Glossary¶

Agent. A decision-making participant in a shared PettingZoo environment.

Focal agent. The agent whose policy, safety label, induced MDP, or shield is currently being constructed. It defaults to player_0 in most workflows.

Opponent team. Every non-focal agent, modelled jointly when more than one is present.

Public Markov state. The flattened state bits exposed by the environment and used consistently by WM/OM data, graph nodes, and safety projection.

State key. A stable hash identifying public state bits in a joint graph.

Legal joint graph. Exact reachable topology: public states, full joint actions, and possible successors, without learned environment probabilities.

Environment transition graph. The legal joint graph annotated with learned environment probabilities and per-agent rewards.

Joint action. One ordered tuple containing every environment agent’s action.

Focal MDP. The single-agent transition system obtained by fixing a focal action and marginalizing the opponent action components.

World model (WM). The learned transition and reward model over the legal joint graph. The default is the successor kernel; the product-Bernoulli WorldModelMLP is the baseline. Selected by wm_variant.

Successor kernel. The default world model, SuccessorEdgeModel: a masked categorical distribution over each (state, joint_action) group's legal successors, with a reward per successor edge.

Product-Bernoulli world model. The bit_mlp baseline, which predicts independent next-state bits and scores legal successors by renormalizing their product likelihood.

Opponent model (OM/IOP). ImaginedOpponent, a neural stack predicting the joint opponent-team action. With several opponents each level is an exact autoregressive factorization of the joint law.

Level 0. The opponent policy fitted from observed behavior.

Reasoning level. A higher opponent policy improved through imagined responses to the preceding level.

Mixture over levels. mix_alpha, the smoothed posterior belief over opponent reasoning levels.

Level floor. The lowest opponent level a shield must currently tolerate. It is monotone within a shield instance.

Floor patience. shield_patience: the number of consecutive supporting posterior updates needed before the floor rises by one level.

Credible floor. An experimental monotone level floor chosen as the largest ordered suffix retaining a scheduled amount of opponent-posterior mass. Safety remains worst-case over the retained suffix.

Safety abstraction. An environment-owned projection of public observation used to label one explicit focal agent safe or unsafe.

Safety cost. The runtime binary cost written by LabelledEnv; distinct from environment reward.

Unsafe reachability. Probability of eventually reaching any unsafe graph state, rather than only failing on the next transition.

Safety budget. The current upper allowance for robust eventual unsafe reachability. max_risk initializes it at reset.

Egalitarian successor budget. The default robust allocation that raises non-saturated successor budgets equally while preserving every expected-budget constraint.

Learned successor budget. A policy-proposed successor allocation that is projected into the same robust feasible set before the transition.

Pairwise mixed action. A focal policy output representing a distribution on at most two primitive actions. The shield projects its mixture coordinate against the active safety constraints before sampling the executed primitive.

Sound value iteration. Interval value iteration that returns conservative lower/upper reachability bounds and sends missing transition mass to unsafe.

Shield bundle. Immutable per-level value-iteration results, transition kernels, and robust successor-budget certificates used by TransitionShield.

Missing coverage. A graph state, action, successor, or probability mass not represented by the learned/induced model. The shield handles it conservatively.

True-policy shield (TrueShielded). The oracle reference shield built from exact environment transitions and refreshed from live opponent policies.

Coordinated shield. MultiAgentTransitionShield, which protects several roles in one environment step, each with its own opponent model, posterior, floor, and budget.

Coverage assumptions (A1, A2). The conditions that carry a learned certificate over to the real game: the learned kernel is within eps_WM of the true kernel (A1), and some retained level is within eps_OM of the real opponent (A2).

Padded certificate. A certificate computed against a total-variation ball of radius shield_padding_radius around each retained level's row rather than the row itself.

Robustness margin. The largest uniform padding radius whose certificate still fits the budget at reset.

Closest-retained-level gap. At one step, the smallest total variation between the real opponent's action law and any level at or above the current floor. Its worst case over a run bounds the eps_OM that A2 needs.

Ambiguity set. A statistically valid set of transition kernels for one row: a total-variation ball around the empirical counts, with a time-uniform radius.

delta_cov. The chosen probability budget for any ambiguity set failing to contain the true kernel. The shield then runs at p_total - delta_cov.

Step excess (eta_t). How far the true expectation of the committed certificate exceeds the carried budget at one step. Its expected sum bounds how far unsafe probability can exceed max_risk; the risk-dominance audit measures it.

Exact-anytime certificate. A per-episode, focal-agent reachability certificate built from a validated serialized exact graph and worst-cased over every declared joint opponent action. Its current delta_cov=0 claim is relative to that finite graph kernel and its declared reset support. It is an original-simulator guarantee only when the graph rows are independently bound to the simulator transition kernel.

Tagged campaign. An experiment run with experiment_tag set, isolating its durable condition artifacts below experiments/runs/<tag>/ for later aggregate reporting.

Frozen final policy. The policy at the last optimizer update, evaluated after training on fresh seeds without further learning.

U_H. The probability that an episode of the frozen policy becomes unsafe within H steps, reported per seed with a Wilson interval. Campaign tables report the worst seed.

Controller snapshot. The post-training OmSh posterior, floor, and patience state stored in the checkpoint and restored before every frozen evaluation episode, so replicates are IID.

Predeclared screen. A comparison whose arms, seeds, and promotion rule are fixed before any result exists, and applied by script.

Episode cap. A limit read from max_steps and converted into explicit truncation semantics even when the base environment keeps it internal.

Marimo notebook. A reactive Python notebook saved as .py, editable with marimo edit and runnable as a normal Python script.