Simulation of competition in NGNs with a Game Theory model

Like in a real competitive market situation, Next Generation Networks (NGN) competitors need to adapt their strategy to face/react the strategies from other players. To better understand the effects of interaction between different players, we build a Game Theory model, in which the profit of each operator will be dependent not only on their actions, but also on the actions of the other operators in the market. This paper analyzes the impact of the price (retail and wholesale) variations on several output results: players’ profit, consumer surplus, welfare, costs, service adoption, etc. We assume that two competing FTTH networks (incumbent operator and new entrant) are deployed in two different areas. We also propose an adoption model use in a way that reflects the competition between players and that the variation of the services prices of one player has an influence on the market share of all players. Finally, model use the Nash equilibrium to find the best strategies.


INTRODUCTION
The main objective of a game-theory model is providing a mathematical description of a social situation in which two or more players interact, and every player can choose from different strategies. (J. P. Pereira & Ferreira, 2012;Yongkang, Xiuming, & Yong, 2005) define game theory as a collection of mathematical models formulated to study situations of conflict and cooperation, and concerned with finding the best actions for individual decision makers. (Machado & Tekinay, 2008) argue that game theory is a theory of decision making under conditions of uncertainty and interdependence. The players compete for some good or reward, and often in business cases, the customer will be the aim of the competition (J. P. R. Pereira, 2013a;Verbrugge, Casier, Ooteghem, & Lannoo, 2009). The object of study in game theory is the game, where there are at least two players, and each player can choose amongst different actions (often referred to as strategies). The strategies chosen by each player determine the outcome of the game -the collection of numerical payoffs (one to each player). So, the game has three main key parts (Easley & Kleinberg, 2010): a) a set of participants; b) each player has a set of options for how to behave; we will refer to these as the player's possible strategies; and c) for each choice of strategies, each player receives a payoff that can depend on the strategies selected by everyone (in our model, the payoff to each player is the profit each provider gets). After the calculation of the several payoffs, game theoretic concepts can be used for retrieving the most likely (set of) interactions between the players (Verbrugge et al., 2009). There are several different equilibrium-definitions of which probably the Nash equilibrium is the most commonly known -A broad class of games is characterized by the Nash equilibrium solution. In 1950, John Nash demonstrated that finite games always have a Nash equilibrium, also called a strategic equilibrium (Yongkang et al., 2005). A Nash equilibrium is a list of strategies, one for each player, which has the property that no player can unilaterally change his strategy and get a better payoff -each player's strategy is an optimal response to the other players' strategies. Even when there are no dominant strategies, it should be expected that players use strategies that are the best responses to each other. This is the central concept of noncooperative game theory and has been a focal point of analysis since then. For example, if player 1 chooses strategy S1 and player 2 chooses S2, the pair of strategies (S1 and S2) is a Nash equilibrium if S1 is the best response to S2, and S2 is the best response to S1. So, if the players choose strategies that are best responses to each other, then no player has an incentive to turn to an alternative strategy, and the system is in a kind of equilibrium, with no force pushing it toward a different outcome (Easley & Kleinberg, 2010). One of the main goals of regulated access is to prevent the incumbent from abusing a dominant market position (J. P. R. Pereira, 2013b). It is necessary to make sure that alternative operators can compete effectively. It is fundamental that incumbent operators give access to the civil works infrastructure, including its ducts, and to give wholesale broadband access (bitstream) to the local loop (be it based on copper, new fiber, etc.). However, at the same time, alternative operators should be able to compete on the basis of the wholesale broadband input while they progressively roll out their own NGAN infrastructure. In some areas, especially with higher density, alternative operators have rolled out their own infrastructure and broadband competition has developed. This would result in more innovation and better prices to consumers (J. P. Pereira & Ferreira, 2011). Many European incumbents and some alternative operators are starting to plan and in some cases deploy large-scale fiber investments, which has resulted in important changes for fixed-line markets (Amendola & Pupillo, 2007). The risk of alternative operators will take longer to deploy their own infrastructure and will give to incumbents the possibility to create new monopolies at the access level. The technologies used and the pace of development vary from country to country according to existing networks and local factors. Based on the different underlying cost conditions of entry and presence of alternative platforms, it may be more appropriate to geographically differentiate the access regulatory regime.

