4 Yale students are going on a trip abroad where they will have to live close to each other. At the destination, there is a disease that spreads easily among those who live close to each other. The value of the trip is 8 for a student who is not sick. The value of the trip is 0 for a student with the disease.

There is a vaccine against this disease. The cost of the vaccine is different among the students (perhaps they have different health insurance). Let's call the students 1, 2, 3, 4. The cost of the vaccine is 1 for student 1, 2 for student 2, and so on.

If a student is vaccinated, he or she will not get the disease. However, if he/she is not vaccinated, then the probability of getting the disease depends on the total number of unvaccinated people in the group. If he/she is the only one not vaccinated, then the probability of getting the disease is 1/4. If there is one other person who is not vaccinated (2 in total, including him/her), then the probability of getting the disease is 2/4. If there are two other unvaccinated people (3 in total, including oneself), then the probability of contracting the disease is 3/4, and so on.

[1] For example, let's say that only 1 and 4 of the students were vaccinated. Then the expected return of 1 is 4 - [2], where [2] is the cost of the vaccination. In this case, the expected payoff of student 4 is 4 - [4]. The expected payoff for student 3 (recall that he was not vaccinated) is (2/4) x 4 + (4/4) x 0 = x, where 4/4 is his probability of contracting the disease].

To make it a game, suppose that students want to maximise their expected return. Students decide individually and simultaneously whether or not to be vaccinated. • Students briefly explain whether the situation in which 1, 2 are in excess and 3 and 4 are not is a Nash equilibrium.

• Which players in this game have fully or weakly dominated strategies? Be careful in your descriptions to indicate whether the dominance is complete or weak.

• If we delete all fully and weakly dominated strategies from all players, which players (in order) have fully or weakly dominated strategies in the current game? Explain carefully.

• Find all ND (can be mixed) in this game. Explain

Answer rating: 100% (QA)

The analysis of the vaccination game 1 Is 1 vaccinated 2 vaccinated 3 not vaccinated 4 not vaccinated a Nash equilibrium No this is not a Nash equilibrium Student 3 can deviate and get vaccinated to increase their expected payoff from 1 to 431 2 Which strategies are dominated Student 1 Not vaccinating is weakly dominated by vaccinating since cost is 1 but payoff is 4 if vaccinated Student 2 Not vaccinating is weakly dominated by vaccinating since cost is 2 but payoff is 4 if vaccinated Student 3 Not vaccinating is weakly dominated by vaccinating since probability of illness goes from 24 to 04 if vaccinated Student 4 Not vaccinating is weakly dominated by vaccinating since probability of illness goes from 34 to 24 if vaccinated 3 In the updated game after deleting dominated strategies there are no more dominated strategies 4 The only Nash equilibrium is for all students to vaccinate ... View the full answer