Q5. You and your n -1 friends working on a project together Every student in the

Question

Q5. You and your n -1 friends working on a project together Every student in the group has other exams to study, hence, each of them can spent at most 6 hours on that project. Every student in the group does not prefer working on the project, yet, they all want to get a good grade from the project. Each student's payoff is the two times total hours spent by everyone at project minus a constant c times the hours spent individually on the project. That is, where each s,is the hours spent on the project by student (note that s, (0,1,2,3,4,5.6)). Assume everyone chooses smultaneously how much time to spend working (a) Find the pure Nash equilibrium Ifo 2

Explanation / Answer

Consider the given problem here there are “n” worker working on a project together, => their payoff function is given by.

=> Ui = 2*sum(sj) – c*si, where “j” is running from “1” to “n”. Now, given this payoff function the “ith” worker wants to maximize their payoff.

=> Ui = 2*sum(sj) – c*si, => dUi/dsi = 2 – c. Now if “c” is between “0” and “2”, => “dUi/dsi > 0”.

So, here it is optimum to increase the hours spent on project. So, here all the worker will increase their working hour and as we know that each workers have maximum of “6 hours”, => here the optimum hour in the pure strategy NE is “si=6” for "0 < c < 2".

b).

Now, dUi/dsi = 2 – c. Now if “c” is more than “2”, => “dUi/dsi < 0”. So, here as “si” increases implied “Ui” decreases implied it is optimum to reduce “si”. So, here the pure strategy NE is given by “si=0”, for "c > 2".

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