MODEL OVERVIEW
This part of the work focuses the development of a tool that simulates the impact of retail and wholesale price variation on provider's profit, welfare, consumer surplus, costs, market served, network size, etc. The programming language used to implement the model was C language. The application runs on several platforms and was developed to use multiprocessing capacity. The choice of C language was because the tool needs to compute higher quantity of data very quickly, and this programming language has the characteristics required (see Figure 1. ). In the proposed model, "Retail Prices" represents the set of retail prices charged by providers for each service to consumers in a given region/area. We assume that retail providers cannot price discriminate in the retail market. "Wholesale Prices" represents the prices that one provider charges to other provider to allow the later to use the infrastructure to reach consumers. We assume that wholesale price can be different in each area. Also, we assume that when a provider buys infrastructure access in the wholesale market, it cannot resell to another provider. The shared infrastructure consists of: conduit and collocation facilities; cable leasing (dark fiber requires active equipment to illuminate the fiber -for example repeaters); and bit stream. For example, one or several wholesaler providers can sell Layer 0 access (conduit and collocation facilities) and/or Layer 1 access (cable leasing) or Layer 2 access (bitstream -network layer unbundling -UNE loop) only to retail providers and not directly to consumers. UNE loop is defined as the local loop network element that is a transmission facility between the central office and the point of demarcation at an end-user's premises.

INPUT PARAMETER ASSUMPTIONS
As we can see in Figure 1. , our tool has several input parameters (one of them came from the technoeconomic model), computes several results and finds the strategies that are Nash equilibrium. The results are represented in tables and graphics. In this section, we describe the inputs: fixed and marginal costs and retail/wholesale variation values.

Fixed and marginal costs
In our model, we assume that providers incur in fixed costs to build network infrastructure to provide access to a region and in marginal costs to connect each consumer separately. The fixed and marginal costs are calculated in the techno-economic tool. The fixed costs are detailed by provider, region and infrastructure layer. So, we assume that the fixed costs of each provider can be different in different regions -for example, if a provider has part of the infrastructure deployed in a region, and in the other is required all the infrastructure, the costs are different.
For marginal costs, we assume that each provider has different costs for deployment in each infrastructure layer. In each region, the marginal cost could be different for each provider depending of the total number of subscribers -scale economies. This means that the marginal cost can decrease when a specific provider buys higher quantities of equipment, cable, etc. (see TABLE III. ).

Pricing strategy
Both suppliers and consumers aim at maximizing the benefit or surplus they receive (ITU-T, 2008). The suppliers aim at maximizing the profit, which is the difference between revenue and cost. The consumers aim at maximizing the consumer surplus, which is the difference between consumer value (also known as utility or maximum willingness to pay) and price. As discussed previously, some of the factors that are important in the design of pricing scheme include technology risks, availability of resources, competition, supplier and consumer behavior, price discrimination and regulation.

Definition of the variation in retail prices
The definition of retail prices and trend was explained previously. For the game-theoretic tool, we need to define the variation in retail prices which we want to simulate. So, for each service, we define the price values we wish to simulate -the tool gives the possibility to simulate n values.
In the example presented in the next table, the tool simulates the results obtained when the value of service 1 is P r S1,Value1, P r S1,Value2, P r S1,Value3, and P r S1,Value4 for all players (providers).

Definition of the variation in wholesale prices
For wholesale prices, we define the variation in wholesale price layers that we want to simulate. Similarly, for retail price, for each layer we define the price values we wish to simulatethe tool gives the possibility to simulate n values. For infrastructure, the definition of which layer or combination of layers we would like to simulate is also required (next table). For example, if a provider wants to use (lease) the conduit from another provider, we choose option 0.

SIMULATION MODEL (MODELING COMPETITION)
The simulation model can be sub-divided into seven main parts: retail and wholesale modeling, calculate total costs (build and lease infrastructure), calculate revenues (retail and wholesale market), calculate profit, calculate consumer surplus, and calculate welfare. The next sections describe all these parts.

