.
- 4.
Essentially, \(\mathsf{P}^{\pi }_{\varLambda }\) projects each action to a part of it.
- 5.
Note that the rational players are exactly players in the coalition, capturing that the rational players assume the unknown players (not in C) are altruistic by default.
- 6.
In this example, it happens (uncommonly) that given any fixed choice of \(d_1\) and \(d_2\), that \(d_3\) chooses the lowest price p leads to the guaranteed pay-off of \(d_1\) and \(d_2\) in every case.
- 7.
\(a_C\) is short for \(\mathsf{p}(a,C)\), \(a_L\) is short hand for \(\mathsf{p}(a,L) \).
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Acknowledgement
This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Corporate Laboratory@University Scheme, National University of Singapore, and Singapore Telecommunications Ltd.
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Fernando, D., Dong, N., Jegourel, C., Dong, J.S. (2018). Verification of Strong Nash-equilibrium for Probabilistic BAR Systems. In: Sun, J., Sun, M. (eds) Formal Methods and Software Engineering. ICFEM 2018. Lecture Notes in Computer Science(), vol 11232. Springer, Cham. https://doi.org/10.1007/978-3-030-02450-5_7
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