Game Theory: Concept, Application, and Example

2 years ago

Atiqur Rahman

12 minutes

Introduction to Game Theory

An oligopoly is an industry characterized by few dominant firms. Oligopoly is said to prevail when there are few firms in the market producing or selling a product. Oligopoly is also referred to as “competition among the few”. A special case of oligopoly is a duopoly where two firms are competing with each other.

Under an oligopoly, each firm has enough market power to prevent itself from being a price taker but each firm is facing inter-firm rivalry which is preventing it to consider the market demand curve as its own demand curve. Competition among oligopolistic firms is so severe those economists call it a ‘cut-throat competition’.

All the firms in the oligopoly industry are behaving strategically. But when firms are behaving strategically, the oligopolistic firm faces a basic dilemma i.e.

To cooperate with other firms or
To compete with other firms.

The Cooperative Solution:

If the firms in an industry cooperate among themselves to produce the monopoly output, then they can maximize their joint profits. The cooperation among firms may be either tacit or formal. Formal cooperation among firms is one when all the firms under an oligopolistic industry, after consultation and discussion, agree to observe certain rules of conduct with respect to price, output etc. This formal cooperation is also known as a collusive oligopoly.

On the other hand, tacit cooperation among firms occurs when all the firms under an oligopolistic industry, without consultation and discussion, have developed some understanding between them and pursue a uniform policy with regard to price, output etc. So, if the firms tacitly or formerly cooperate, they will reach a cooperative solution.

The Non-Cooperative Equilibrium or Nash Equilibrium:

The non-cooperative equilibrium is an equilibrium that is reached by firms when they proceed by calculating only their own gains without cooperating with others. It is called a non-cooperative equilibrium or a Nash Equilibrium.

Nash Equilibrium: Each firm is doing the best it can, given what its competitors are doing, and firms proceed by calculating only their own gains without cooperating with others.

The concept of Nash equilibrium was first explained by Mathematician John Nash in 1951, and received the Nobel Prize in Economics for his work. Nash Equilibrium, in which each firm’s best strategy is to maintain its present behaviour, given the present behaviour of the other firms. The concept of Nash equilibrium is widely used in game theory.

Oligopoly as a Game

In 1994, Mathematician John Von Neumann and Economist Oskar Morgenstern published a work in which they analyzed a set of problems or games in which two or more people pursues their own interests and in which no one of them can dictate the outcome.

In an oligopolistic market, each individual firm is faced with the problem of choosing a rational course of action from many possible actions while keeping in view the possible reactions of its rivals whose counter moves would affect him.

Game theory analyzes oligopolistic behaviour as a series of strategic moves and reactive countermoves among rival firms. A strategy is a course of action which a player in a game will choose during the play of the game. Players are choosing strategies without knowing with certainty what strategy their rival is going to play.

Here it is assumed that firms are anticipating rival reactions. Each player’s aim is to maximize their own payoff by choosing specific actions, but the actual outcome also depends upon what other players are doing. The payoff matrix shows the profit of each player, given its decision and the decision of its competitors.

Types of Games: In normal (or strategic) form games, players are moving moves simultaneously, whereas, in extensive form games, players are moving moves in some order over time. ‘One shot’ game is a game that is played only once and a super game that is repeated an infinite number of times.

Game Theory Application to Oligopoly

When game theory is applied to oligopoly, the players are firms, and their game is played in a market. Their strategies are their price and output decisions, and their payoffs are profits. Let us explain the concept of equilibrium with the help of an example.

We are taking the case of two firms who are producing a homogeneous product, and each firm has two possible strategies i.e.

Each firm can produce an output equal to one-half of the monopoly output (the passive strategy) and jointly share the monopoly profit.
Each firm can produce an output equal to two-thirds of the monopoly output (the aggressive strategy).

The strategic form of the game can be written as follows:

Players: There are two players- Firm A and Firm B.

Game: Both A and B are producing output in the market.

Strategy: Each firm can produce an output equal to either one-half of the monopoly output or two-thirds of the monopoly output.

Payoff Matrix: Profit of each firm given its decision of its competitors.

The Payoff Matrix is given in the Table below:

In Table 1, the figure in the four boxes shows the profit of firm A and firm B. Let us explain these payoffs.

The upper left-hand corner of the payoff matrix tells us that if both firms are producing one-half of the monopoly output, both firms would make a profit of Rs 20 each.
The upper right-hand corner tells us that if firm B produces one-half of the monopoly output and firm A produces two-thirds of the monopoly output, firm B would make a profit of Rs 15 and firm A Rs 22.
The lower left-hand corner tells us that if firm B produces two-thirds of the monopoly output and firm A produces one-half of the monopoly output, firm B would make a profit of Rs22 and firm A Rs15.
The lower right-hand corner tells us that if firm B produces two-thirds of the monopoly output and firm A also chooses to produce two-thirds of the monopoly output, both would make a profit of Rs17 each.

Now the question is what the two firms would do, i.e. to cooperate with each other or compete with each other.

