Logistics operators provide freight services using different transport modes and, in many corridors, can choose between road, rail, and inland waterways. Mode choice and investment decisions are typically driven by cost, reliability, frequency, transit time, and infrastructure accessibility [11]. In the Brazilian context, river-based operations face additional uncertainty linked to hydrological variability, uneven port performance, and limited intermodal integration [3]. Given these constraints, operators may avoid committing capacity to inland waterways unless infrastructure and demand conditions reduce perceived risks and improve the expected payoff relative to alternative modes.

Public agencies influence freight systems through investment planning and regulatory instruments, including infrastructure spending, taxation, subsidies, concessions, and environmental standards [11]. In Brazil, the development of inland waterways has been affected by governance arrangements and the ability to execute budgets for navigation, dredging, and port-related projects [4]. Game-theoretic modeling provides a structured way to test how alternative policy packages change private incentives and can reduce market failures that keep freight in higher-emission modes. Recent work on emission regulation illustrates how strategic behavior responds to fee-based mechanisms and dynamic enforcement, reinforcing the need to represent regulation explicitly in payoff structures [13].

Freight-generating companies determine effective demand for transport services and influence modal split through contracting practices, service-level requirements, and willingness to bear switching costs [11]. In emerging economies, shippers often perceive inland waterways as less flexible and less reliable because of service discontinuities, seasonal navigability, and limited terminal availability [3]. At the same time, sustainability targets and stakeholder pressure can increase the attractiveness of lower-emission modes when reliable service is available, which can be represented through environmental terms in the utility function [1]. The strategic interactions described above are summarized in Figure 1, which illustrates the relationships among the main agents and the role of policy instruments in shaping equilibrium outcomes.

Figure 1.

Conceptual framework of the strategic interaction among logistics operators, government agencies, and freight-generating companies. Government policies, including subsidies and carbon taxation, influence incentives and shape modal choice and equilibrium outcomes.

Each player adopts strategies aimed at maximizing utility that is, profit, efficiency, cost reduction, positive environmental impact, and other relevant aspects. Therefore, it is first necessary to define:

• The players (actors in the system);

• The sets of available strategies;

• The payoff functions (return/cost for each actor);

• The resolution of the Nash equilibrium.

These players, in turn, may choose among different strategies. Table 3 summarizes the players, strategies, and notation adopted in the model:

The next step consists of defining, for each player, a specific payoff according to the chosen strategies. Thus, the utilities are represented by linear functions, as shown in eqs. (1) - (3):

where are:

• R_O(s_O, s_E): operator revenue (affected by contracted demand).

• C_O(s_O): operator cost by mode.

• I_O(s_O, s_G): incentive/subsidy under policy s_G.

• K_O(s_O): integration/switching/investment cost.

• B_G(s_O, s_E): government benefit (environment + infrastructure/efficiency).

• S_G(s_G): public expenditure under policy s_G.

• τ_G(s_O, s_G, s_E): net fiscal effect (e.g., tax revenue minus admin cost), if modeled.

• OC_E(s_O): logistics cost for the company.

• PV_E(s_O): perceived value (e.g., ESG/reputation or cost advantage).

• DT_E(s_O): delivery-time penalty.

• CT_E(s_O, s_G): carbon charge/tax under policy s_G.

Thus, the Nash equilibria are defined. It is important to emphasize that a Nash Equilibrium corresponds to a state in which no player has an incentive to unilaterally change their strategy. Therefore, a set of strategy profile denoted by s* represents the equilibrium condition, as shown in Equation (4):

For clarity, all variables used in the payoff formulation are explicitly defined. For the logistics operator, ROR represents operational revenue, COC denotes operational cost, GI corresponds to government incentives, and IC represents integration or switching costs. For the freight-generating company, OC denotes operational cost, PV represents perceived value, DT corresponds to delivery time, and CT denotes the carbon tax applied to road transport. These variables capture both economic and operational factors influencing the strategic decisions of each agent.

To illustrate the mathematical formulation of the game, we can add simulated values for the agents’ payoffs, based on real or estimated data. This will help demonstrate how different variables affect the equilibrium. Table 4 presents an illustrative example of the payoff structure for the logistics operator under the proposed policy scenario.

Table 5 shows the corresponding payoff structure for the freight-generating company, highlighting the impact of policy instruments on decision-making. Together, Table 4 and Table 5 illustrate how policy instruments reshape incentives across agents:

This section presents the results of the simulations in a summarized manner, as well as the analysis of the strategic behaviors observed among the agents. Different combinations of policies and modal choices are explored to identify stability and evaluation when publics investments are made. The model relies on simplified payoff structures and simulated parameters, which may introduce measurement inaccuracies. In particular, uncertainties related to hydrological variability, seasonal navigability, and operational disruptions are not explicitly modeled and may affect real-world outcomes. Additionally, non-cooperative game theory assumes rational decision-making and may not fully capture institutional frictions such as licensing disputes, political constraints, or perceived socio-environmental risks. These limitations should be considered when interpreting the results. For clarity, the main abbreviations used throughout the model are listed in the Abbreviations section before the References.

