Analysis of the competition of the South-Eastern Railway of Peru through a timetable auction

Aliaga-Miranda, Augusto et al. Article Analysis of the competition of the South-Eastern Railway of Peru through a timetable auction Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Aliaga-Miranda, Augusto et al. (2025) : Analysis of the competition of the SouthEastern Railway of Peru through a timetable auction, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 2, pp. 1-19, https://doi.org/10.3390/g16020016 This Version is available at: https://hdl.handle.net/10419/330130 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Ulrich Berger Received: 15 January 2025 Revised: 24 March 2025 Accepted: 1 April 2025 Published: 7 April 2025 Citation: Aliaga-Miranda, A., FloresVilcapoma, L. R., Raqui-Ramirez, C. E., Claudio-Pérez, J. L., Yanase-Rojas, Y., & Espinoza-Yangali, J. P. (2025). Analysis of the Competition of the South-Eastern Railway of Peru Through a Timetable Auction. Games, 16(2), 16. https://doi.org/10.3390/ g16020016 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article Analysis of the Competition of the South-Eastern Railway of Peru Through a Timetable Auction Augusto Aliaga-Miranda 1, Luis Ricardo Flores-Vilcapoma 2,* , Christian Efrain Raqui-Ramirez 1, José Luis Claudio-Pérez 1, Yadira Yanase-Rojas 2and Jovany Pompilio Espinoza-Yangali 3 1School of Business, Universidad Nacional Intercultural de la Selva Central Juan Santos Atahualpa, La Merced 12856, Peru; [email protected] (A.A.-M.); [email protected] (C.E.R.-R.); [email protected] (J.L.C.-P.) 2School of Business, Universidad Tecnológica del Perú, Lima 15046, Peru; [email protected] 3School of Business, Universidad Continental, Junín 12000, Peru; [email protected] *Correspondence: [email protected] Abstract: Our research analyzes the design of an auction model for railway transportation on the South-East Railway of Peru, managed by Ferrocarril Transandino S.A. (Fetransa) and operated by PeruRail. Initially, the regulatory framework aimed to promote competition in railway transportation through timetable auctions and infrastructure access. However, the concession has resulted in a vertically integrated structure that favors PeruRail, which faces minimal direct competition, controls high-demand time slots, and hinders the entry of other operators due to strategic and structural access barriers. To address these distortions, we propose reforming the auction mechanism to neutralize these advantages and enhance competition. In this revised framework, the track usage fee will serve as the competitive factor, with the highest bid above a minimum base rate securing the allocation. Additionally, we propose the implementation of asymmetric tariffs to compensate for the higher costs faced by operators with fewer economies of scale, technological optimizations to facilitate equitable access to time slots, and stricter oversight mechanisms to ensure transparency in timetable allocation. These measures aim to balance the market and safeguard competition through a more equitable and efficient auction design. Keywords: auction theory; game theory; railway market; railway timetable 1. Introduction The question regarding the consequences of implementing auctions in Peru’s railway service is crucial, as the South-East 1 Railway has demonstrated significant limitations, particularly in achieving equitable access to railway infrastructure. This lack of competition not only increases fees and limits service options for users but also impacts public perception and the effectiveness of privatization policies in the country. Until the 1980s, the railway sector in Peru was exclusively controlled and operated by a state entity within a highly regulated system with no private participation. This model aligned with the natural monopoly structure characteristic of the railway industry, where economies of scale favor operation by a single provider to minimize costs and maximize operational efficiency. However, as modernization policies progressed and more competitive market models were promoted, the privatization and concession of strategic sectors were encouraged to improve service quality and achieve greater economic efficiency. This transition aimed to adapt the railway system to a limited competition framework, where private operators would have incentives to enhance the quality of services offered to users. Games 2025,16, 16 https://doi.org/10.3390/g16020016 Games 2025,16, 16 2 of 19 Since 1999, concessions for the management of railway infrastructure have been implemented, allowing private companies to participate in the sector. In the case of the South-East Railway, infrastructure management was entrusted to Ferrocarril Transandino S.A. (Fetransa), while railway transportation operations were assigned to PeruRail S.A. (PeruRail). This concession scheme has been subject to controversy, as contrary to the intended goals of privatization policies, PeruRail continues to operate in a market with limited direct competition 2 . This situation has affected public perception due to the high fares and lack of service options. The absence of direct competitors and the vertically integrated structure of Fetransa and PeruRail have resulted in a scenario where the profits