Congestion dynamics and safety-objective options¶

What the current benchmark establishes¶

The current two-agent game has a high-reward, zero-risk stationary convention: player 0 always takes Road A and player 1 always takes Road B. Each receives -2 per step and neither is ever labelled unsafe. A checkpoint/exact-graph test now verifies that this deterministic joint policy has zero finite- and infinite-horizon unsafe reachability.

Standard OmSh nevertheless admits only Detour and receives -6. Its focal certificate is decentralized and retains nonzero opponent probability on a mismatching road or detour. Repeating any positive mismatch hazard forever makes eventual unsafe reachability one. The gap is therefore a coordination and opponent-support issue, not a physical impossibility of the environment or a successor-budget allocation failure.

The final_cpo_risk_v1 frozen CPO audit makes the horizon mismatch measurable. Across three replayed seeds, CPO receives exactly -2 with zero violations in 3,000 live H=200 episodes. The exact policy-induced chain nevertheless has mean H=200 unsafe reachability .000556 (worst seed .001527) and eventual reachability exactly one in every seed. CPO has learned the valuable split convention, but its stochastic softmax policy retains enough mismatch support for failure to occur almost surely over unbounded repetition.

Changes that preserve infinite-horizon reachability¶

Route commitment¶

Make route selection a one-time or persistent commitment. After both agents choose, a safely split A/B assignment becomes an absorbing safe operating state whose rewards continue indefinitely. A mismatch is resolved once through the safety budget or by taking the detour. This retains an infinite-horizon reachability objective while removing an identical fresh mismatch lottery from every timestep.

This is the cleanest environment-dynamics variant for the paper. It tests whether OmSh spends a finite risk budget on a valuable irreversible choice, rather than whether soft opponent probabilities can ever become exact zeros. It should be added as a separately named condition, not silently replace the current repeated-routing stress test.

Public reservations or a correlation device¶

Add a public role/reservation signal that assigns different roads, then make actions conditional on that signal. This only solves the certificate problem if compliance is guaranteed by the dynamics or jointly certified. If an opponent can ignore the reservation with nonzero probability and the focal agent is still blamed for the resulting imbalance, eventual risk remains one.

Infrastructure-enforced overflow routing¶

The environment can automatically send excess road users to the detour. This gives every agent a robustly safe action, but infrastructure now performs the shielding. It is useful as a systems baseline, not as primary evidence for opponent-modelling shielding.

Safety-objective changes¶

Persistent congestion rather than one bad round¶

Label unsafe only after k consecutive congested rounds and expose the counter in public state. This provides recovery and a more physical failure event, but it does not by itself solve eventual reachability: under a stationary nonzero hazard, a run of k failures still occurs almost surely over an infinite horizon.

Jam-only safety¶

Remove the "strictly busier road" violation and retain only capacity/jam failures. With the current two-agent jam_threshold=5, this makes safety trivial because two agents cannot jam a road. Lowering the threshold to two restores a meaningful split-routing constraint but leaves the same repeated mismatch issue.

Discounted, rolling, or average safety¶

Constrain discounted cost, violation rate, stationary unsafe occupancy, or a rolling finite-horizon chance. These objectives can value recovery and make a small recurrent hazard feasible. They are legitimate constrained-control benchmarks, but they are not Edwin-style eventual-unsafe reachability and do not support the paper's current anytime claim.

Responsibility-based safety¶

Label a focal agent unsafe only when it violates its assigned route, regardless of congestion caused by the opponent. This creates a zero-risk compliant action, but changes safety from a physical outcome to rule compliance. It is appropriate only if that is the intended normative question.

Recommendation¶

Keep the current Congestion condition as an explicit decentralized-support stress test and add a separately named route-commitment variant. In parallel, the algorithmically faithful solution is a joint/correlated shield that certifies the A/B pair. Do not solve the headline result by switching to discounted or finite-horizon cost while continuing to call the result an infinite-horizon reachability guarantee.