Game theory

Game theory is the important tool when people face the competition. It can help people to analyze the situation to decide the strategy which he/ she should take when he/ she need to make a decision to compete with his/her competitors. Situations modeled as games typically involve several parties having different interests, who need to decide how to behave. The level of benefit that each party gains depends not only on its own actions, but also on the choices of the other parties. The mathematical formulation of all games is similar, either explicitly or implicitly, to an optimization problem that includes more than one objective, and the decision variables are shared by the different objectives. Defining a game requires identification of the players, their alternative strategies and their objectives. Formulating a problem as a game is worthwhile if the solution, such as Nash equilibrium or Stackelberg equilibrium, leads to new insights on the analyzed problem.

3.1 Development of Game theory

The first known discussion of game theory occurred by James Waldegrave in 1713. Cournot (1838) publicated a general game theoretic analysis, considers a duopoly and presents a solution that is a restricted version of Nash equilibrium. But the major development of the theory began in the 1920s with the work of the mathematician Emile Borel and the polymath John von Neumann (1928). A decisive event in the development of the theory was the book public by Von Neumann and Morgenstern (1944), which established the foundations of the field. In the early 1950s, Nash’s (1950) Ph.D. thesis, 28 pages in length, introduces the equilibrium notion now known as “Nash equilibrium” as the following equation

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  



* 1

*

*: , ,

,

,   _  _

i x_i S_i x_i x_i f_i x_i x f_i x_i x

Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by

changing his or her strategy while the other players keep their unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

Game theory is the most popularity tool when people tend to make the decision in competition. The next section will introduce the application of the game theory in the transportation field.

3.2 The application of game theory in transport field

This part introduces the application of game theory in the transport application.

Colony(1970) formulates a route choice problem as a zero-sum game. One of the players is a driver that chooses whether to use an arterial road, the other is an imaginary entity which chooses the level of service on the road, and tries to disturb the driver’s journey. Rosenthal (1973) and James(1998) formulates a general game between n- individuals who choose the road segment out of a given set, where the cost of each road segment increases if more individuals choose it. The former formulated a programming problem, which solution is always a pure-strategies Nash equilibrium of the game, and shown that a solution always exists. Fisk(1984) mentions that the user equilibrium principle, introduced by Wardrop(1952), is in fact a game since it meets the conditions of Nash equilibrium. Van Vugt et al. (1995) present a two-player

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strategic form game, where each player chooses either car or public transport. The conclusion is that the selfish way travelers make their choices is bad for everyone, that is , the prisoner's dilemma in game theory.

Table 1 The game theory application in traffic field

Source Year Players Strategies/decision variable

Wardrop 1952 Drivers Route choice

Colony 1970 Driver Route choice: arterial road or

motorway Level of service

Rosenthal 1973 Drivers Route choice

Fisk 1984 Authority

Drivers

Traffic control settings Route choice

Van Vugt et al. 1995 Travellers Car/public transport

James 1998 Drivers Travel/ not travel

Lucking et al. 2004 Drivers Routing game

Sun 2007 Drivers Routing game/Mode choice

Gairing 2008 Drivers Routing game

In the recent paper, Sun(2007) construct a urban transit non-cooperative static game, and assume the prefect information, to find the generalized Nash equilibrium, which is to describe both the competitions among different transit operators and the interactive influences among passengers. Lucking et al. (2008) use the concept of Nash equilibrium to construct a self routing non-cooperative network model. In the hybrid model which consist of KP model and Worst-cast model, each of n users is using a mixed strategy to ship it unsplittalbe traffic over a network consisting of m

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parallel links. Gairing et.al (2008) use the simaliar concept to discuss a discrete routing game.

3.3 The Non-cooperative Game

From the Wardrop’s principle, the concept of user equilibrium, every user

pursuit one’s maximum utility (we can also said, minimum travel cost). We know that users all non-cooperative to pursuit one’s maximum utility. That is, the character of the route choice game is non-cooperative game. By this reason, the application of game theory in transportation field almost non-cooperative game.

Game theory is divided into two branches, co-operative and non-cooperative game theory. The distinction can be fuzzy at time but, essentially, in non-cooperative game theory the unit of analysis is the individual participant in the game who is concerned with doing as well for himself as possible subject to clearly defined rules and possibilities. In comparison, in co-operative game theory the unit of analysis is most often the group or, in the standard jargon, the coalition; when a game is specified, part of the specification is what each group or coalition of players can achieve, without reference to how the coalition would effect a particular outcome or result.

N.N. Vorb’ev(1977) give this kind of game a briefly definition:

   

I _{ }



 , ,

(3.3-1)

S represent the situation i

S is the situation set, and







I i

Si

S .

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 

H_i is the payoff function of player i in the situation s.

H represent the payoff function of player i i. Iis the player set.

Definition 3.3-1: Let a constantc, for sS, we define constant-sum game if the following game exist

 

I i

i 



If c0, We called this type of game zero-sum game.

3.4 summary

We aim to use the view of game theory to analyze the network assignment process. First we have to understand the basis concept of the game theory. In this chapter briefly reviews the game theory, and introduces the history of the game theory.

The second part of the chapter review the application of game theory in the issue relate to traffic assignment problem. Most of them is a concept game assume a entity which aims to reduce the user’s utility. The last part of the chapter introduces several important definition of game. We define the Nash equilibrium, the non-cooperative game, and the zero-sum game, which are the special cases in constant game.

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