Question

Consider a version of the Cournot duopoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. 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 $p=100-2q_1-2q_2$. Suppose that each firm produces at a cost of 20 per unit. Further, assume that each firm's payoff is defined as its profit. (If you completed Exercise 5 of Chapter 3, then you have already dealt with this type of game.) Suppose that player 1 has the belief that player 2 is equally likely to select each of the quantities 6,11 , and 13 . What is player l's expected payoff of choosing a quantity of 14 ?

Short answer

Player 1's expected payoff when choosing 14 is 448.

Step-by-step solution

Calculate Market Price

Firstly, determine the market price when firm 1 chooses a quantity of 14 and firm 2 chooses quantities 6, 11, and 13. The market price equation is given by: $$p=100-2q_1-2q_2$$ Substitute $q_1=14$ and each $q_2$ value:- If $q_2=6$, $p=100-2(14)-2(6)=60$.- If $q_2=11$, $p=100-2(14)-2(11)=50$.- If $q_2=13$, $p=100-2(14)-2(13)=46$.

Calculate Profit for Each Scenario

Next, calculate the profit for firm 1 in each scenario. Profit is given by:$$\text{Profit}=(p-20)q_1$$Substitute $q_1=14$ and each previously calculated market price:- If $p=60$, then Profit = $(60-20)\times 14=560$.- If $p=50$, then Profit = $(50-20)\times 14=420$.- If $p=46$, then Profit = $(46-20)\times 14=364$.

Calculate Expected Payoff

The expected payoff is calculated by considering the probability distribution of firm 2's quantity choices. Since player 2 is equally likely to choose each quantity (6, 11, 13), each has a probability of $\frac{1}{3}$. Thus, the expected payoff is:$$\text{Expected Payoff}=\frac{1}{3}(560)+\frac{1}{3}(420)+\frac{1}{3}(364)$$Calculating this gives:$$\text{Expected Payoff}=\frac{1}{3}(1344)=448$$.

Key concepts

Game Theory

Game theory is a structured way of analyzing strategic interactions between different players or firms. In a Cournot duopoly, which is a specific type of game theory model, two firms decide how much quantity to produce in a market where they both sell identical products. Each firm's decision affects the market price and their own profits, creating a competitive environment. These firms make decisions simultaneously without knowing the choice of the other, resembling a strategic game where each player's choice could affect the other's payoff. In our case, firms 1 and 2 decide on quantities to produce, and these choices impact their profits based on the market demand equation. The aim is to find a "Nash Equilibrium," where neither firm would benefit from changing its decision, given the quantity chosen by the other firm. This concept allows businesses to predict competitors' potential decisions and adjust their strategies accordingly, making game theory a vital tool in market competition analysis.

Expected Payoff

The expected payoff is an essential outcome in game theory that considers the probabilities of different scenarios occurring based on varying strategies. In our example, player 1 forms a belief about player 2's choices. Expecting that player 2 might select quantities of 6, 11, or 13, each with equal probability of 1/3. Player 1's expected payoff is then the average profit from all these scenarios weighted by their probabilities. This analysis helps players decide on their best course of action by calculating their expected gains or losses. By weighing the potential outcomes, firms can make more informed production decisions, reducing risks and optimizing profits. Understanding expected payoffs allows businesses to navigate the uncertainties of market dynamics and competitive interactions.

Market Price

Market price in a Cournot duopoly is influenced directly by the quantities produced by the competing firms. The market price formula given is $p=100-2q_1-2q_2$, where $q_1$ and $q_2$ are the quantities produced by firms 1 and 2, respectively. When both firms submit their production quantities, these values determine the price customers will see. By adjusting the quantities, firms can influence the market price, potentially increasing their competitive edge and market share. As seen, when player 1 chooses a quantity of 14, and player 2 chooses various quantities, each combination results in a different market price. Understanding how to utilize this equation helps firms anticipate their competitors' actions and react appropriately, ultimately aligning supply with market demand to maximize profitability.

Quantity Competition

Quantity competition is a critical aspect of the Cournot duopoly model. Firms compete by deciding the quantities they produce rather than choosing prices directly. The interplay of these production decisions leads to a particular market equilibrium. Each firm considers its profit-maximizing quantity by analyzing the potential outcomes based on the quantities chosen by its rival. This competition can lead to intense strategic decisions as each firm wants to produce just enough to maximize its profit without oversupplying the market, which can drive prices down. The optimal strategy involves understanding both the market demand and the competitor’s reactions. Successfully navigating quantity competition requires firms to predict competitors’ production levels and adjust their strategies accordingly, ensuring they maintain optimal profit conditions in a shared market environment.