Question

Consider a version of the Cournot duopoly game, which will be thoroughly analyzed in Chapter 10 . Two firms (1 and 2) compete in a homogeneous goods market, where the firms produce exactly the same good. The firms simultaneously and independently select quantities to produce. The quantity selected by firm \(i\) is denoted \(q_{i}\) and must be greater than or equal to zero, for \(i=1,2\). The market price is given by \(p=2-q_{1}-q_{2}\). For simplicity, assume that the cost to firm \(i\) of producing any quantity is zero. Further, assume that each firm's payoff is defined as its profit. That is, firm \(i\) 's payoff is \(p q_{i}\), where \(j\) denotes firm \(i\) 's opponent in the game. Describe the normal form of this game by expressing the strategy spaces and writing the payoffs as functions of the strategies.

Short answer

Strategy sets are \(S_1 = [0, \infty)\), \(S_2 = [0, \infty)\). Payoffs: Firm 1: \(2q_1 - q_1^2 - q_1q_2\), Firm 2: \(2q_2 - q_1q_2 - q_2^2\).

Step-by-step solution

Understanding the Game

The Cournot duopoly game involves two firms choosing quantities to produce in a homogeneous goods market. The quantities are denoted by \(q_1\) and \(q_2\) for firms 1 and 2, respectively. The market price is influenced by these quantities and is given by the equation \(p = 2 - q_1 - q_2\), with costs assumed to be zero.

Defining Strategy Spaces

The strategy space for each firm is the set of possible quantities they can produce. Since the production cost is zero and quantities must be non-negative, the strategy space for each firm \(i = 1, 2\) is \(S_i = [0, \infty)\).

Calculating Payoffs

The payoff for each firm is the profit, which is the product of the market price and the quantity produced by that firm. For firm 1, the payoff is \(\pi_1(q_1, q_2) = p \cdot q_1 = (2 - q_1 - q_2) q_1 = 2q_1 - q_1^2 - q_1q_2\). Similarly, for firm 2, the payoff is \(\pi_2(q_1, q_2) = p \cdot q_2 = (2 - q_1 - q_2) q_2 = 2q_2 - q_1q_2 - q_2^2\).

Expressing the Normal Form

In the normal form of the game, you write the payoff functions for both firms based on their strategies. The game can be represented as: - Firm 1 chooses \(q_1 \in [0, \infty)\)- Firm 2 chooses \(q_2 \in [0, \infty)\)- Payoffs: - For Firm 1: \(2q_1 - q_1^2 - q_1q_2\) - For Firm 2: \(2q_2 - q_1q_2 - q_2^2\)Thus, each firm's choice of quantity affects both its own profit and the competitor's.

Key concepts

Game Theory

Game theory is a field of study that examines strategic interactions between different decision-makers, known as players. In the case of a Cournot duopoly, we have two firms acting as players in a market. Each firm must decide how much of a product to produce, and their choices directly affect the market outcome and their profits.

This setting allows us to analyze concepts such as competition, strategic decision-making, and equilibrium.

• The main aim of each firm is to maximize its profit given the action of the other firm.
• Both firms try to make rational decisions based on the expected action of their competitor.
• This can lead to a stable outcome known as the Nash Equilibrium, where no player has an incentive to unilaterally change their strategy.

The key idea in game theory is understanding how the choices of one player influence the outcomes for others. In real-world problems, this helps in predicting the strategies competitors might adopt.

Normal Form Game

A normal form game neatly organizes the players, their strategies, and the resulting payoffs in a tabular form or mathematical functions. For the Cournot duopoly game, it involves specifying each firm’s strategy set and payoffs.

In normal form:

• Players are listed, in this case, the two firms.
• Their respective strategies involve choosing a quantity to produce, with no negative quantities.
• Their payoffs are clear functions of the chosen strategies, such as i. For Firm 1: defined by the function \(2q_1 - q_1^2 - q_1q_2\)
  
  ii. For Firm 2: defined by the function \(2q_2 - q_1q_2 - q_2^2\)

The normal form game simplifies the analysis by providing a structured view of how firm strategy choices affect their respective payoffs. It serves as a foundation to predict equilibrium outcomes through strategic thinking.

Market Competition

Market competition in a Cournot duopoly refers to how two firms compete by setting quantities in a homogeneous or alike product market. The interaction in the market competition generates the following dynamics:

• Each firm influences market price with their output decisions.
• The price is determined by the total quantity available in the market, captured by the expression \(p = 2 - q_1 - q_2\).
• Since costs are assumed to be zero, firms aim to set a quantity that maximizes their revenue, balancing against the competitor’s production level.
• Each firm's understanding of market dynamics affects its strategy, aiming to gain a competitive advantage over the rival.

The Cournot model helps analyze the situation where firms cannot alter market conditions but adjust their outputs to optimize profits, showing competition's nature within oligopolistic markets.

Strategy Spaces

In game theory, the strategy space for each player identifies all the possible actions they can take. In the Cournot duopoly, strategy spaces for both firms consist of the quantities they can choose to produce.

• For each firm, the strategy space is all non-negative real numbers (i.e., \(S_i = [0, \infty)\)).
• This allows firms to choose any quantity from zero to an infinite amount of the goods.
• Strategy spaces are infinite in this game, illustrating theoretical action possibilities.
• In practice, firms face constraints like production capacity or market saturation, even though not explicitly noted.

Each firm's choice within its strategy space not only determines its profit but also impacts the other firm, making understanding strategy spaces crucial for anticipating competitor actions and strategizing responses.