of privatization, in terms of fees and service diversity, remain limited. Since the inception of the railway concession, the state-owned traction and rolling stock assigned to Fetransa for its operation has been leased exclusively to PeruRail. This agreement, which forms part of the concession’s assets, has been renewed on four occasions, with the most recent extension valid until 2021. As a result, PeruRail’s dominant position in the market has been further consolidated, significantly limiting opportunities for competition and creating a scenario where access to essential infrastructure is exclusively dependent on this single operator. This situation presents considerable challenges for potential new entrants, who must contend with high initial costs and significant structural barriers to compete in a market where state resources continue to benefit a single company. The exclusive leasing of state-owned traction and rolling stock to PeruRail has erected substantial barriers to competition, primarily by restricting new operators’ access and raising market entry costs. This, in turn, hampers the railway market’s dynamism. However, these obstacles could diminish in the long term due to the natural obsolescence of the leased equipment, which will inevitably need to be replaced. Introducing a transparent and inclusive access mechanism could encourage new participants and foster a fairer distribution of railway infrastructure. Unlike the leasing of traction equipment, access fees for railway infrastructure are regulated under the terms of the concession contract and are adjusted annually through an indexation mechanism. This process is overseen by the Supervisory Agency for Investment in Public Transport Infrastructure (OSITRAN), which ensures compliance with the established regulatory framework. Although the concession allows railway operators to access the infrastructure either through direct negotiation or auction, only the former has been employed to date. The lack of incentives, combined with an ineffective regulatory design for auctions, may intensify competition-related challenges, as the current system does not guarantee neutrality or non-discrimination. The prevailing auction conditions, along with the existing structural and strategic barriers, undermine market performance by reducing opportunities for new operators and limiting the overall efficiency of the railway network. Our analysis reveals that the design of the timetable auction is inherently anticompetitive. The primary operator, PeruRail, can offer high bids without fully internalizing the associated costs, thereby reducing the likelihood of success for other potential operators. Furthermore, structural barriers, such as the vertical integration between the infrastructure manager and the transport service provider, reinforce PeruRail’s dominant position. The operational advantages held by this company allow it to manipulate the allocation of time slots, undermining the competitiveness of independent operators and limiting the potential for service diversification in non-auctioned time slots. The auction reform emerges as the most viable alternative to renegotiating the rolling stock lease contract due to its greater legal feasibility, faster implementation, and impact on competition. Modifying the concession contract would require negotiations with Fetransa and PeruRail, potentially leading to litigation over acquired rights and significantly delaying its execution. In contrast, auction reform could be implemented through regulatory Games 2025,16, 16 3 of 19 adjustments without the need for complex contractual renegotiations. Moreover, its implementation would allow for the correction of distortions in the short term, whereas opening the leasing process would require an operational and financial restructuring of Fetransa, extending the timeline. In terms of competition, while granting other operators access to rolling stock would reduce certain entry barriers, structural obstacles such as access to key information and schedule distribution would persist, which could be more effectively addressed through a well-designed auction reform. In response to the current challenges, a reform of the timetable auction system is proposed, incorporating measures to level the playing field through a base fee. As complementary conditions, we suggest implementing asymmetric fees per wagon-kilometer traveled by the winning railway operator during the auctioned time slot to offset the additional costs faced by smaller operators, as well as adopting technological solutions to facilitate equitable access to time slots. Additionally, stricter regulatory oversight is recommended to ensure neutrality and prevent the concentration of consecutive time slots in the hands of a single operator. Our research provides a novel approach by highlighting the importance of adapting auction designs to contexts of market dominance, such as the Peruvian case. It also emphasizes the need for regulatory policies to evolve in response to market dynamics and proposes practical solutions that could be applied to other infrastructure sectors. The combination of differentiated tariffs, access technology, and continuous oversight offers an adaptive regulatory model that could transform the competitiveness and efficiency of the South-East Railway in Peru and similar contexts. 