/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 An island has two reefs that are... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 15

An island has two reefs that are suitable for fishing, and there are twenty fishers who simultaneously and independently choose at which of the two reefs ( 1 or 2 ) to fish. Each fisher can fish at only one reef. The total number of fish harvested at a single reef depends on the number of fishers who choose to fish there. The total catch is equally divided between the fishers at the reef. At reef 1 , the total harvest is given by \(f_{1}\left(r_{1}\right)=8 r_{1}-\frac{r_{1}^{2}}{2}\), where \(r_{1}\) is the number of fishers who select reef 1 . For reef 2 , the total catch is \(f_{2}\left(r_{2}\right)=4 r_{2}\), where \(r_{2}\) is the number of fishers who choose reef 2 . Assume that each fisher wants to maximize the number of fish that he or she catches. (a) Find the Nash equilibrium of this game. In equilibrium, what is the total number of fish caught? (b) The chief of the island asks his economics advisor whether this arrangement is efficient (i.e., whether the equilibrium allocation of fishers among reefs maximizes the number of fish caught). What is the answer to the chief's question? What is the efficient number of fishers at each reef? (c) The chief decides to require a fishing license for reef 1 , which would require each fisher who fishes there to pay the chief \(x\) fish. Find the Nash equilibrium of the resulting location-choice game between the fishers. Is there a value of \(x\) such that the equilibrium choices of the fishers results in an efficient outcome? If so, what is this value of \(x\) ?

Short Answer

Expert verified
(a) Nash equilibrium: 8 fishers on Reef 1; 12 on Reef 2. Total catch: 80 fish. (b) Equilibrium is not efficient. Efficient allocation: 4 on Reef 1; 16 on Reef 2. Total catch: 88. (c) Licensing fee of 2 fish on Reef 1 achieves efficiency.

Step by step solution

01

Determine the Individual Catch for Each Fisher

The first step is to determine the number of fish each fisher catches if they choose a particular reef. The total catch at each reef is divided equally among the fishers there. For Reef 1, the catch per fisher is:\[ f_{1}(r_1)/r_1 = \frac{8r_1 - \frac{r_1^2}{2}}{r_1} = 8 - \frac{r_1}{2} \]For Reef 2, the catch per fisher is:\[ f_{2}(r_2)/r_2 = \frac{4r_2}{r_2} = 4 \]
02

Establish Nash Equilibrium Condition

In Nash equilibrium, no fisher can unilaterally improve their catch by moving to another reef. So, each fisher should be indifferent between choosing Reef 1 or Reef 2. Set the catch per fisher for both reefs equal:\[ 8 - \frac{r_1}{2} = 4 \]Solve for \( r_1 \):\[ 8 - 4 = \frac{r_1}{2} \rightarrow \ r_1 = 8 \]Since there are 20 fishers total, \( r_2 = 20 - r_1 = 12 \).
03

Calculate Total Fish Caught in Equilibrium

Now calculate the total number of fish caught when \( r_1 = 8 \) and \( r_2 = 12 \). For Reef 1:\[ f_{1}(8) = 8\times8 - \frac{8^2}{2} = 64 - 32 = 32 \]For Reef 2:\[ f_{2}(12) = 4 \times 12 = 48 \]Total catch = 32 + 48 = 80.
04

Determine Efficient Allocation

To determine if the arrangement is efficient, we maximize the total number of fish caught, which is \( f_1(r_1) + f_2(20 - r_1) \).The total catch function is:\[ T(r_1) = 8r_1 - \frac{r_1^2}{2} + 4(20 - r_1) \]Differentiate and set the derivative to zero:\[ \frac{dT}{dr_1} = 8 - r_1 - 4 = 4 - r_1 \]\[ 4 - r_1 = 0 \Rightarrow r_1 = 4 \]Thus, \( r_2 = 20 - 4 = 16 \).
05

Assess Nash Equilibrium Efficiency

The Nash equilibrium was \( r_1 = 8 \) and \( r_2 = 12 \) with a total catch of 80. The efficient allocation, however, indicates \( r_1 = 4 \) and \( r_2 = 16 \) which should be checked if different.Calculate total catch at efficient allocation:\[ f_{1}(4) = 8 \times 4 - \frac{4^2}{2} = 32 - 8 = 24 \]\[ f_{2}(16) = 4 \times 16 = 64 \]Total catch = 24 + 64 = 88. This means the equilibrium is not efficient as 88 > 80.
06

