Introduction to Game Theory

People interact with each other in daily life, regardless of the interaction is cooperative or competitive. In either case, it means that one person’s action affects another person’s well-being, no matter positively or negatively. This is a situation of interdependence, also known as strategic setting. In other word, when a person made his best decision, he must have considered how the others responses. A systematic study of this sort yields a theory of strategic interaction.

Early in 1800s, Augustine Cournot explored equilibrium in models of oligopoly. Then lots of scholars dedicate in this field. In 1944, the general theory of strategic setting, also called the theory of games, was launched by John von Neumann (Watson, 2002). Currently, the theory is employed by practitioners from a variety of fields, including economics, political science, law, international relations, and mathematics.

Games are formal descriptions of strategic setting, so game theory is a methodology of formally studying situations of interdependence. That is to say, game theory is concerned with how rational individuals make decisions when they are mutually interdependent (Romp, 1997).

It is usual to distinguish two separate branches of game theory. These are cooperative and non-cooperative game theory. In non-cooperative game theory the individuals are unable to enter into binding and enforceable agreements with one another. In contrast, cooperative game theory focuses on how groups of individuals commit to each other’s rational decisions. Here the individual can be an organization such as firms, governments, or countries. All individuals are assumed to be rational.

In non-cooperative game theory, there are two ways to represent a game. The first type

is called a normal form game or strategic form game. The second type is called an extensive form game. In a normal form game, we can identify the three things:

(1) The players: The individuals who make decisions in the game. For there to be interdependence we need to have at least two players in the game.

(2) The strategies available to each player: A strategy is a complete description of how a player can play a game. It describes how the player’s actions are dependent on what he observes other players in the game to have done.

(3) The payoffs: A payoffs is what a player will receive at the end of the game, which is dependent on the actions of all the players in the game. Players are assumed to be rational when they try to maximize their payoffs. The payoffs may correspond to monetary rewards, such as profits, or the utility each player obtains at the end of the game.

The normal form game expresses the players choosing their strategies independently and simultaneously. It is usually represented with a matrix. We depicted an example of normal form in Figure 2 by a well-known game called “the prisoner’s dilemma” below: Two players are ‘prisoner 1’ and ‘prisoner 2’; and ‘confess’ and ‘don’t confess’ are their strategies. The numbers in the cell are payoffs of two players.

Prisoner 2

Confess Don’t confess

Prisoner 1 Confess -6,-6 0,-9

Don’t Confess -9,0 -1,-1

Figure 2 The normal form of “prisoner’s dilemma”

The extensive form game is represented by a game tree, which includes four elements:

(1) Nodes: A position where a player makes a decision.

(2) Branches: To represents the alternative choices or actions.

(3) Vectors: To represents the payoff of each player, listed in the order of players.

(4) Information sets: When two or more nodes are joined together by a dashed line.

This means that the player whose decision it is dose not know which node he is at.

When this situation occurs, the game is characterized as one of imperfect information.

The extensive form of the same example is shown below:

Figure 3 The extensive form of “prisoner’s dilemma”

In extensive form games, more attention is placed on the sequence and on the amount of information available to each player when he is making a decision.

A solution to a game is a prediction of what each player in the game will do. There are many different solution techniques that have been proposed for different types of games. For static games, two kinds of solution techniques are widely applied. The first is using the concept of dominance. This technique is to obtain the solution by ruling out the strategies that a rational person would never play. The second is based on the concept of equilibrium. In a

non-cooperative game the equilibrium occurs when none of the player has an incentive to deviate from the predicted solution. The Nash equilibrium was introduced by John Nash, which pointed out that in equilibrium of each player’s strategy is optimal given that other player chooses the equilibrium strategy. If this is not the case, then at least one player would wish to choose a different strategy and so the situation would not be in the equilibrium any more.

Static games are what we can think of players making their decision simultaneously.

However, sometimes the games are played over a number of time periods, it would become dynamic game. In a dynamic game, players are able to observe the actions of other players before making their optimal response. The extensive form can be a good way to describe a dynamic game. To solve the game, the backward induction is widely employed. Every finite game with perfect information can be solved by backward induction. It is the process of analyzing a game from back to front. At each information set, one strikes from consideration actions that are dominated, given the terminal nodes that can be reached (Watson, 2002).

