Model Checking Nash Equilibria in MAD Distributed Systems

Abstract

Abstract-We present a symbolic model checking algorithm for verification of Nash equilibria in finite state mechanisms modeling Multiple Administrative Domains (MAD) distributed systems.Given a finite state mechanism, a proposed protocol for each agent and an indifference threshold for rewards, our model checker returns PASS if the proposed protocol is a Nash equilibrium (up to the given indifference threshold) for the given mechanism, FAIL otherwise. We implemented our model checking algorithm inside the NuSMV model checker and present experimental results showing its effectiveness for moderate size mechanisms.

Model Checking Coalition Nash Equilibria in MAD Distributed Systems

“…2 we show how a MAD distributed system with coalitions of players can be modeled as a Coalition Schema, that is a suitable synchronous product of Finite State Machines. This framework extends Finite State Mechanisms presented in [20] which do not account for coalitions.…”

“…A symbolic model checking algorithm to automatically verify that a given protocol is a Nash eaquilibrium for a given MAD distributed system has been presented in [20]. However the model checker presented in [20] only addresses the case in which agents do not collude.…”

“…A symbolic model checking algorithm to automatically verify that a given protocol is a Nash eaquilibrium for a given MAD distributed system has been presented in [20]. However the model checker presented in [20] only addresses the case in which agents do not collude. On the other hand, it is well known from game theory that coalitions of agents may have an advantage in deviating even when no single agent may get any advantage by deviating alone (e.g.…”

“…From a mathematical point of view, given a coalition schema Q and a coalition Q, we can build a mechanism M Q in which the coalition is just one of the agents. As a consequence, a first approach (that we call explicit) to verify that a mechanism is ε-f -q-Nash consists of adapting the symbolic algorithm in [20] to check that M Q is a Nash equilibrium for all Q, such that 0 < |Q| ≤ q. If Q has n players, following this approach entails calling O(n q ) times a variation of the algorithm in [20] (more specifically, q k=1 n k ).…”

“…As a consequence, a first approach (that we call explicit) to verify that a mechanism is ε-f -q-Nash consists of adapting the symbolic algorithm in [20] to check that M Q is a Nash equilibrium for all Q, such that 0 < |Q| ≤ q. If Q has n players, following this approach entails calling O(n q ) times a variation of the algorithm in [20] (more specifically, q k=1 n k ). To overcome this exponential growth, we propose an alternative approach (that we call symbolic) by extending the algorithm in [20] so as to represent symbolically (i.e.…”