Question

Two neighboring homeowners, \(i=1,2,\) simultaneously choose how many hours \(l_{i}\) to spend maintaining a beautiful lawn. The average benefit per hour is \\[ 10-l_{i}+\frac{l_{j}}{2} \\] and the (opportunity) cost per hour for each is \(4 .\) Homeowner \(i\) 's average benefit is increasing in the hours neighbor \(j\) spends on his own lawn because the appearance of one's property depends in part on the beauty of the surrounding neighborhood. a. Compute the Nash equilibrium. b. Graph the best-response functions and indicate the Nash equilibrium on the graph. c. On the graph, show how the equilibrium would change if the intercept of one of the neighbor's average benefit functions fell from 10 to some smaller number.

Short answer

Question: Analyze a two-player game involving the decisions of neighboring homeowners and find the Nash equilibrium. Graph the best-response functions and indicate the Nash equilibrium on the graph. Show how equilibrium changes if the intercept of one neighbour's average benefit function falls from 10 to a smaller number. Answer: The Nash equilibrium occurs at (4, 4) with the given benefit functions. When the intercept of one neighbor's average benefit function falls from 10 to 8, the new Nash equilibrium can be found at (l'_1, l_2). Graph the best-response functions and the changes in the Nash equilibrium to see the effect.

Step-by-step solution

Find the Best-Response Functions

To find the best-response functions, we need to maximize the net benefit for each homeowner. Let's denote the net benefit for homeowner \(i\) as \(B_i\). Then, we have: \(B_i = (10 - l_i + \frac{l_j}{2})l_i - 4l_i = 10l_i - l_i^2 + \frac{l_il_j}{2} - 4l_i\) Next, we take the first derivative of the net benefit with respect to \(l_i\) and set it equal to \(0\): \(\frac{d B_i}{d l_i} = 10 - 2l_i + \frac{l_j}{2} - 4 = 6 - 2l_i + \frac{l_j}{2} = 0\) Solve the above equation for \(l_i\) to find the best-response functions: \(l_i = BR_i(l_j) = 3 + \frac{l_j}{4}\) (for each \(i=1,2\))

Find the Nash Equilibrium

To find the Nash equilibrium, we need to find the point where both best-response functions intersect. We set the first homeowner's best-response function equal to the second homeowner's best-response function: \(l_1 = 3 + \frac{l_2}{4}\) \(l_2 = 3 + \frac{l_1}{4}\) Solve for \(l_1\) in the first equation and substitute in the second equation: \(l_1 = 3 + \frac{3 + \frac{l_1}{4}}{4}\) \(l_1 = 3 + \frac{3}{4} + \frac{l_1}{16}\) \(l_1 - \frac{l_1}{16} = 3 + \frac{3}{4}\) \(\frac{15}{16}l_1 = \frac{15}{4}\) \(l_1 = 4\) Now solve for \(l_2\): \(l_2 = 3 + \frac{4}{4} = 4\) The Nash equilibrium is \((l_1, l_2) = (4, 4)\).

Graph the Best-Response Functions and the Nash Equilibrium

First, plot the best-response functions for both homeowners: \(BR_1: l_1 = 3 + \frac{l_2}{4}\) \(BR_2: l_2 = 3 + \frac{l_1}{4}\) Second, plot the Nash equilibrium found in Step 2: \((l_1, l_2) = (4, 4)\).

Analyze the Effect of a Drop in the Intercept

Now, let's analyze the situation when the intercept of homeowner 1's average benefit function drops from 10 to a smaller number, say, 8. The new benefit function for homeowner 1 is now: \(B'_1 = (8 - l_1 + \frac{l_2}{2})l_1 - 4l_1\) Next, calculate the new best-response function for homeowner 1: \(\frac{d B'_1}{d l_1} = 4 - 2l_1 + \frac{l_2}{2}\) Set the derivative equal to \(0\) and solve for \(l_1\): \(4 - 2l_1 + \frac{l_2}{2} = 0\) \(l_1 = BR'_1(l_2) = 2 + \frac{l_2}{4}\) Now, calculate the new Nash equilibrium by setting homeowner 1's new best-response function equal to homeowner 2's original best-response function: \(2 + \frac{l_2}{4} = 3 + \frac{l_1}{4}\) Solve for \(l_1\) and \(l_2\). The new Nash equilibrium is \((l'_1, l'_2) = (l'_1, l_2)\). Finally, graph the Nash equilibrium changing due to the drop in the intercept of one of the neighbor's average benefit functions.

Key concepts

Best-Response Functions

When discussing strategic interactions among players in a game, best-response functions play a pivotal role in understanding each participant's decision-making process. These functions essentially detail the optimal strategy for a player, assuming the strategies of other players are known.

For instance, in the lawn maintenance scenario, homeowners determine how long to tend their lawns based not only on their preferences but also while considering their neighbor’s actions. The best-response function is the technical representation of their response. Mathematically, one calculates this by setting the derivative of the benefit with respect to the player's action equal to zero to find the maximum point of benefit given the other player's choices. In our exercise, each homeowner maximizes their net benefit, resulting in equations that represent their best responses to the other.
Furthermore, when graphing these best-response functions, the point where they intersect represents a state where neither homeowner can improve their outcome by unilaterally changing their strategy. This intersection is the sought-after Nash equilibrium in this game. To clarify these concepts for students, graphical visualizations of such functions can be incredibly helpful. By plotting the corresponding lines on a graph, the visual intersection clearly designates the Nash equilibrium, driving home its significance in the context of best-response functions.

Game Theory

Game theory is a systematic framework for analyzing strategic interactions among rational decision-makers. It is used to understand the behavior of individuals or groups who are aware that their actions affect each other. The quintessential element of game theory is that it does not look at actions in isolation but rather in conjunction with the strategies of others.

In the realm of game theory, we often deal with multiple players, as in the case of two homeowners, who must choose strategies based on the likely responses of others. Each homeowner is a 'player' in this 'game', and they must account for their own benefit maximization while not ignoring the potential choices of their neighbor. By evaluating the Nash equilibrium, game theorists can predict stable outcomes where no player has an incentive to deviate unilaterally.
Understanding this framework can empower students to analytically approach and solve real-world problems that involve strategic interactions, such as company pricing wars, political negotiations, or even family dynamics where the actions of one member affect all others.

Opportunity Cost

Opportunity cost is a critical concept in economics and decision-making, representing the value of the best alternative foregone when a choice is made. Whenever one commits time or resources to one activity, they implicitly decide against another use of that time or resource that could have provided a benefit.

In our exercise, homeowners spend time on lawn care. Each hour spent gardening is an hour not spent on another activity that could have yielded earnings, relaxation, or any other benefit. If we quantify this alternative benefit as a monetary value, say $4 per hour, that becomes the opportunity cost of their time spent gardening. When the homeowners weigh the benefit of having a beautiful lawn against this cost, they maximize their net benefit by choosing an optimal number of hours to dedicate to lawn maintenance.
Teaching students about opportunity cost helps them understand that every choice has a consequence and that rational decision-making involves considering these trade-offs. By incorporating this understanding when plotting the best-response functions, students will realize how opportunity cost shapes the slope and the location of these curves in the graphical analysis of a game.