Retail modeling
In our model, we assume that consumers choose the service from the provider with the lowest price. However, consumers only buy a service if the price is less than their willingness to pay. This means that if there are two or more providers, consumers choose the service from the provider with the lowest price. Moreover, if several providers have the same price, we use the provider ranking. We also assume that consumers have a different willingness to pay for each service (e.g., voice, video and data). First, the tool identify the retail provider for each service in the regions in study using information from providers, retail prices, consumer willingness to pay, and provider rank. Next, as we know which provider will provide each service, we can compute the total subscribers per region, service, and provider (market segment). The structure used is presented in next figure.

Wholesale modeling
In wholesale modeling, we determine the infrastructure chosen by each provider to reach consumers. To model the wholesale market, we assume that if a provider does not have infrastructure, it uses the infrastructure (or part of the infrastructure, such as a conduit cable) of another provider if the price charged to access it is lower than the cost to build an infrastructure. To achieve that goal, the algorithm uses information about wholesale prices, fixed costs, and marginal costs to identify the best solution (lease or build infrastructure) for each region and service. The algorithm also utilizes the information produced in retail modeling to determine which providers offer services to consumers in all the regions. The fixed and marginal costs are calculated in the techno-economic tool.

Calculate total costs (build and lease infrastructures)
The calculation of the total costs incurred by each provider is divided in two main parts: wholesale costs and build-out costs. As sees in next figure, in order to compute the total wholesale costs, we use the wholesale infrastructure design computed previously and the wholesale prices charged by the infrastructure owners (i.e., payments that a specific provider gives to the infrastructure owner to buy wholesale access in order to reach consumers). We assume that the network owner charges the same wholesale price to all providers.
To calculate the build-out costs, the algorithm uses the fixed and marginal costs parameters with region parameters to compute the total costs required to deploy an entire or part of an infrastructure. The total number of consumers per region and per provider is also used to add the effect of economies of scale. When a provider buys a large quantity of equipment, the probability of attaining better prices is higher.

Calculate revenues (retail and wholesale market)
To compute the total revenues per provider, we first calculate the revenues from the retail market. These are primarily based on the retail prices charged by providers and the total number of consumers per provider and services computed in the retail modeling. Revenues from the retail market are equal to the product of the retail price of each service and the total customers of the service. Next, we calculate the revenues from the wholesale market. The wholesale infrastructure provides information about the number of access leased. The revenues of a provider are the sum of all payments received from other providers that use its infrastructure to reach consumers. Finally, the total revenues of a given provider are the sum of the revenues from the retail and the wholesale market.

Calculate profit
After computing the total costs and revenues in the previous algorithms, the formula we use to calculate total profit is the difference between total revenues and total profit. The total profit is also used in the identification of the Nash equilibrium strategies.

Calculate consumer surplus
Consumer surplus (CS) is the difference between the total amount that consumers are willing and able to pay for each service and the total amount that they actually pay (i.e., the retail price). So, the CS of a specific market is the sum of the individual consumer surpluses of all those customers in the market who actually bought the service at the going retail price "Pr" (ACMA, 2009). To compute CS, we need information about consumer willingness to pay and retail prices for each service (next figure).

Calculate total welfare
Total welfare is computed on base of the formula: welfare = consumer surplus + total profit. Like the previously calculations, the CS and the profit are computed in the algorithms presented above. The block diagram is presented in the next figure. (2) analysis the impact of retail and wholesale price variations on profit, consumer surplus, welfare, and retail/wholesale market (different wholesale prices in each region). For the game-theoretic evaluation, the model calculates the NPV and operator's profit for both operators' pricing strategies. Operators' NPVs are used as payoffs for the players in the first and second game, and operators' profits for the third game. From the several assumptions, we posit: (a) the price that players charge for their services (retail and wholesale) will be varied; (b) the retail price setting will influence the market share of both players (resulting in a higher or lower market share); and (c) consumers only buy a retail service if the price is less than their willingness to pay. As stated above, we assume that when one player increases/decreases the retail price, the market share of all players will be affected. For example, if one player offers cheaper services, it will be able to capture a higher market share. If a price decreases to nearly zero, everyone will use the service, and the market share of this operator will be close to 100% (total market). On the other hand, if an operator charges a higher price for a service, no one will subscribe to the service from this player, and its market share will decrease to 0%. The impact of changing prices in the market share (i.e., the estimate of the impact of the price on the service adoption) is modeled using the Boltzmann equation: (2) in which the variables are defined as follows: • X0: is the mean base (or center) • dx: is the width • Vi: is the initial value of y • Vf: is the final value of y The next graph shows the s-curves of three functions. The slope (parameter b) of the market penetration should give an advantage or disadvantage to one of the players (Katsianis, Gyürke, Konkoly, Varoutas, & Sphicopoulos, 2007).