2. Literature Review In the upstream market, the railway industry exhibits distinctive characteristics that configure it as a natural monopoly due to the economies of scale, scope, and density required for infrastructure provision. Baumol et al. (1983) and Brock (1983) highlighted the sources of subadditivity in railway operations, identifying specific areas where competition can be introduced without compromising the overall efficiency of the system. ForemanPeck (1987) conducted a historical analysis of 19th-century railway policies, evaluating how natural monopolies influenced the development of transportation and consumer choices. In the downstream market, focused on freight and passenger transportation services, Bitzan (2003) emphasized the importance of introducing competition in railway networks to mitigate monopolistic power and enhance efficiency. However, such intervention requires robust regulatory oversight to ensure equitable access and prevent anticompetitive practices. The high concentration in the relevant market grants the related company to Fetransa a dominant position, making it difficult for new competitors to enter. Peira et al. (2022) conducted a systematic review on the use of railway transportation in tourism, highlighting how monopolies on specific routes can impact tourism development, accessibility, and passenger costs. The main barriers to entry in railway transportation include structural barriers, such as the high sunk costs associated with acquiring traction and rolling stock compared to leasing, and strategic barriers, such as the allocation of uncompetitive schedules that affect freight operations and reduce demand. To address the latter, various authors have proposed first-price auction models as a solution to enhance competitiveness and optimize resources. Several studies, including those by Joskow and Tirole (2005), Burkart (1995), and Ettinger (2008), examined auction models for capacity allocation, demonstrating how these tools can impact efficiency and profitability in the sector. Klemperer (1999) analyzed auctions in regulated sectors, highlighting their ability to allocate resources to agents who value them the most. Yarrow (2003) examined auctions Games 2025,16, 16 4 of 19 with reserve prices in energy networks, while McDaniel (2003) focused on informational asymmetry and correlated values in gas networks, demonstrating how iterative auctions enhance efficiency in capacity allocation. Similarly, van Koten (2012) employed a combination of first-price and second-price auctions, along with toehold auction models, to demonstrate that when a generator owns the interconnector, its bidding aggressiveness increases its profits and auction revenues. However, this comes at the expense of efficiency and the earnings of independent generators. In the railway context, Harrod (2013) and Kuo and Miller-Hooks (2015) developed models that explore capacity allocation through iterative and combinatorial auctions, addressing challenges associated with efficiency and high demand in complex networks. Perennes (2014) emphasized the need for centralized planning, considering the technical limitations of large railway systems. Other studies, such as those by Svedberg et al. (2017), Broman et al. (2022), and Stojadinovi´c et al. (2019), examined the implementation of auctions and simulations to manage capacity and promote competition in railway markets. These works highlight how hybrid auctions and open access can increase social welfare and reduce fees, although they entail challenges such as the need for temporary subsidies. While this study focuses on railway competition and draws on auction models from the energy sector, the underlying issue is broader and closely related to the literature on public procurement auctions. In particular, extensive documentation highlights how the participation of bidders linked to the auctioneer can impact competition in regulated markets and infrastructure concessions (Busu & Busu,2021). The design of auctions in vertically integrated environments presents significant challenges for efficiency and competition, similar to those observed in public procurement processes and public–private partnerships (Gao,2018). The literature has shown that asymmetries in cost structures and access to strategic information can create advantages for certain operators, undermining fairness and fostering a less competitive environment (Hanák et al.,2020). In this regard, incorporating references to public procurement auction models would strengthen the theoretical framework of this study, providing a broader perspective to assess the distortions arising from the current design of railway auctions. The application of auction theory to regulated markets has been widely studied in the literature. Joskow and Tirole (2005) analyze how asymmetries in market power can distort competition in capacity allocation mechanisms, demonstrating that strategic interactions between bidders linked to the auctioneer can reduce efficiency. Similarly, Busu and Busu (2021) highlight the risks of bid rigging in public procurement auctions, emphasizing the role of regulatory oversight in