Best responses across the strategy profiles are analyzed to identify Nash equilibria, defined as outcomes in which no player can improve its payoff through unilateral deviation. Table 6 summarizes the equilibrium payoffs under the policy scenario combining subsidies and carbon taxation. The results indicate that the equilibrium configuration emerges when inland waterway transport is supported by subsidies and selected by both logistics operators and freight-generating companies. In this configuration, the incentives align across agents, resulting in higher payoffs and a stable strategic outcome.

Nash Equilibrium (Scenario with Subsidies): Based on the simulation, the Nash equilibrium occurs when the payoffs cannot be improved by any agent changing their strategy unilaterally, given the behavior of the others.

To further explore the model behavior, alternative policy scenarios were simulated by varying subsidy levels and carbon taxation. In the illustrative case, the inland waterway subsidy is set to 60 units and the carbon tax applied to road transport is set to 15 units.

Under these conditions, the payoff structure shifts significantly, favoring inland waterway transport. The results reported in Table 6 demonstrate how coordinated policy instruments can alter incentives and drive the system toward a more sustainable equilibrium.

• IWT Subsidy: 60

• Road Transport Carbon Tax: 15

With these values, the payoff for each agent is recalculated, as shown in the tables above.

To extend the analysis beyond discrete payoff tables, a parametric evaluation of policy instruments was conducted. The subsidy level for inland waterway transport and the carbon tax applied to road transport were treated as continuous variables.

Figure 3 shows the variation in the logistics operator’s payoff as a function of the subsidy level. The results reveal a nonlinear relationship, with diminishing returns as subsidies increase. An optimal range can be identified around intermediate subsidy levels, indicating that excessive incentives may reduce overall efficiency.

Figure 3.

Logistics operator payoff as a function of subsidy level. The nonlinear concave relationship reveals diminishing returns to subsidies. The red marker denotes the optimal subsidy level (S* = 50), corresponding to the maximum payoff, highlighting the existence of a policy optimum.

This implies that policy effectiveness is subject to diminishing marginal returns, meaning that increasing subsidies beyond a certain threshold may lead to inefficient allocation of public resources without proportional gains in system performance. This also suggests that optimal policy design should prioritize efficiency over scale of intervention.

Figure 4 shows the payoff of freight-generating companies as a function of carbon taxation. The results reveal a nonlinear decreasing relationship, indicating that increases in carbon taxation progressively reduce the payoff associated with road-based transport.

Figure 4.

Payoff of freight-generating companies as a function of carbon taxation. The nonlinear decreasing relationship indicates that higher taxation consistently reduces payoff, reinforcing its role as a regulatory instrument.

Unlike the logistics operator case, no internal optimum is observed within the analyzed range. Instead, the highest payoff occurs at the lowest taxation level, suggesting that carbon taxation operates primarily as a regulatory constraint rather than an optimization instrument. This reinforces its role in discouraging high-emission transport modes rather than enhancing system efficiency.

Figure 5 presents the strategic payoff landscape as a function of subsidy levels and carbon taxation. The color gradient represents the magnitude of the logistics operator’s payoff, while the highlighted region denotes policy combinations that achieve at least 90% of the maximum payoff.

Figure 5.

Strategic payoff landscape as a function of subsidy and carbon taxation. The color gradient represents the magnitude of the logistics operator’s payoff. The shaded region indicates policy combinations that achieve at least 90% of the maximum payoff, defining a robust zone of near-optimal performance. The red marker denotes the optimal policy point (S*, T*).

The results reveal that optimal outcomes are not confined to a single point but instead emerge within a well-defined region in the policy space. This region is centered around moderate subsidy levels and low carbon taxation, indicating that effective policy design does not require precise parameter tuning but rather the identification of robust combinations of incentives.

The shape of the highlighted region reflects the nonlinear structure of the payoff function. In particular, the curvature along the subsidy axis indicates diminishing returns, while the sharper contraction along the carbon tax axis suggests a stronger negative impact of taxation on system performance. This asymmetry implies that increases in carbon taxation reduce the payoff more rapidly than equivalent increases in subsidies improve it.

The red marker indicates the global maximum of the payoff function, corresponding to the optimal policy configuration. However, the existence of an extended near-optimal region suggests that policymakers may operate within a range of acceptable parameter values while still achieving high levels of performance.

From a strategic perspective, this result reinforces the importance of coordinated policy instruments. Rather than relying on a single mechanism, such as subsidies or taxation alone, the combination of moderate incentives and regulatory pressure creates a stable environment in which inland waterway transport becomes the dominant strategy.

