This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way to solving following question.
Question: Six students are going on a foreign trip on which they will live close together. Where they are going, there is a disease which spreads easily among people who live close together. The value of the trip to a student who does not get the disease is 6. The value of the trip to a student who gets the disease is 0.
There is a vaccination against the disease. The vaccination costs different amounts for different students (perhaps they have different health plans). Let's call the students 1, 2, 3, 4, 5, and 6 respectively. The vaccination costs 1 for student 1; it costs 2 for students 2; etc...
If a student gets vaccinated, she will not get the disease. But, if she is not vaccinated then her probability of getting the disease depends on the total number in the group who are not vaccinated. If she is the only person not to get vaccinated then the probability that she gets the disease is 1/6. If there is one other person who is not vaccinated (i.e., two in all including her) then the probability that she gets disease is 2/6. If there are two other people who are not vaccinated (i.e., three including her)then the probability that she gets the disease is 3/6, etc...
For example, suppose only students 2 and 4 get vaccinated. Then 2's expected payoff is 6- where the  is the cost of the vaccination. Student 4’s expected payoff in this case is 6-. Student 5’s expected payoff in this case (recall she did not get vaccinated) is 6-. Student 5’s expected payoff in this case (recall she did not get vaccinated) is (2/6)6+(4/6)0=2 where the fraction (4/6) is the probability that she gets the disease. To make this into a game , suppose that each student aims to maximize her expected payoff. The student decide, individually and simultaneously, whether or not to get a vaccination.
(a) Explain concisely whether or not it is a Nash equilibrium for students 1,2, 3 and 4 to get vaccinated and students 5 and 6 not to get vaccinated.
(b) Explain concisely whether or not it is a Nash equilibrium for students 1, 2 and 3 to get vaccinated and student 4, 5, and 6 not to get vaccinated.
(c) Which players in this game have weekly dominated strategies? Explain your answer.