Game theory reveals which Naples road closures hurt the network most

When a road closes in a big city, the damage is not always where you expect it. A new study of the metropolitan road network of Naples, Italy, shows that the disruption affecting the largest number of routes is not necessarily the one that exerts the greatest strategic influence on how travellers should spread their path choices.

The study, led by Tahseen Bashir with Michael G. H. Bell, Glenn Geers, Angela Stefania Bergantino and Francesca Pagliara, addresses a long-standing problem in transport planning: how to assess network vulnerability when the probabilities of different disruptions are unknown or cannot be estimated with confidence. Probabilistic reliability approaches are valuable when uncertainty can be characterised empirically, but for rare, heterogeneous or location-specific events such information is often missing. The researchers therefore built their analysis on Bell’s minimax formulation, a two-player non-cooperative zero-sum game in which users select probability distributions over available paths to minimise expected trip cost, while an adversarial entity selects among network-performance states to maximise it. Crucially, the resulting equilibrium weights assigned to adverse states are strategic quantities, not empirical estimates of how often each disruption will actually occur.

The framework was applied to the Metropolitan City of Naples, an area of roughly 1,179 square kilometres with more than three million inhabitants across 82 municipalities. The road network, derived from OpenStreetMap and implemented in the PTV VISUM transport modelling package, was represented by 112 traffic analysis zones. Before any disruption analysis, the model was validated against Google Maps free-flow travel times for corresponding origin-destination movements at an off-peak departure time of 01:30 on 15 May 2025. The fitted relationship showed a coefficient of determination of 0.9046, with a slope above unity indicating that the model generally produced somewhat higher travel-time estimates, providing support for using the modelled network as the reference condition.

For the game itself, the researchers selected 25 origin-destination pairs associated with the investigated disruption corridors. Between three and five alternative paths were generated for each pair from the intact baseline network using a link-penalty approach, which temporarily increases the search cost of links belonging to already identified paths to encourage genuinely different alternatives. Paths were screened to avoid excessive overlap, with a maximum path overlap of 75 percent and a maximum detour of 30 percent. Each candidate path was then retained as a fixed sequence of directed links and evaluated under five road-closure scenarios, producing a final strategy set of 122 fixed paths and 610 path-scenario combinations.

The five scenarios represented different functional components of the network. They were the closure of Galleria Laziale, an approximately 500-metre tunnel in western Naples; Via Amerigo Vespucci, an approximately 600-metre corridor providing port access; Corso Umberto I, an approximately 1.2-kilometre central corridor linking the historic centre with the area around Napoli Centrale railway station; Corso Vittorio Emanuele, an approximately 4.5-kilometre hillside corridor; and an interruption of the Tangenziale di Napoli between Exits 4 and 6, an approximately 3-kilometre motorway segment. For each scenario, a user-equilibrium assignment was performed in PTV VISUM, and the scenario-specific congested link travel times were used to build origin-destination-specific payoff matrices. When a fixed path intersected a link rendered unavailable, it was assigned an origin-destination-specific finite failure cost, scaled by a multiplier whose value was tested in sensitivity analysis at 1.5, 2, 3 and 5.

Solving the resulting linear programs produced a striking asymmetry between physical exposure and strategic importance. Corso Umberto I, scenario q3, produced the largest number of disruption-infeasible combinations, affecting 46 path-scenario combinations across 18 of the 25 origin-destination pairs. Yet when the adversarial equilibrium weights were aggregated across demand, the Tangenziale di Napoli closure, scenario q5, consistently received the largest weight, ranging from 58.50 to 59.16 percent across all tested failure-cost multipliers. Corso Umberto I came second at 13.42 to 13.86 percent, followed by Via Amerigo Vespucci, Galleria Laziale and Corso Vittorio Emanuele. The ordering by equilibrium weight thus differed from the ordering by physical exposure, demonstrating that the number of affected paths or origin-destination pairs alone does not fully characterise network vulnerability.

