Question

Consider the Cournot duopoly game with incomplete information. First, nature chooses a number \(x\), which is equally likely to be 8 or 4 . This number represents whether demand is high \((x=8)\) or low \((x=4)\). Firm 1 observes \(x\) because this firm has performed market research and knows the demand curve. Firm 2 does not observe \(x\). Then the two firms simultaneously select quantities, \(q_{1}\) and \(q_{2}\), and the market price is determined by \(p=x-q_{1}-q_{2}\). Assume that the firms produce at zero cost. Thus, the payoff of firm 1 is \(\left(x-q_{1}-q_{2}\right) q_{1}\), and the payoff of firm 2 is \(\left(x-q_{1}-q_{2}\right) q_{2}\). (a) Note that there are two types of firm 1, the high type (observing \(x=8\) ) and the low type (observing \(x=4\) ). Let \(q_{1}^{\mathrm{H}}\) and \(q_{1}^{\mathrm{L}}\) denote the quantity choices of the high and low types of firm 1 . Calculate the players' bestresponse functions. (b) Find the Bayesian Nash equilibrium of this game. (c) Does firm l's information give it an advantage over firm 2 in this game? Quantify this.

Short answer

The Bayesian Nash Equilibrium involves solving best response equations together. Firm 1's information advantage results in potentially higher expected payoffs.

Step-by-step solution

Identify Firm 1's Best Response for High Demand

For high demand, where \(x=8\), Firm 1's payoff function is \(\pi_1 = (8-q_1^H-q_2)q_1^H\). To find the best response, take the derivative with respect to \(q_1^H\) and set it to zero: \[ \frac{d\pi_1}{dq_1^H} = 8 - 2q_1^H - q_2 = 0 \] This simplifies to Firm 1's best response: \[ q_1^H = 4 - 0.5q_2 \]

Identify Firm 1's Best Response for Low Demand

For low demand, where \(x=4\), Firm 1's payoff function is \(\pi_1 = (4-q_1^L-q_2)q_1^L\). Similarly, take the derivative with respect to \(q_1^L\) and set it to zero: \[ \frac{d\pi_1}{dq_1^L} = 4 - 2q_1^L - q_2 = 0 \] This gives Firm 1's best response: \[ q_1^L = 2 - 0.5q_2 \]

Identify Firm 2's Best Response

Firm 2 does not observe \(x\), so it considers the expected payoff function based on probabilities: \[ \pi_2 = \frac{1}{2}[(8 - q_1^H - q_2)q_2] + \frac{1}{2}[(4 - q_1^L - q_2)q_2] \]This becomes: \[ \pi_2 = (6 - 0.5q_1^H - 0.5q_1^L - q_2)q_2 \]Taking the derivative with respect to \(q_2\) and setting it to zero: \[ \frac{d\pi_2}{dq_2} = 6 - 0.5q_1^H - 0.5q_1^L - 2q_2 = 0 \]Thus, Firm 2's best response is:\[ q_2 = 3 - 0.25q_1^H - 0.25q_1^L \]

Solve for the Bayesian Nash Equilibrium

The Bayesian Nash Equilibrium involves finding \(q_1^H\), \(q_1^L\), and \(q_2\) such that each best response function is satisfied simultaneously. Substituting Firm 2's best response into Firm 1's high type best response:\[ q_1^H = 4 - 0.5(3 - 0.25q_1^H - 0.25q_1^L) \]This simplifies to:\[ q_1^H = \frac{16 - 0.5q_1^L}{3} \]Similarly, substitute into Firm 1's low type best response:\[ q_1^L = 2 - 0.5(3 - 0.25q_1^H - 0.25q_1^L) \]This simplifies to:\[ q_1^L = \frac{4 - 0.5q_1^H}{3} \]These equations along with Firm 2's best-response need to be solved together to find the equilibrium quantities.