Main assumptions
We assume that the willingness to pay for each retail service is different in both regions. In the urban area (region 1) the maximum amount subscribers would be willing to pay for service 1 is 26 euros and 65 euros for service 2. In the rural area we assume a willingness value of 22 euros for service 1 and 55 euros for service 2 (see TABLE VII. ). For the wholesale infrastructure we assume a duct availability of player 1 100% in the urban area and 90% in the rural area. We also assume that operator 2 (new entrant) leases 100% of the ducts available in the urban area and 100% of the ducts available (operator 1 has only 90% and the remaining 10% are deployed by operator 2) in the rural area from operator 1 (incumbent operator). In the other hand, player 1 leases the 10% remaining (in region 2) from operator 2. The wholesale prices assumptions are: 9.1€ (month / km / cm2) for urban area and 7.5€ (month / km / cm2) for the rural area. The wholesale infrastructure assumptions and described in The next sections present the three games results and analyses. In the first game, retail prices vary between tariff multiplier 0.7 and 1.3 (in increments of 0.1). For the second game, retail prices vary between 0.8 and 1.2, and wholesale prices between 0.5 and 1.5.

RESULTS
Based on the numerous input parameters described, our tool computes several results, including profit, consumer surplus, welfare, market served, network size, costs, and revenues, and finds the strategies that are Nash equilibriums. The results are saved in text files (see Figure 11. ). The final results with all strategies computed are saved in a file named "results_f.txt". In addition, the Nash equilibriums are saved in a file named "equilib.txt". Figure 11. Structure of the results produced (output from tool) Figure 11. show the structure of the results that correspond to a scenario of two providers, two retail services, two infrastructure layers, and two regions. Each line is a strategy. We consider a strategy to be a set of retail and wholesale prices. For each combination of prices, the tool calculates profit, CS, welfare, market served, network size, and total costs. In addition to the results presented in the tables, the tool creates several types of graphs. For that, we incorporated a Gnuplot program in our C code. Gnuplot is a portable command-line-driven graphing utility for Linux, OS/2, MS Windows, OSX, VMS, and many other platforms. The source code is copyrighted but freely distributed. Next figures show two examples of the graphs produced. The graph shows the impact on profit of both providers and variation in wholesale and retail prices. This representation gives users a tool to gain a better perspective of the results.

Game 1: Impact of retail prices variation on NPV
In this game we assume that wholesale prices are fixed and that operators choose retail prices to maximize their profit. The impact of varying retail prices on market shares is estimated using the Boltzmann equation (described above). The main goal of this analysis is to determine the optimal retail price strategy for both players. The retail prices vary between -30% and 30%, with increasing steps of 10% (next table). The combination of the two retail prices and seven multiplier factors leads to 49 possible strategies for each player (49x49 matrix) in each region (2,401 total strategies). The next table presents the structure of the combinations and calculated NPV. The results (payoff matrix) of this game are presented in TABLE XI. for region 1 (urban area). TABLE XIII. shows the sum of the payoffs of each player in both regions. The following tables present the NPV for both players for each possible combination of strategies (one strategy for each player); Nash equilibrium strategies are also identified. The first two rows represents the prices multiplier factor of player 2 (for services 1 and 2) and the first two columns show the variation (multiplier factors) of player 1. Each cell contains two values: The left value corresponds to the NPV of player 1, and the value on right side corresponds to the NPV of player 2. For example, the first value calculated (16,512,089 €) corresponds to the NPV of player 1 when the strategy of player 1 is to decrease the price of service 1 and service 2 by about 30% (multiplier factor 0.7), and the strategy of player 2 is also to decrease the price of service 1 and service 2 by about 30%.  We have analyzed the different strategies of the two players to find the game's Nash equilibria. (Our model includes a function for searching NE in the games). The NE strategies are formatted with a black background. From the analysis of these results, we find two NE strategies that are detailed in the next table. The next graph shows the impact of service 2 price variation on the NPV of both operators (for the urban area).