The possible solution could be:

i) The Passive (or Cooperative) Solution:

If both firms cooperate with each other i.e. both are producing one-half of the monopoly output, then they both are getting a profit of Rs 20 each. Under a cooperative solution, both jointly produce the monopoly output and share profit equally among them. The cooperative solution is achieved by playing strategy (one-half monopoly output, one-half monopoly output).

ii) The Aggressive (Non-Cooperative) or Nash Equilibrium:

Nash Equilibrium is the equilibrium where each firm is doing the best it can give what its competitors are doing. Each firm proceeds by calculating only its own gains without cooperating with others.

Definition: A strategy 𝑠^∗ is the best response to a strategy vector 𝑠^∗_{_-𝑖} of the other player if 𝜋_𝑖(𝑠^∗_𝑖, 𝑠^∗_-𝑖) ≥ 𝜋_𝑖(𝑠_𝑖, 𝑠^∗_-𝑖 ) For all 𝑠_𝑖.

Here 𝑠^∗_{_𝑖} is a “dominant strategy” in the sense that it is the best strategy to play provided the other players do in fact play the strategy combination 𝑠^∗_{_-𝑖}. There must be a condition to ensure that player i is correct in his inference and that the other players are correct in their inferences. This gives us the following definition.

The strategy vector 𝑠^∗ = 𝑠^∗₁, 𝑠^∗₂ , … … … . , 𝑠^∗_𝑛 is a Nash equilibrium if 𝜋_𝑖(𝑠^∗_𝑖 , 𝑠^∗_−𝑖) ≥ 𝜋_𝑖(𝑠_𝑖 , 𝑠^∗_−𝑖) For all 𝑠_𝑖 and all _𝑖.

There is one Nash equilibrium in Table 1. The best decision for each firm is to produce two-thirds of the monopoly output, where each firm is earning a profit of Rs 17. Here neither firm has the incentive to depart from this position except through cooperation with the other firm. In any other cell, each firm has the incentive to change its output, given the other firm’s output.

In Table 1, suppose if firm A chooses to produce one-half of the monopoly output, and then firm B would choose to produce two-thirds of the monopoly output. This is so because the profit earned by firm B when producing two-thirds of the monopoly output is Rs 22, which is more than Rs 20 (when producing one-half of the monopoly output).

But if firm A chooses to produce two-thirds of the monopoly output, then firm B would also choose to produce two-thirds of the monopoly output as it is giving a profit of Rs 17, which is more than Rs 15 (when he is producing one-half of the monopoly output). So there is one Nash Equilibrium when both firms are producing two-thirds of the monopoly output. Neither firm has the incentive to depart from this position.

Nash Equilibrium: Both firms are producing two-thirds of the monopoly output and earning a profit of Rs 17 each.

iii) Strategic Behavior:

When each firm is behaving strategically, i.e. choosing its optimal strategy, taking into account what the other firm may do, we attain Nash equilibrium. What would firm A would do if it knew what firm B is doing? Firm B can do two things, i.e. either he can produce one-half of the monopoly output or two third of the monopoly output.

First, suppose firm B is producing one-half of the monopoly output, then what would be the best decision for firm A. When firm B is producing one-half of the monopoly output and firm A also chooses the same, then firm A will get a profit of Rs 20. But if firm A chooses to produce two-thirds of the monopoly output, then firm A will get a profit of Rs 22.

Secondly, suppose firm B is producing two-thirds of the monopoly output and firm A also chooses the same, then firm A will get a profit of Rs 17 But if firm A chooses to produce one-half of the monopoly output, then firm A will get a profit of Rs 15.

In either case, the best strategy for firm A is to produce two-thirds of the monopoly output. Aggressive behaviour is the dominant strategy. A dominant strategy is a strategy that is best no matter what the rival does. In this game, both firms have a dominant strategy which is to produce two-thirds of the monopoly output. Firm B will reason in the same way. So, as a result, both end up jointly producing one and third times the monopoly output and earning a profit of Rs 17.

iv) Break Down of Cooperation:

Nash equilibrium is attained when firms are behaving strategically, but then why tacit cooperation tends to break down? Suppose both firms are cooperating with each other and producing one-half of the monopoly output and earning a profit of Rs 20 each. If firm A cheats by increasing its output, its profit will increase, but firm B’s profit will decline.

This behaviour of firm A drives down the price in the industry, so firm B earns less from its unchanged output. This cheating by firm A causes a fall in the joint profits by taking firms away from the joint maximizing monopoly output. So Firm B’s profits would fall by more than the rise in Firm A’s profits.

Game Theory: Concept, Application, and Example

Introduction to Game Theory

The Cooperative Solution:

The Non-Cooperative Equilibrium or Nash Equilibrium:

Oligopoly as a Game

Game Theory Application to Oligopoly

The possible solution could be:

i) The Passive (or Cooperative) Solution:

ii) The Aggressive (Non-Cooperative) or Nash Equilibrium:

iii) Strategic Behavior:

iv) Break Down of Cooperation:

Read More- Microeconomics

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Introduction to Game Theory

The Cooperative Solution:

The Non-Cooperative Equilibrium or Nash Equilibrium:

Oligopoly as a Game

Game Theory Application to Oligopoly

The possible solution could be:

i) The Passive (or Cooperative) Solution:

ii) The Aggressive (Non-Cooperative) or Nash Equilibrium:

iii) Strategic Behavior:

iv) Break Down of Cooperation:

Read More- Microeconomics

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