The user side of the game told a complementary story. Mixed strategies, in which positive equilibrium weight is assigned to more than one candidate path, emerged for 20 of the 25 origin-destination pairs at every tested failure-cost multiplier, and these pairs achieved lower worst-case costs than the corresponding best pure-path worst-case benchmarks. The remaining five pairs, including San Giuseppe-Afragola and four movements originating in Ponticelli, produced pure strategies across all tested values. In aggregate terms, the mean equilibrium travel cost across the 25 pairs rose from 64.88 minutes at a failure-cost multiplier of 1.5 to 132.53 minutes at 5.0, while the mean pure-path benchmark rose from 76.88 to 212.35 minutes, widening the mean benefit of mixed strategies from 12.00 to 79.82 minutes. Demand-weighted results showed the equilibrium cost remaining below the benchmark for all tested values, with the difference growing from 11.07 minutes, or 14.05 percent, to 74.17 minutes, or 36.22 percent. Origin-destination pairs with mixed strategies represented approximately 67.71 percent of the demand in the selected sample.

The authors are careful about what these numbers do and do not mean. The sensitivity analysis shows that increasing the failure-cost multiplier raises both the equilibrium cost and the apparent benefit of diversification, simply because higher costs are assigned to paths that become unavailable; the larger differences at higher multipliers should therefore not be read as evidence of greater real-world benefits from path diversification. At the same time, the support sets returned by the solver remained unchanged for all 25 pairs across the tested range, the independently solved primal and dual formulations were numerically consistent across all 100 origin-destination-specific games with a maximum primal-dual gap of 1.49 times ten to the minus twelve, and the Tangenziale retained its roughly 59 percent demand-weighted equilibrium weight throughout. The absolute values are sensitive to the assumed consequence of path failure, while the qualitative patterns were comparatively robust.

Several limitations define the scope of the findings. The analysis covers 25 selected origin-destination pairs rather than all movements in the 112-zone metropolitan model, so the demand-weighted results characterise the selected sample and should not be interpreted as metropolitan-wide vulnerability measures. Candidate paths are generated on the intact network and held fixed across scenarios, meaning the framework represents ex-ante path choice under disruption uncertainty rather than adaptive rerouting after a closure is observed. The finite cost assigned to infeasible paths is a modelling assumption rather than an empirically observed behavioural response, although the sensitivity analysis makes its influence explicit. The adversarial entity represents adverse network conditions rather than an intentional attacker, and its equilibrium weights express relative strategic influence within the specified game, not disruption probabilities.

For transport planners, the practical message is that physical exposure and strategic importance are complementary rather than interchangeable measures. Physical exposure identifies how extensively a closure directly affects the predefined path set, whereas the equilibrium analysis indicates how strongly each disruption influences path-choice strategies when all scenarios are considered jointly. In Naples, the high physical exposure of Corso Umberto I and the large demand-weighted equilibrium weight of the Tangenziale di Napoli identify different dimensions of vulnerability, helping to distinguish corridors whose closure affects a large portion of the available path set from those whose closure is harder to accommodate through alternatives. Because the framework requires no predefined disruption occurrence probabilities, the authors argue it can complement conventional disruption assessment wherever reliable frequencies for disruptive events are unavailable, and future work could extend it to larger origin-destination sets, adaptive path choice, partial capacity reductions, simultaneous disruptions and multimodal networks.

Subject of Research: Game-theoretic assessment of road-network vulnerability under road closures in Naples, Italy

Article Title: A game-theoretic assessment of urban road-network vulnerability: a case study of Naples, Italy

Article References: Bashir, T., Bell, M. G. H., Geers, G., Stefania Bergantino, A., & Pagliara, F. (2026). A game-theoretic assessment of urban road-network vulnerability: a case study of Naples, Italy. European Transport Research Review, 18(1), Article 93. https://doi.org/10.1186/s12544-026-00860-y

Image Credits: AI Generated

DOI: 10.1186/s12544-026-00860-y

Keywords: road-network vulnerability, game theory, minimax equilibrium, Naples, PTV VISUM, disruption scenarios, path choice, transport planning, zero-sum game, strategic importance, physical exposure, European Transport Research Review