From model checking to equilibrium checking: Reactive modules for rational verification

Abstract

Model checking is the best-known and most successful approach to formally verifying that systems satisfy specifications, expressed as temporal logic formulae. In this article, we develop the theory of equilibrium checking, a related but distinct problem. Equilibrium checking is relevant for multi-agent systems in which system components (agents) are assumed to be acting rationally in pursuit of delegated goals, and is concerned with understanding what temporal properties hold of such systems under the assumption that agents select strategies in equilibrium. The formal framework we use to study this problem assumes agents are modelled using REACTIVE MODULES, a system modelling language that is used in a range of practical model checking systems. Each agent (or player) in a REACTIVE MODULES game is specified as a nondeterministic guarded command program, and each player's goal is specified with a temporal logic formula that the player desires to see satisfied. A strategy for a player in a REACTIVE MODULES game defines how that player selects enabled guarded commands for execution over successive rounds of the game. For this general setting, we investigate games in which players have goals specified in Linear Temporal Logic (in which case it is assumed that players choose deterministic strategies) and in Computation Tree Logic (in which case players select nondeterministic strategies). For each of these cases, after formally defining the game setting, we characterise the complexity of a range of problems relating to Nash equilibria (e.g., the computation or the verification of existence of a Nash equilibrium or checking whether a given temporal formula is satisfied on some Nash equilibrium). We then go on to show how the model we present can be used to encode, for example, games in which the choices available to players are specified using STRIPS planning operators.

Natural strategic ability

“…The above decision problems come very close to the recent work on equilibrium checking by Wooldridge, Gutierrez and colleagues [116,128,63]. The difference is that they look at arbitrary combinatorial strategies with perfect recall, whereas we consider natural strategies under bound k (both perfect recall and memoryless).…”

“…Finally, rational verification (also known as equilibrium checking) is an extension of model checking to verify behavior of agents limited to rational choices. Typically, this kind of verification is based on Nash equilibrium, and applied to concurrent games with players' objectives defined by formulas of linear time logic [116,128,63]. Preliminary approaches to similar analysis based on other solution concepts were presented in [65,41,100,99].…”

“…, ϕ n i ). The idea is that the agent is happiest if ϕ 1 i is achieved, otherwise would prefer to obtain ϕ 2 i , and so on [41,62,63]. In the simplest case of n = 1, this boils down to specification of the agent's winning condition ϕ i , i.e., each agent gets a binary payoff: either win or lose [51,24].…”

“…The difference is that they look at arbitrary combinatorial strategies with perfect recall, whereas we consider natural strategies under bound k (both perfect recall and memoryless). We will see in Section 9 that this reduces the computational complexity: from 2EXPTIME for combinatorial strategies [128,63] to EXPSPACE for natural strategies with recall, and between Σ P 2 and PSPACE for natural memoryless strategies. We also note that a similar set of decision problems can be defined for many other solution concepts, such as undominated strategies, Stackelberg equilibrium, subgameperfect Nash equilibrium, and so on.…”

“…Besides redefining strategy-based requirements, natural strategies provide a new, different view of concurrent multi-player games. In concurrent games, the overall behavior of the system emerges from interaction of individual behaviors driven by individual goals of agents [116,128,63]. In the new view, the emergence results from simple individual play.…”