Quantify Firm 1's Information Advantage

The information advantage can be determined by comparing payoffs. If Firm 1 can choose different quantities based on demand (high \(x=8\) and low \(x=4\)), its expected profit can be higher than if it did not have this information. In equilibrium, check the value of \(\pi_1\) and \(\pi_2\) for both high and low types and compare it with a scenario where Firm 1 does not observe \(x\).

Key concepts

Cournot Duopoly

The concept of Cournot Duopoly arises from the foundational models in the study of oligopolistic markets, was introduced by Antoine Augustin Cournot in 1838. It describes a market structure where two firms compete with one another by selecting the quantity of output that they will produce independently and simultaneously, which then determines the market price.
The main assumption here is that each firm aims to maximize its own profit, assuming the quantity chosen by the competitor is fixed. This creates an interdependence, as the decision of one firm affects the payoff of the other, compelling them to strategically think about how much to produce.

• This model leads to a strategic equilibrium where neither firm can improve its profit by changing its own output unilaterally, given the output of the other firm. This equilibrium is known as the Nash Equilibrium.
• The model assumes that each firm knows its own cost but not that of its rival, and there is no product differentiation.
• To find this equilibrium, each firm computes a 'best response' function, pinpointing the output that maximizes its profit given the other firm's output.

Under Cournot competition, firms consider only current quantities, ensuring that the decisions remain static and based on current information. This classic setup underpins much of game theory analyses in duopolistic markets, providing students with a clear framework to understand competitive strategies.

Incomplete Information

In game theory, the concept of incomplete information pertains to situations where competitors in a market are not aware of some aspects of their surroundings or the payoffs and strategies available to other players. In the Cournot Duopoly scenario provided, this translates into Firm 2 not having full information about the demand represented by the factor \(x\). Such situations model real-world settings where competitors might lack complete market intelligence.
In our example:

• Firm 1 has access to market research and observes \(x\), giving it a clear advantage in understanding demand.
• Firm 2 needs to make decisions without knowing whether the demand is high (\(x=8\)) or low (\(x=4\)).
• Incomplete information leads to uncertainty in decision-making and forces Firm 2 to base its strategies on expected outcomes.

Managing incomplete information often involves strategic decision-making that accounts for potential variations and uncertainties in other players' strategies. This can significantly impact the equilibrium of the game, affecting not only the strategies but also profits of firms involved.

Bayesian Nash Equilibrium

A significant concept in game theory for handling situations with incomplete information is the Bayesian Nash Equilibrium. This is an extension of the Nash Equilibrium to settings where players have beliefs about the types of their opponents based on incomplete information. It involves each player maximizing their expected utility, taking into account their beliefs about other players' private information.
To identify a Bayesian Nash Equilibrium:

• Firms must form beliefs regarding the state of nature, such as the demand being high or low, and accordingly adjust their strategies.
• In our exercise, Firm 2's strategy is crafted under uncertainty, expecting either a high or low demand scenario with equal probability.
• The equilibrium is found when each player’s strategy is optimal, given their beliefs about the unobserved types and the strategies adopted by other players.

A Bayesian Nash Equilibrium helps clarify how firms in oligopolistic markets can operate effectively even when some aspects of their environment are unknown, leading to a rigorous approach in strategic decision-making under uncertainty.

Market Research

Market research forms the cornerstone of strategic advantage in competitive markets, specifically illustrated in this exercise where Firm 1 conducted it to ascertain the demand (\(x\)). This allows the firm to quantify its production to align with the market demand effectively.
Benefits of conducting market research include:

• Understanding market demand precisely, enabling informed quantity decisions, such as Firm 1 being able to discern whether demand is high or low.
• Gaining a competitive edge over rivals who lack this critical insight. Firm 1, with its research, can adapt its strategy optimally.
• Reducing uncertainty in decision making, which directly contributes to achieving higher payoffs and potentially dominating market segments.

By leveraging market research, firms can transition from reactive to proactive strategies, anticipating customer needs and competitors' moves more effectively. For students, deciphering the importance and techniques of market research in game theory illuminates how real-world businesses strive to bridge information gaps and sculpt winning strategies in uncertain environments.