Overall, the analysis highlights that the transition toward more sustainable transport systems is governed by nonlinear interactions between policy variables, and that robust policy regions are more relevant than isolated optimal points in practical decision-making contexts. These findings are consistent with theoretical expectations from nonlinear systems, in which equilibrium configurations emerge from the interaction of competing forces rather than from linear optimization of isolated variables.

Figure 6 provides a three-dimensional representation of the payoff surface, offering a comprehensive view of the interaction between subsidy levels and carbon taxation. The surface exhibits a well-defined peak corresponding to the optimal policy configuration.

Figure 6.

Three-dimensional representation of the logistics operator’s payoff as a function of subsidy and carbon taxation. The surface illustrates the nonlinear structure of the payoff function, with a well-defined peak corresponding to the optimal policy combination (S* ≈ 50, T* ≈ 0). The red marker indicates the global maximum within the analyzed domain, while the surrounding smooth region highlights the robustness of near-optimal outcomes.

The curvature of the surface confirms the presence of diminishing returns with respect to subsidies and a stronger negative effect associated with carbon taxation. The steeper gradient along the taxation axis indicates that increases in carbon taxes reduce system performance more rapidly than equivalent increases in subsidies improve it.

The red marker identifies the global maximum of the payoff function, representing the optimal combination of policy instruments. The optimal solution can also be derived analytically from first-order conditions, reinforcing the consistency of the graphical representation.

These results highlight that equilibrium outcomes emerge from the interaction of policy variables, reinforcing the importance of coordinated and balanced policy design in promoting sustainable transport systems.

From a policy perspective, this indicates that effective intervention does not require precise calibration of parameters, but rather the identification of robust regions in the policy space where desirable outcomes emerge. This reduces implementation uncertainty and enhances the robustness of policy design in real-world applications.

The results demonstrate that the transition toward inland waterway transport is driven by coordinated policy mechanisms rather than isolated interventions. The combination of subsidies and carbon taxation alters the strategic environment by simultaneously increasing the attractiveness of waterway transport and penalizing less sustainable alternatives.

From the perspective of logistics operators, subsidies reduce perceived risks and improve expected returns, encouraging investment in inland waterway infrastructure and services. For freight-generating companies, carbon taxation increases the relative cost of road-based logistics, shifting preferences toward lower-emission transport options.

The nonlinear structure of the payoff functions indicates that policy effectiveness is not proportional to the magnitude of intervention. Instead, there exists a range of policy configurations that maximize system performance, reinforcing the importance of calibrated and balanced policy design.

Moreover, the results highlight that equilibrium outcomes emerge from the interaction of agents operating under institutional and economic constraints. This supports the applicability of game theory as a framework for analyzing sustainable logistics transitions.

These findings are consistent with the theory of induced innovation, according to which external incentives influence technological and operational change. More broadly, the results suggest that policy effectiveness in freight transport systems is inherently nonlinear and depends on the alignment of incentives across multiple stakeholders.

Importantly, the proposed framework can be directly mapped into measurable sustainability indicators, such as emission intensity (g·CO₂/(t·km)), logistics cost efficiency, and modal share. This enables alignment with SDG monitoring frameworks and supports the translation of strategic outcomes into quantifiable policy-relevant metrics.

To further operationalize this connection, the payoff structure can be associated with emission-based indicators. Let emission intensity be defined as Em (g·CO₂/(t·km)) for each transport mode m. The expected emissions associated with a given strategy profile can be approximated as follows:

where S_m represents the share of freight allocated to mode m, endogenously determined by the equilibrium of the game. Since the payoff functions incorporate costs, incentives, and carbon taxation, changes in the equilibrium strategy profile directly affect S_m, thereby influencing total emission intensity. This establishes a direct link between strategic interaction outcomes and measurable sustainability indicators, allowing the model to be interpreted not only in terms of economic performance but also environmental impact.

In comparison with existing studies in the literature, the present model introduces relevant conceptual and methodological advances. Previous works, such as [11], primarily focus on competition between transport modes under predefined regulatory conditions, typically considering government intervention as an external parameter. In contrast, this study explicitly models government agencies as strategic players within the game, allowing policy instruments such as subsidies and carbon taxation to emerge as endogenous elements of the equilibrium structure.

Furthermore, while most existing approaches are limited to bimodal or operator–shipper interactions, the proposed framework incorporates a tripartite structure, capturing the interdependence between logistics operators, freight-generating companies, and public authorities. This enables a more realistic representation of decision-making processes in freight transport systems, particularly in emerging economies where institutional and policy dynamics play a central role. As a result, the model moves beyond static comparisons of transport modes and provides a framework for analyzing how coordinated policy design influences equilibrium outcomes, modal shifts, and sustainability performance. This distinction is particularly relevant in the context of sustainable transport planning, where policy effectiveness depends on dynamic interactions rather than isolated optimization of individual agents. In this sense, the model provides a bridge between micro-level strategic decisions and macro-level sustainability performance metrics.