1 M 2
a x y b
Figure 3. Hotelling’s “Main Street”
Each customer has a decision-making for the products in the market. Most markets operate complicated network of dispersed buyers and sellers. In addition, the market actives are executed at spread points in space. Each firm finds not only the advantage of their benefits location but also further away other competitors. Not all customers like the location that the firm has been set up. The customers attempt to match the firm’s location as possible as they can. The distance between firm’s location and customers’ location means transportation cost. The distance is too far, the customer pays more transportation costs.
Furthermore, D’Aspremont, Gabszewicz, and Thisse (1979) revised the Hotelling’s example for which there exists a price equilibrium solution for any pair of locations.
Accordingly, competition exist among the space, thus conduct an analysis of DRTS problem as a game of strategy. Many of literatures with respect to the economic relevance of location games are discussed. I mainly discuss two issues that are product differentiation and price discrimination.
3.3 Game Theory
Traditional economic models start from the assumption that the decision makers being studied are rationally seeking some aim. von Neumann and Morgenstern (1944) originally published that theory of games was applied in social science. In particular, they mentioned that economics theory could combine rational behavior with theory of games. Furthermore, numerous savants (Deming, 1945; Hawkins, 1945; Kattsoff, 1945; Wald, 1947) deem that the advanced development of economics was spread by John von Neumann and Oscar Morgenstern. Due to exist of economic theretofore no suitable treatment of the question of rational behavior they firstly have developed theory of game in their strategic aspect as elementary social phenomena that included political science, sociology, or even military strategy. Afterward, Hurwicz (1945) and Marschak (1946) introduced an easier approach to the application of this book. Wald (1945a) established a modern theory of statistical
estimation that is nearly connected to the theory of the zero-sum two-person game and Wald (1945b) extended this theory of the zero-sum two-person games to definite continuous infinite studies. Loomis (1946) investigated the rudimentary proof of this theorem. Moreover, Kaplansky (1945) drew on the completely-mixed strategies in the zero-sum two-person game.
Marshall (1890) established that demand and supply interplay to decide the equilibrium price and the quality that are exchanged in the market; furthermore, the price is increased then the quality is decreased. Likewise, in road-transport pricing, Small and Verhoef (2007) discussed that first-best pricing in the context of congested highway traffic. Verhoef, Bliemer, Steg, and Wee (2008) interpreted that the number of road users increase then speed will decrease and average user cost will rise because travel times rise. However, actually, in rural area or remote districts in Taiwan, the public bus operations are burdened with heavily cost beacuse the frequency of used public bus transport routes is seldom.
Many of bus routes were abolished. Tranditionally, the demand side is recession then the supply side is suspended.
In this paper, I attempt to use game theory to solve these problems. Kreps (1990) and Romp (1997) explained that a game theory is applied to a modarn economics. A game theory is a different sight and mathmatical method to solve the existing problems. Two branches of game theory are separated. These are co-operative and non-cooperative game theory. I use the method of non-cooperative game theory because it is inherently individualistic and in a game are unable to enter into blinding and enforceable agreements with one another. In addition, the another characteristic of game theory is mutually interdependent.For instance, in the existing game, one individual is determined by the actions of other players; furthermore, he or she adopts strategic decision-making and purpuse to anticipate the effect their own actions are going to have on the behaviour of others. Chosen the decision each individual then decides his or her best response in order to accomplish the most pleasing outcome. Likewise, our research model is interpreted by the game theory. The government appropriates their owned budgets for Fu-Kang bus transport firms. How do the firms pursue to acquire their budgets? One firm have to consider what strategic with respect to routing types must take is benefit itself. The other firm needs to consider how to enter this Fu-Kang bus transport market.
3.4 Duopoly Model
Due to narrow a paratransit duopoly market with respect to DRTS in Taiwan, the private firm hesitates to launch this market and the public firm trend to use the “Purchase of Service Contracting, POSC” method to operate the Fu-Kang bus transport service. The government must consider how to incentive the paratransit markets in order to promote the social welfare with respect to disabled persons’ Fu-Kang bus transport, and furthermore continue in operation for a long time. Therefore, I suppose both firms were existed, and then consider the oligopoly theory in the beginning. Cournot (1838), the earliest ancestor, has been established his first model of oligopoly theory. The duopoly model is the simplest in oligopoly. If the duopoly model could be solved then it could be derivate triple, quadruple or N-tuple buyers’ models.
Friedman (1983) indicated that the Hotelling model of spatial duopoly is the simple model which can explain the relationship between two firms, in particular, public and private firm. Each chooses its own-setting price and provides to all riders at that price and at its own-setting location.
3.5 Optimal Control
Because precisely solve the financial resource allocation, I consider that the government will allot their budgets to the public transport firm and the private firm;
furthermore, constricted budgets are optimized by government. Therefore, I attempt to use the optimal control theory to solve our setting problems. Most resources are restricted, in particular, nature resources. Ramsey (1928) first tackled the problem of how much of its income should a nation save. He assumed that the income is taken to be a function of the total of labor and capital. Hotelling (1931) tackled the problem of the economics of exhaustible assets and used the method of the calculus of variations. Bellman (1957) developed the modern formalism in the calculus of variations and Pontryagin, Boltyanskii, Gamkrelidze, and Mishchenko (1962) provided the Pontryagin Maximum Principle.
Afterward, the method of optimization over time was already used in an economic context.
Arrow and Kurz (1970) manipulated the discount rate of optimal resource allocation;
furthermore, due to decision-making in investment between a government and a private sector, they use this method to calculate the outcome of the publicly optimal policy. The policy with respect to all variables is involved in public investment, private investment, and consumption that would be adopted by an ideally altruistic government with
unconstraint powers. Kirk (1970) introduced the optimal control theory is to decide the control signals that will lead a process to gratify the physical restraints and at the same time. Seierstad and Sydsaeter (1987) introduced the theory of deterministic optimal control and argued the theoretical results for sufficient conditions. Chiang (1992) introduced the new development that implies the optimal control formulation of a dynamic optimization problem focuses upon one or more control variables. Leonard and Van Long (1992) approached the modern technique between the method of classical programming and those of optimal control theory. Sethi and Thompson (2006) argue an optimal control theory to apply it to a wide change of different situations producing in management science. Weber (2011) provide a continuous-time systems and methods for solving dynamic optimization problems at three different level such as personal decision-making, game theory and mechanism design.
3.6 Differential Games
Roos (1925 ; 1927) introduced mathematical and dynamic theory to economics. Issacs (1965) developed the formal theory of differential games; furthermore, his work enhanced by a grand range of practical military and other applications with solutions. After that, Friedman (1971), Basar and Olsder (1982), Mehlmann (1988) provided intuitive interpretations in regard to differential games. Feichtinger and Jorgensen (1983), Clemhout and Wan (1989) provided researches of applications of differential games in economics and management sciences. The total cost function of our model is based on Kamien and Schwartz (1991).