Games hung on Petri-net places: a newly found phishing scenario joins an existing campaign through one entry value, no re-solve

Large strategic systems are no longer flattened into one global game; they are split into local games whose outcomes decide which later interactions become possible, who takes part, and with what resources. Earlier treatments either wrote the whole system as an extensive-form game that grows multiplicatively with the number of stages, or as a stochastic game over global states in which the identity of sub-interactions and their interfaces is lost; Petri games ask only a win-or-lose question and carry no payoffs or equilibrium selection. In a 39-segment enterprise study, the decomposed model needs on the order of a few tree nodes, against roughly another order of magnitude for a monolithic extensive form.

The construction attaches a local game to a game place of a colored Petri net, and a game-to-token interface turns the binding, the action profile, and the outcome into typed tokens when the game resolves. Those tokens then enable ordinary net transitions, including guarded branches, forks, and synchronizations, so one outcome can spawn several games and several completed games may be jointly required before a successor is enabled. The model separates a structural net, local games, and a solution layer, carried by the incidence matrix, the local payoff and outcome-probability matrices, and the routing matrix between games; guards act only on the incidence matrix and incentive instruments only on the local matrices.

In sequential nets, global equilibria are exactly collections of local equilibria, one per place, of the local games augmented by the continuation values that the collection itself generates. Equilibrium values can then be computed place by place by solving the strongly connected components of the place graph in reverse topological order, at a cost additive rather than multiplicative in the sizes of the local games. In a maintenance workflow, graded routing reverses the unique myopic equilibrium into cooperation while binary routing cannot; in an intrusion campaign, a decoy posture lowers the attacker's value at an upstream place although no action there has changed.

The equilibrium and computation results apply to sequential nets, in which at most one game is active at a time, with a finite unfolding; concurrent interaction is represented by the general semantics, but its analysis falls back on the marking-level stochastic game, whose state space can be large. Local games are in normal form with complete information, and the scheduler is exogenous. The next steps stated are a toolchain that compiles such nets from workflow or attack-graph descriptions, computes certificates and values, and reports where coarse summaries are inaccurate, and a calibration of the examples to operational data.

The equilibrium and computation results rely on the sequential-net assumptions and on a finite unfolding, and concurrent strategic interaction falls back on the marking-level stochastic game. The cost of building the unfolded net lies outside the complexity bound, and that step can itself be large when color or binding spaces are large. Local games are normal form with complete information, and the scheduler is exogenous. The numerical examples are illustrative and not calibrated to data. The authors also state a conjecture about selection-independent safety under interface contracts, which is not proved.