On a peaceful neighborhood in Glendale live two types of people: those who like buying a...

 

 

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On a peaceful neighborhood in Glendale live two types of people: those who like buying a Regular number of grapes, and those who like buying Many grapes. On any given day, any of the many stores in the neighborhood can be visited by two random individuals looking to buy grapes. The number of grapes they end up buying and the enjoyment they get from their purchase depends on the types of the individuals visiting the store: If the two individuals are of the type who buy a Regular number of grapes, both will be able to buy the number they need, and therefore both will get an enjoyment of 12. If the individuals are of different type, the one who likes buying Many grapes will selfishly get all the bananas available, for an enjoyment of 8, while the one who buys Regular amounts will go home grapeless and her enjoyment will be 0. If the two individuals like buying Many grapes, both will obtain only a fraction of the number they're looking for, meaning both will receive an enjoyment level of only 4. Part 1. Build a payoff matrix to summarize this problem. Part 2. Let r represent the proportion of individuals who like buying a Regular number of grapes. Write, in terms of r, the fitness functions of each type of individual. Part 3. Graph the fitness functions you found in the previous question with r on the horizontal axis. The lines you draw must be properly labeled. If they intersect make sure to indicate the value of r at which this happens. Part 4. Indicate, with arrows on the horizontal axis of your graph, the evolutionary trends, aka population dynamics, for r. Part 5. What is (are) the equilibriums) for this system? What type of evolutionary equilibrium is it (are they)? Part 6. One day, a hurricane floods the grape fields and people on the news start reporting that grapes will soon become very expensive. The consequence of this "gratastrophe" is that it suddenly alters the enjoyment levels of people who like buying Many grapes. Now, due to fear of missing out, when they go to the store at the same time as a Regular type, their enjoyment from buying all the grapes is 10 instead of 8. And when they go at the same time as another Many-buyer, their enjoyment from getting at least some grapes is 6 instead of 4. The enjoyment levels experienced by the Regular buyers doesn't change. So, the hurricane suddenly changes the position of the fitness function of those who buy Many grapes, but not the function of Regular buyers. Suppose that, before the hurricane, the neighborhood was at the equilibrium in which both types of individuals coexist. Immediately after the hurricane, the position of the polymorphic equilibrium changes, but the population is, at least initially, still at the previous equilibrium. Explain what will happen, over time, to the proportions of Regular and Many-avocado buyers. Hint: draw the new fitness functions and the new population dynamics (arrows) and place your initial population at the polymorphic equilibrium defined by the pre-hurricane fitness functions.) Explain in one or two sentences how you think this problem relates to the excessive purchases of toilet paper observed for a while when COVID started. On a peaceful neighborhood in Glendale live two types of people: those who like buying a Regular number of grapes, and those who like buying Many grapes. On any given day, any of the many stores in the neighborhood can be visited by two random individuals looking to buy grapes. The number of grapes they end up buying and the enjoyment they get from their purchase depends on the types of the individuals visiting the store: If the two individuals are of the type who buy a Regular number of grapes, both will be able to buy the number they need, and therefore both will get an enjoyment of 12. If the individuals are of different type, the one who likes buying Many grapes will selfishly get all the bananas available, for an enjoyment of 8, while the one who buys Regular amounts will go home grapeless and her enjoyment will be 0. If the two individuals like buying Many grapes, both will obtain only a fraction of the number they're looking for, meaning both will receive an enjoyment level of only 4. Part 1. Build a payoff matrix to summarize this problem. Part 2. Let r represent the proportion of individuals who like buying a Regular number of grapes. Write, in terms of r, the fitness functions of each type of individual. Part 3. Graph the fitness functions you found in the previous question with r on the horizontal axis. The lines you draw must be properly labeled. If they intersect make sure to indicate the value of r at which this happens. Part 4. Indicate, with arrows on the horizontal axis of your graph, the evolutionary trends, aka population dynamics, for r. Part 5. What is (are) the equilibriums) for this system? What type of evolutionary equilibrium is it (are they)? Part 6. One day, a hurricane floods the grape fields and people on the news start reporting that grapes will soon become very expensive. The consequence of this "gratastrophe" is that it suddenly alters the enjoyment levels of people who like buying Many grapes. Now, due to fear of missing out, when they go to the store at the same time as a Regular type, their enjoyment from buying all the grapes is 10 instead of 8. And when they go at the same time as another Many-buyer, their enjoyment from getting at least some grapes is 6 instead of 4. The enjoyment levels experienced by the Regular buyers doesn't change. So, the hurricane suddenly changes the position of the fitness function of those who buy Many grapes, but not the function of Regular buyers. Suppose that, before the hurricane, the neighborhood was at the equilibrium in which both types of individuals coexist. Immediately after the hurricane, the position of the polymorphic equilibrium changes, but the population is, at least initially, still at the previous equilibrium. Explain what will happen, over time, to the proportions of Regular and Many-avocado buyers. Hint: draw the new fitness functions and the new population dynamics (arrows) and place your initial population at the polymorphic equilibrium defined by the pre-hurricane fitness functions.) Explain in one or two sentences how you think this problem relates to the excessive purchases of toilet paper observed for a while when COVID started.

Answer rating: 100% (QA)

Part 1 Payoff matrix Regular Many Regular 12 12 8 0 Many 0 8 4 4 Part 2 Fitness functions Regular fR ... View the full answer