preventing collusion and ensuring fairness. Their findings are particularly relevant to the Peruvian railway case, where the link between Fetransa and PeruRail creates an environment susceptible to strategic bidding behaviors that hinder competition. Gao (2018) further explores sustainable winner determination mechanisms in public–private partnership auctions, proposing methodologies that could be adapted to railway capacity allocation. The insights from these studies reinforce the importance of designing an auction mechanism that neutralizes market distortions and ensures a level playing field for all participants. Finally, Adler et al. (2021), Ali and Eliasson (2022), and Trifunovi´c et al. (2024), explored the use of game theory models and asymmetric auctions to optimize capacity allocation and foster competition in European railway markets. The findings reveal that, while these strategies enhance consumer welfare and encourage the entry of new participants, they also pose challenges related to economic sustainability and transparency in resource management. Games 2025,16, 16 5 of 19 3. Methodology We present a detailed analysis of the timetable auction design and its effects on the competitive dynamics of the passenger transportation market in the South-East Railway in Peru. Following the proposal of Adler et al. (2021), the use of game theory-based models is essential for developing auction mechanisms that foster competition, even in highly concentrated markets such as the railway sector. The analysis has been coherently divided into two parts. The first describes the current design of the timetable auction. The second outlines the main distortions arising from the current auction design. 3.1. Description of the Timetable Auction Design According to the concession contract, the bidder offering the highest track usage fee will gain access to the infrastructure to operate during the auctioned time slot. In return, they must pay the offered fee per wagon-kilometer moved until the expiration of the access contract term. Likewise, if upon the expiration of the aforementioned contract term there is once again a concurrence of schedule requests, the affiliated operator retains the right to initiate an auction under any circumstances, provided that the requested schedule is not already allocated as the result of an ongoing auction process with a valid term. The current characteristics of the auction reflect a first-price model involving related bidders and independent bidders. In this framework, the costs associated with PeruRail’s track usage fee are not fully internalized. Although there is partial internalization of these costs within the economic group, 37.25% of Fetransa’s gross revenues are transferred to the state. However, the non-internalized portion of costs can create significant asymmetries compared to an auction model involving independent companies. In this regard, van Koten (2012) identifies a set of distortions in first-price auctions with related bidders to the auctioneer, which can lead to strategic behavior by the related company, even when the cost internalization by the economic group is only partial.3 3.2. Distortions in the Auction Design Based on the previously described auction mechanism, we have developed an analytical model to identify the distortions generated by the auction system established in the Concession Contract. This model aims to provide a detailed view of how the mechanism’s design affects allocations and economic outcomes for the actors involved. The detailed development of the mathematical model and its fundamental assumptions can be found in Appendix A. In this analysis, we have modeled the payment flows corresponding to four key agents within the auction mechanism in a two-bidder scenario as follows: one related bidder and one unrelated bidder. The agents considered are the auctioneer, Fetransa (A); the related bidder, PeruRail (R); the unrelated bidder, Inka Rail (U); and the state (G). This approach allows for an in-depth analysis of the interaction between the agents and the potential economic and regulatory implications. It is important to highlight that, to properly identify the incentives of both the related company and the auctioneer (R and A), they must be analyzed jointly. In this regard, the strategy of the related company in the auction will be aimed at maximizing the expected profits of the Economic Group (EG), which are calculated as the sum of the expected profits of both actors, namely the related company and the auctioneer. Additionally, to maintain consistency with the latest auction announcement conducted by Fetransa, it is assumed that the auction pertains to a newly identified time slot. The profits for each agent will depend on the auction outcomes. Table 1summarizes the payments corresponding to each scenario in the timetable auction. Games 2025,16, 16 6 of 19 Table 1. Payoff matrix by agent. U R A G R wins 0 ΠR−CRbR(1−θ)CRbR−CAθCRbR U wins ΠU−CUbU0(1−θ)CUbU−CAθCUbU In the table, ΠR and ΠU represent the present value of the profits for the related and unrelated bidders, respectively, when winning the auction. These values are standardized within a range from 0 to 1 and do not include the payment corresponding to the track usage fee. On the