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📗 [4 points] In GoogSoft, software engineers A and B form a two-person team. Their year-end bonus depends on their relative performance. The bonus outcomes are summarized in the following table. The value of slacking to each person is \(s\) = . The total payoff to each person is the sum of the bonus and the value from slacking. What is the smallest value of \(x\) such that both players will work hard in a Nash equilibrium?
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B works hard
B slacks
A works hard
\(x, x\)
A slacks
📗 Answer: .
📗 [4 points] There are people living in the suburbs and all of them commute to work in the city. Every morning, each individual decides which way to drive to the city simultaneously: the Direct Way or the Long Way. The Long Way takes 1 hour of driving. The time spent on the Direct Way depends on the traffic is equal to \(\dfrac{n}{c}\) hours, where \(n\) is the total number of cars taking the Direct Way, and \(c\) = is the capacity. Each individual wants to minimize the driving time, and break ties by choosing the Direct Way. What is the number of people taking the Long Way in the Nash equilibrium?
📗 Answer: .
📗 [3 points] \(N\) = firms sharing the use of a river decide whether to filter (F) or release (R) pollutants (a poisonous substance) into the river. If \(n\) firms choose to pollute the river (R), each of these \(n\) firms incurs a cost of dollars, and each of the remaining firms that choose to install filters (F) incurs a cost of (cost due to pollution plus the cost of the filter). Every firm wants to minimize costs. What is the number of firms that choose to install filters (F) in a pure strategy Nash equilibrium? Note: remember to enter an integer.
📗 Answer: .
📗 [4 points] Imagine a population of \(N\) = individuals. Each of them simultaneously chooses between taking the vaccine and not. All individuals have the same payoffs. Suppose there are \(n\) people who choose not to take the vaccine, then the payoff from not taking the vaccine is \(- \alpha \cdot \dfrac{n}{N}\), and the payoff from taking the vaccine is \(- c - \beta \cdot \dfrac{n}{N}\), \(\alpha\) = is the herd immunity coefficient, \(\beta\) = measures the ineffectiveness of the vaccine, and \(c\) = is the cost of getting the vaccine. In a Nash equilibrium, what is the largest number of individuals who choose NOT to take the vaccine?
📗 Note: \(n\) is the number of people NOT taking the vaccine, and the question is asking for the largest number of individuals who choose NOT to take the vaccine.
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