A subgame is defined as a smaller part of the whole game starting from any one node and continuing to the end of the entire game, with the qualification that no information set is subdivided. In many dynamic games there are multiple Nash equilibria. These equilibria involve incredible threats or promises that are not in the interests of the players making them to carry out. The concept of subgame perfect Nash equilibrium rules out these situations. By requiring that a solution to a dynamic game must be a Nash equilibrium, it must comprise a Nash equilibrium in each of these subgame. That is to say, each player must act in his self-interest in every period of the game.

To sum up, the game theory is one of the theories for strategic decision. Under some basic assumptions, it applies the mathematical model to infer the behaviors between the players.

The most important assumption is that all the participants are rational. However, when a participant makes a decision, he would like to maximize his benefits by taking the opponent’s best policy into consideration. A lot of models have been developed to explain the behaviors of oligopolistic firms. In the next two subsections, we will introduce Cournot model and Stackelberg model that are related to our work.

2.1.1 Cournot Model

French economist Cournot developed the model in 1838. In this model, the players include two firms and produce homogeneous products. The available strategies are the quantities of product each firm can supply to the market. Here the both firms are assumed to be able to supply any positive level of output. The payoffs are the profits each firm receives.

Both firms make their quantities decisions before knowing how many products the other firm has supplied to the market. Once each firm has chosen its optimal level of supply, the market price is also determined, and the firms can gain the corresponding level of profit.

To find the solution of this model, it begins with the reaction function. Each firm’s reaction function can be obtained by differentiating the firm’s profit function with respect to its own output level and then setting it to zero. Then rearranging the equation of this first-order condition, it leads to find a maximum solution. The second-order condition is checked to ensure that a maximum has indeed been found. In Nash equilibrium both firms must be maximizing profits simultaneously, given that their beliefs about the other firm’s level of supply.

The solution of this model is Pareto inefficient. It implies that there are other level of supply where at least one firm can be better off and the other firm no worse off. It also has some comments on Cournot model, such as it postulate that each firm can react to the other firm’s output level. This is inconsistent with the initial structure of the game where both firms set their

output level simultaneously. Besides, if the firms interact repeatedly, each firm assumes the other does not respond to the output changes would not be a rational conjecture.

2.1.2 Stackelberg Model

Stackelberg introduced this model in 1934. In this model, the players, the available strategies and the payoff setting are quite the same as in the setting of Cournot model. In Cournot model, both firms make their quantities decisions before knowing how many products the other firm has supplied to the market, that is, both firms choose their desired levels of supply simultaneously. Contrast to the Cournot model, the Stackelberg model assumed that at least one of the firms in the market is able to claim itself to a level of supply before the other firm in the market has decided its level of supply. The other firm observes the leader’s supply and then responds with its output decision. The firm that has the ability to claim its level of output initially is called the market leader, and the other firm is called the follower.

In the Stackelberg model, two firms involve to take actions in sequence. To solve this situation of the model, the backward induction approach will be employed. It starts with the last period first, initially determines the follower’s output decision. Given that the follower is rational, he will attempt to maximize his profit level, subject to the leader’s known level of supply.

By differentiating the follower’s profit function with respect to the quantity and setting the equation to zero, it gives the first-order condition for a maximum. It shows the follower’s optimal response for any level of supply chosen by the leader, also known as the follower’s reaction function. By knowing the eventual outcome of the game in the last period, the leader will maximize his profits subject to this constraint. Replacing the output level of leader’s profit function with follower’s reaction function, it can obtain the first-order condition by differentiating the leader’s profit function with respect to it output level. This is subgame

perfect Nash equilibrium level of supply of the leader. Then applying this result into the follower’s reaction function, the follower’s best response can be acquired. The process can be summarized below:

(1) The follower’s reaction function is known by the leader.

(2) The leader applies the follower’s reaction function into his own profit function, in order to maximize his profit.

(3) The follower decides his own best action by applying the leader’s best policy.

With only one leader and one follower, the leader will produce the monopoly output level but not earn the monopoly profits because of the follower’s positive level of output. The Stackelberg equilibrium entails higher profits for the leader and smaller profits for the follower.

It is said to be the first-move advantage.