other hand, CRbR and CUbU represent the present values of the costs associated with the track usage fee payment, distinguishing between the related bidder (bR) and the unrelated bidder (bU) . Additionally, CA reflects the operating and maintenance costs borne by the auctioneer, while θ represents the percentage of remuneration paid to the state by the auctioneer for track usage. The parameter θ , which represents the degree of cost-sharing within the economic group, plays a crucial role in determining auction outcomes. Its definition and regulation must be clearly communicated to all participants before the auction process begins to ensure transparency and predictability. Potential regulatory approaches: Ex-Ante fixed regulation—OSITRAN sets a predetermined value for θ based on market studies, ensuring a stable bidding environment. Dynamic adjustment mechanism— θ is adjusted over time based on auction outcomes, maintaining a target level of market competitiveness. Self-regulating bidding structure—a θ -indexed rebate system returns excess fees to disadvantaged bidders, preventing systemic exclusion without direct regulatory intervention. We assume a uniform distribution for the private information variables Πi , defined within the interval [0, 1] . This choice allows for the modeling of a market where information is distributed equally among participants, eliminating potential informational asymmetries. Under this assumption, the density remains constant at f(Πi)= 1 throughout the entire interval, ensuring a consistent and transparent analysis of bidding strategies. Based on the presented payoff matrix, the expected profits for the Economic Group (πEG)and the unrelated company (πU)have been calculated as follows: πEG =Pr(R wins)E{[ΠR−θCRbR−CA]|bR>bU}+Pr(U wins)E{[(1−θ)CUbU−CA]|bR<bU}(1) πU=Pr(U wins)E{[ΠU−CUbU]|bR<bU}(2) where Pr(R wins) is the probability that the related bidder wins the auction, and Pr(U wins) is the probability that the unrelated bidder wins the auction. The expression E{·|bR>bU} represents the expected profits of each bidder when the related company wins the auction, that is, when the bid of the related bidder (bR) is greater than the bid of the unrelated bidder (bU) . In this regard, note that when the related company wins the auction, the unrelated company does not obtain any profits. On the other hand, the expressions E{·|bR<bU} represent the expected profits of each bidder when the unrelated company wins the auction, that is, when the bid of the related bidder (bR) is lower than the bid of the unrelated bidder (bU) . In this regard, note that when the unrelated company wins the auction, the related company receives a proportion of the track usage fee. To simplify the results, it is assumed that the number of wagon-kilometers moved is equal to 1 for both bidders. 4 Thus, the expected profits for each bidder are given by the following equations: πEG =Pr(R wins)E{[ΠR−θCRbR−CA]|bR>bU}+Pr(U wins)E{[(1−θ)bU−CA]|bR<bU}(3) Games 2025,16, 16 7 of 19 πU=Pr(U wins)E{[ΠU−bU]|bR<bU}(4) Following the analysis developed by van Koten (2012), the bids submitted by the related and unrelated bidders ( bR and bU ) can be expressed as functions of ΠR and ΠU , respectively. Thus, the related company will win the auction as long as bR>bU(ΠU). Let the inverse bidding function for the related company be U(·)=b−1 U(·) . Replacing this equality in the condition for the related company to win the time slot, it follows that for the related company to win the auction, it must hold that ΠU>U(bU) . Thus, (3) can be rewritten as follows: πEG =ZU(bR) 0[ΠR−θbR]dz +Z1 U(bR)[(1−θ)bU(z)]dz −CA(5) Solving the first integral, substituting ΠU≡U(bR) , and integrating the second integral by parts, the following expression is obtained, where bmax is the maximum bid as follows: πEG =U(bR)[ΠR−θbR]−CA+(1−θ)bmax −bRU(bR)−Zbmax bR U(q)dq(6) To determine the first-order condition in the optimization process for the related bidder, (6) is differentiated with respect to bR , and then ΠR≡R(bR) is substituted, where R(·)= b−1 R(·). The optimality condition for the related company will be given by the following: [R(bR)−bR]U′(bR)=(1−θ)[U(bR)] (7) The optimization process can be similarly applied starting from (4) for the unrelated company. Thus, the optimality condition for the unrelated company will be given by the following: [U(bU)−bR]R′(bU)=R(bU)(8) 4. Results The results obtained show that the current structure of auctions, characterized by the participation of related and unrelated bidders, presents significant distortions. Additionally, we explain the potential anticompetitive effects resulting from the identified distortions. 