Introduce Licensing Fee and New Equilibrium

To achieve the efficient outcome, a fishing license fee \( x \) needs to be introduced such that fishers are inclined to distribute as per efficient allocation. Update the profit per fisher at Reef 1 with license fee:\[ 8 - \frac{r_1}{2} - x = 4 \]Substitute \( r_1 = 4 \) (efficient allocation for Reef 1):\[ 8 - \frac{4}{2} - x = 4 \Rightarrow 6 - x = 4 \]\[ x = 2 \]Thus, licensing each fisher at Reef 1 2 fish can achieve the efficient outcome.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

game theory
Game theory is a fascinating branch of mathematics focused on strategizing in situations of competition and cooperation. Imagine you are playing a game with several participants, each making decisions to maximize their outcomes. In our example, the "fishing game model" represents such a scenario. Here, each fisher independently decides which reef to fish in to maximize their catch.

Nash Equilibrium is a crucial concept in game theory. It signifies a situation where no player can benefit from changing their strategy while the others keep theirs unchanged. In this fishing game, the Nash Equilibrium determines the allocation of fishers to reefs such that none can improve their individual catch by unilaterally changing their reef choice.

Understanding the Nash Equilibrium helps us analyze the fishers' strategic behavior and predict their patterns without any external coordination.
efficiency in resource allocation
Efficiency in resource allocation is about making the best use of available resources to achieve the highest possible output. In our fishing game, the focus is on maximizing the total number of fish caught by optimally allocating the fishers between the two reefs.

Initially, each fisher seeks to maximize personal benefit, which often doesn't lead to the most efficient distribution for the group as a whole. The exercise shows that while the Nash Equilibrium leads to a total catch of 80 fish, an efficient allocation could yield 88 fish. This difference highlights how individual optimal strategies (Nash Equilibrium) may not necessarily result in the best collective outcome.

Addressing this inefficiency requires intervention, such as introducing a fishing license fee, to align individual strategies with the group鈥檚 optimal outcome.
strategic decision making
Strategic decision-making involves choosing actions by evaluating the potential reactions of other participants. Every fisher in the model must decide strategically which reef to fish, considering both the actions of others and the availability of fish.

This decision-making process is highly dynamic. Fishers enter a continuous feedback loop, assessing which reef promises a higher individual catch and adjusting tactics accordingly until reaching an equilibrium. It's crucial that each participant understands the impact of their choices on the collective catch and vice versa.

The licensing fee introduces an additional strategic layer, compelling fishers to reconsider the potential cost versus benefit of choosing Reef 1. Consequently, strategic decision making becomes pivotal in driving towards an efficient resource allocation outcome.
fishing game model
The fishing game model is an illustrative example within game theory contexts. It simulates real-world scenarios where limited resources must be shared among multiple users, resembling common resource allocation challenges.

In this model, 20 fishers choose between two reefs, each with different fish production functions. Reef 1's catch depends on the number of fishers squared, resulting in diminishing returns as more fishers converge. Conversely, Reef 2 offers a consistent catch regardless of the fisher count.

Through this model, we explore concepts like Nash Equilibrium and efficiency, observing how fishers' independent strategies impact overall resource utilization. The model serves as a great tool to understand strategic interactions and to design policies like licensing to guide participants toward preferred, efficient outcomes.

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Most popular questions from this chapter

Consider a game in which, simultaneously, player 1 selects a number \(x \in[0,6]\) and player 2 selects a number \(y \in[0,6]\). The payoffs are given by: $$ \begin{aligned} &u_{1}(x, y)=\frac{16 x}{y+2}-x^{2} \\ &u_{2}(x, y)=\frac{16 y}{x+2}-y^{2} \end{aligned} $$ (a) Calculate each player's best-response function as a function of the opposing player's pure strategy. (b) Find and report the Nash equilibrium of the game. (c) Suppose that there is no social institution to coordinate the players on an equilibrium. Suppose that each player knows that the other player is rational, but this is not common knowledge. What is the largest set of strategies for player 1 that is consistent with this assumption?

Consider a game in which, simultaneously, player 1 selects a number \(x \in[0,6]\) and player 2 selects a number \(y \in[0,6]\). The payoffs are given by: $$ \begin{aligned} &u_{1}(x, y)=\frac{16 x}{y+2}-x^{2} \\ &u_{2}(x, y)=\frac{16 y}{x+2}-y^{2} \end{aligned} $$ (a) Calculate each player's best-response function as a function of the opposing player's pure strategy. (b) Find and report the Nash equilibrium of the game. (c) Suppose that there is no social institution to coordinate the players on an equilibrium. Suppose that each player knows that the other player is rational, but this is not common knowledge. What is the largest set of strategies for player 1 that is consistent with this assumption?