Figure 12. NPV variation: Provider 1 and 2/Region 1/Retail service 2
In the previous tables and graphs we presented the results for region 1 and region 2 when isolated. However, operators are also interested in the results for both regions. So, the next table analyzes the sum of the payoffs of each player in both regions.

Impact of retail prices variation (0.7 to 1.3) on NPV (€)-Region 1 and 2
Price S1 ,,, Price S2 ,,, , Operator 2, in face of the imposed wholesale prices, decreases the price of service 1 and service 2 by30% and 20%, respectively. A new entrant has to pay the wholesale to the incumbent, but if increase the retail prices their market share will decrease (see model above). The next figure shows the impact of service 1 variation on NPV of both operators. We can verify that the variation of the retail price of service 1 does not have the same impact on NPV that it has on service 2. From the analysis of the next figure we can conclude that the variation of retail prices of service 2 has a greater influence in the NPV than the variation of service 1 price. Service 2 price variation can drop the NPV of operator 1 to negative. On the other hand, operator 2 can turn the NPV positive when the tariff of service 2 increases.

Game 2: Impact of retail and wholesale prices variation on NPV
In this game we assume that wholesale prices are not pre-imposed and we investigate what is the reaction of operators when they can also choose different wholesale prices in different regions (see next table). In game 2 we assume that has the same variation for both regions. Retail prices vary between 0.8 (-20%) and 1.2 (20%) (in increments of 0.1). For wholesale price we assume a variation between 0.5 and 1.5 (in increments of 0.25). In this context, the combination of the three prices and variation multipliers leads to 625(5^4) possible strategies for each player (625x625 matrix) in each region (390625 strategies in both regions). As the matrix is to bigger, for this game we decide to present the NE strategies (players profit is used as payoff) and the graphs that show the impact of variation in the several results (presented in the previous table). The analysis of the results finds five NEs strategies. As player 2 do not operates in the wholesale market of region 1, the variation of this price is not significant.  The comparison of the two games above shows that when the regulator defines wholesale prices, operators increase retail prices to maximize profit. However, when wholesale prices are not regulated, operators maximize profit by decreasing retail prices and increasing wholesale prices. However, without regulation, the higher wholesale prices will limit the entrance of new competitors.
The main results of this game are summarized in the next figures. In the first two graphs we can see the impact of retail prices (left) and wholesale prices (right) on players profit. We can verify that both prices can turn profit positive/negative. Consumer surplus decreases with the increase of prices (left graph). As also expected and modeled above the impact of retail prices variation has higher influence in the market share of competitors (see next figure).

CONCLUSION
Sensitivity analysis shows the impact that changes in a certain parameter will have on the model's outcome. As the interaction between all the players is important, we put the competition component in the business case. With game theory, we want to understand the effects of the interaction between the different players defined in our business case. In the proposed games, the profit (outcome) of each operator (player) will be dependent not only on their actions, but also on the actions of the other operators in the market. The impact of the price (retail and wholesale) variations on several output results: players' profit, consumer surplus, welfare, costs, service adoption, and so on. For that, two price-setting games are played. Players' profits and NPV are used as the payoff for the players in the games analyzed.
In our model we also use the Nash equilibrium to find equilibrium. Proposed tools include a module to search the Nash equilibrium in the game. One strategy is a Nash equilibrium when both competitors play their best strategy related to the other strategies selected (players know each other's strategy in advance).