4.1. Computational Results Consequently, it is possible to analyze (7) and (8) based on the numerical solutions developed by van Koten (2012) for cases where θlies between 0 and 1. According to the analysis presented, the track usage fee offered by the related bidder is consistently higher than that of the unrelated company, as shown in Figure 1. This is explained by distortions identified in the auction model, as described by van Koten (2012), where the related bidder, leveraging cross-subsidies within the economic group, can bid more aggressively without fully internalizing the costs. This advantage significantly reduces the likelihood of success for the unrelated bidder, creating an uneven competitive environment. In response to this disadvantage, the unrelated bidder is incentivized to increase its bid, indirectly raising the expected revenue of the economic group while reducing its own. The strategic complementarity between the bids of both bidders allows the economic group to leverage this dynamic to maximize its profits, reinforcing the incentives for the related bidder to maintain an aggressive bidding stance. This behavior intensifies as the state remuneration rate decreases, as a lower rate reduces the internalization of costs associated with the fees within the economic group. As a result, first-price auctions tend to produce inefficient outcomes if the remuneration rate is Games 2025,16, 16 8 of 19 below 100%, posing a significant challenge in terms of fairness and economic efficiency in the auction model design. Figure 1. Bidding functions for bidders in first-price auctions. The source of the identified distortions is linked to the ability of the related company to increase the auctioneer’s expected profits through higher bids. In the scenario where R wins, A’s revenue directly depends on bR . Conversely, in the scenario where U wins, A’s revenue indirectly depends on bR through its strategic complement, bU . Therefore, if the auctioneer’s revenue were not influenced by the bid of its related company, the distortions observed in the timetable auction for the South-East Railway would disappear. On the other hand, the characteristics of ΠR and ΠU are non-trivial in the process of selecting the optimal bid by the companies. These parameters may vary depending on the specific characteristics of each bidder, as any differences in costs or expected revenues— arising from greater market power during the time slot, economies of scale, or more favorable access to specific assets—impact the profits each bidder associates with the auctioned time slot. To eliminate this distortion, a strategy is proposed to decouple the related operator’s bids from the auctioneer’s revenues. This proposal involves maintaining the track usage fee as the primary competitive factor, with the winner being the bidder offering the highest fee, while also establishing a minimum “base fee”. Under this measure, Fetransa would charge the winning fee per wagon-kilometer and transfer to the state the difference between the winning fee and the base fee for each wagon-kilometer traveled during the auctioned time slot. Thus, by applying this condition, the auctioneer’s profits would become independent of the bids from related operators, eliminating incentives to inflate bids and promoting fairer competition as follows: j wins (1−θ)Cjbmin −CAf or j =R,U(9) The auctioneer’s revenue does not depend on bR in any scenario, as bmin is an exogenous variable. Games 2025,16, 16 15 of 19 each step of the process, including the assumptions made, the equations used, and the implementation procedure. Appendix A.1. Fundamental Assumptions To simplify the analysis and focus on the key distortions, the following assumptions were made: • Distribution of private information: Private signals were assumed to follow a uniform distribution within the interval [0, 1] . This choice reflects the absence of systematic differences in information between market participants. • Cost structure: Fixed maintenance costs and variable costs associated with the distance traveled (measured in wagon-kilometers) were considered. Tariffs were based on official rates specified in the current concession contract. • Operators’ strategic behavior: Operators were modeled as rational agents competing imperfectly. The related operator partially internalizes infrastructure revenues, which allows them to adopt more aggressive bidding strategies. Appendix A.2. Mathematical Model Operators’ strategic behavior was modeled using the following equations: • The profit function of the related operator is expressed as follows: πEG =E[ΠR−θCRbR−CA]+E[(1−θ)bU−CA](A1) • The profit function of the independent operator is expressed as follows: πU=E[ΠU−CUbU](A2) These equations allow the analysis of how varying the level of state participation (θ) influences bidding strategies. The bidding strategies of related and unrelated operators stem from their respective profit maximization conditions, shaped by differences in cost structures and revenue transfers. PeruRail partially internalizes track access costs, recovering a portion through Fetransa, which gives it greater flexibility in its bids. In contrast, Inca Rail fully externalizes these costs, meaning that any increase in its bid directly reduces its net revenue, placing it at a competitive disadvantage in the auction process. To explicitly capture these differences, the optimal bidding strategy for each agent can be rewritten as follows: • For the related operator (PeruRail) bR=arg max b[ΠR−θCRb−CA](A3) • For the unrelated operator (Inca Rail) bU=arg max b[ΠU−CUb](A4) If the model were perfectly competitive, these constraints would generate different bid functions, reflecting the cost asymmetry. However, in the current structure, the internal revenue dynamics partially equalize the functions, explaining why the observed bidding strategies appear more similar than expected. Games 2025,16, 16 16 of 19 Appendix A.3. Computational Implementation The simulation was implemented using Matlab, a widely used tool for numerical analysis. The Runge–Kutta method of fourth order was selected due to its reliability and accuracy when solving differential equations. Simulation Steps: 1. Parameter definition: The following initial parameters were set: • Private information uniformly distributed within [0, 1]. • Fixed and variable costs based on official concession data. • State participation level (θ)varying between 0 and 1. 