Consider a more general Cournot model than the one presented in this chapter. Suppose there are \(n\) firms. The firms simultaneously and independently select quantities to bring to the market. Firm \(i\) 's quantity is denoted \(q_{i}\), which is constrained to be greater than or equal to zero. All of the units of the good are sold, but the prevailing market price depends on the total quantity in the industry, which is \(Q=\sum_{i=1}^{n} q_{i}\). Suppose the price is given by \(p=a-b Q\) and suppose each firm produces with marginal cost \(c\). There is no fixed cost for the firms. Assume \(a>c>0\) and \(b>0\). Note that firm \(i\) 's profit is given by \(u_{i}=p(Q) q_{i}-c q_{i}=(a-b Q) q_{i}-c q_{i}\). Defining \(Q_{-i}\) as the sum of the quantities produced by all firms except firm \(i\), we have \(u_{i}=\left(a-b q_{i}-b Q_{-i}\right) q_{i}-c q_{i}\). Each firm maximizes its own profit. (a) Represent this game in the normal form by describing the strategy spaces and payoff functions. (b) Find firm \(i\) 's best-response function as a function of \(Q_{-i}\). Graph this function. (c) Compute the Nash equilibrium of this game. Report the equilibrium quantities, price, and total output. (Hint: Summing the best-response functions over the different players will help.) What happens to the equilibrium price and the firm's profits as \(n\) becomes large? (d) Show that for the Cournot duopoly game \((n=2)\), the set of rationalizable strategies coincides with the Nash equilibrium.

Consider a strategic setting in which two geographically distinct firms (players 1 and 2) compete by setting prices. Suppose that consumers are uniformly distributed across the interval \([0,1]\), and each will buy either one unit or nothing. Firm 1 is located at 0 and firm 2 is located at 1 . Assume that the firms cannot change their locations; they can only select prices. Simultaneously and independently, firm 1 chooses a price \(p_{1}\) and firm 2 chooses a price \(p_{2}\). Suppose that the firms produce at zero cost and that due to a government regulation, they must set prices between 0 and 6 . As in the standard location game, consumers are sensitive to the distance they have to travel in order to purchase. But they are also sensitive to price. Consumers get a benefit of 6 from the good that is purchased, but they also pay a personal cost of \(c\) times the distance they have to travel to make the purchase. Assume that \(c\) is a positive constant. If the consumer at location \(x \in[0,1]\) purchases from firm 1, then this consumer's utility is \(6-c x-p_{1}\). If this consumer instead purchases from firm 2 , then her utility is \(6-c(1-x)-p_{2}\). If this consumer does not purchase the good, her utility is 0 . (a) Suppose that for given prices \(p_{1}\) and \(p_{2}\), every consumer purchases the good. That is, ignore the case in which prices are so high that some consumers prefer not to purchase. Find an expression for the "marginal consumer" who is indifferent between purchasing from firm 1 or firm \(2 .\) Denote this consumer's location as \(x^{*}\left(p_{1}, p_{2}\right)\). (b) Continue to assume that all consumers purchase at the prices you are analyzing. Note that firm 1's payoff (profit) is \(p_{1} x^{*}\left(p_{1}, p_{2}\right)\) and firm 2's payoff is \(p_{2}\left[1-x^{*}\left(p_{1}, p_{2}\right)\right]\). Calculate each firm's best response as a function of the other player's strategy. Also graph the best-response functions for the case of \(c=2\). (c) Find and report the Nash equilibrium of this game for the case in which \(c=2\). (d) As \(c\) converges to 0 , what happens to the firms' equilibrium profits? (e) What are the rationalizable strategies of this game for the case in which \(c=2\) ? (f) Find the Nash equilibrium of this game for the case in which \(c=8\).

Consider an asymmetric Cournot duopoly game, where the two firms have different costs of production. Firm 1 selects quantity \(q_{1}\) at a production cost of \(2 q_{1}\). Firm 2 selects quantity \(q_{2}\) and pays the production cost \(4 q_{2}\). The market price is given by \(p=12-q_{1}-q_{2}\). Thus, the payoff functions are \(u_{1}\left(q_{1}, q_{2}\right)=\left(12-q_{1}-q_{2}\right) q_{1}-2 q_{1}\) and \(u_{2}\left(q_{1}, q_{2}\right)=\left(12-q_{1}-q_{2}\right) q_{2}-4 q_{2}\). Calculate the firms' best-response functions \(B R_{1}\left(q_{2}\right)\) and \(B R_{2}\left(q_{1}\right)\), and find the Nash equilibrium of this game.
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