2. Generation of private signals: We generated 1000 random observations from a uniform distribution, representing each operator’s subjective valuation of the auctioned time slot. 3. Calculation of bids: The bids were calculated using the following functions: • For the related operator: bR(Π)=bmin +(bmax −bmin)Π. • For the independent operator: bU(Π)=bmin +(bmax −bmin)(1−Π). 4. Solving the Equations: The bidding dynamics were modeled using the following Runge–Kutta scheme: yn+1=yn+h 6(k1+2k2+2k3+k4)(A5) where k1=f(tn,yn) , k2=ftn+h 2,yn+h 2k1 , k3=ftn+h 2,yn+h 2k2 and k4= f(tn+h,yn+hk3). 5. Iteration and Data Collection: The calculations were repeated for all 1000 simulated private signals. Results were recorded for the operators’ bids, profits, and comparative outcomes under different regulatory scenarios. Appendix A.4. Validation and Discussion To validate the reliability of the model, the simulations were repeated with different sample sizes. In all cases, the results remained stable. Furthermore, the simulation outcomes were compared against predictions derived from the analytical model of linear bidding functions, yielding a deviation of less than 2%. Lastly, the model was used to explore how varying the level of state participation (θ) affects market competition. The analysis revealed that the proposed reform, which involves transferring auction surpluses to the state, significantly improves competitive conditions by eliminating incentives for the infrastructure manager to favor its related operator. Appendix B. Numerical Example Figure A1 presents the simulation of expected profits for the operator related to the railway infrastructure manager (panel a) and the independent operator (panel b) under different levels of state participation (θ=0.20, 0.50, 0.80) . In both cases, expected profits decrease linearly as private information (Πi) increases, indicating that a higher valuation of the auctioned time slot by the operators translates into lower expected profits. Additionally, the differences between the curves for different values of θ are minimal, suggesting that the transfer of revenues to the state does not have a significant impact on the operators’ profit structure. Games 2025,16, 16 17 of 19 (a) (b) Figure A1. Simulation of expected profits with different levels of state participation: (a) Profits of the related operator. (b) Profits of the independent operator. To further examine the impact of θ on market competitiveness, we conduct a sensitivity analysis by varying θ between 0 and 1 and evaluating its effect on the probability of success for the unrelated bidder. 1. Simulation setup: •θ= 0.0: No internal cost-sharing; PeruRail bids as if it were an independent operator. •θ= 0.5: Partial internalization, simulating a moderate advantage for the related operator. •θ=1.0: Full internalization, representing a completely distorted auction. 2. Results: • As θ increases, the likelihood of PeruRail winning the auction rises non-linearly, confirming that cost-sharing incentivizes more aggressive bids. • For θ> 0.5, the probability of Inca Rail securing a time slot drops below 30%, demonstrating a strong anticompetitive effect. • A θ -adjusted subsidy or correction factor could mitigate these distortions without imposing strict regulatory interventions. These findings suggest that θ should be regulated within a competitive range (0.2–0.5) to ensure that new entrants have a viable pathway to winning auctions. This result implies that the modification of the payment scheme does not substantially alter the competitive dynamics of the railway market. Both the related and independent operators experience a similar reduction in profits as private information increases, indicating that state participation in the tariff does not create a significant competitive advantage or disadvantage. However, the persistence of this downward trend suggests that operators’ strategic incentives may still favor the dominant company, making it necessary to assess whether additional regulatory reforms could strengthen competition and improve fairness in the allocation of time slots. Games 2025,16, 16 18 of 19 Notes 1 The Southern Railway focuses on freight transportation along an extensive route that connects major cities and ports in southern Peru, while the South-East Railway is primarily dedicated to passenger transportation to Machu Picchu, featuring a shorter route and distinct technical characteristics. 2 Currently, two companies operate in the railway transportation market: Inca Rail S.A.C., the result of the 2013 merger between Inca Rail and Andean Railways Corp. S.A., and PeruRail. 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